FBI Transform in Gevrey Classes and Anosov Flows
FFBI transform in Gevrey classes and Anosov flows
Yannick Guedes BonthonneauMalo J´ez´equel
IRMAR, CNRS, Universit´e de RennesLPSM, CNRS, Sorbonne Universit´e, Universit´e de Paris. a r X i v : . [ m a t h . A P ] S e p ontents Introduction 5Chapter 1. Gevrey microlocal analysis on manifolds 191.1. Gevrey spaces of functions and symbols 191.2. Gevrey oscillatory integrals 391.3. Gevrey pseudo-differential operators 52Chapter 2. FBI transform on compact manifolds 812.1. Basic properties of the FBI transform 842.2. Lifting Pseudo-differential operators 982.3. Bergman projector and symbolic calculus 120Chapter 3. Ruelle–Pollicott resonances and Gevrey Anosov flows 1453.1. I-Lagrangian spaces adapted to a Gevrey Anosov flow 1493.2. Traces and I-Lagrangian spaces 1603.3. Perturbative results 165Bibliography 177 ntroduction Spectral theory and Anosov flows.
Dimitri V. Anosov introduced the flowsthat bear his name in [
Ano67 ]. He wanted to study geodesic flows on the unittangent bundle of compact Riemannian manifolds with ( a priori non-constant)negative sectional curvature. Since then, Anosov flows (and more generally smoothuniformly hyperbolic dynamics) have been widely studied by numerous authors usinga large variety of different tools (Markov partitions, specification, tower constructions,coupling methods, . . . ). In the last two decades, the so-called functional analyticapproach to statistical properties of uniformly hyperbolic dynamics has been a veryactive field of research, due to the introduction in dynamical systems of the notionof
Banach spaces of anisotropic distributions (see [
Bal17, Dem18 ] for surveys onthis subject).Let us recall the basic ideas behind this approach. Let ( φ t ) t ∈ R denote a C ∞ Anosov flow on a compact C ∞ manifold M (the precise definition of an Anosov flowis recalled in the introduction of Chapter 3). As a dynamical system, φ t is verychaotic, so much so that it is hopeless to try to describe the long-time behaviour ofall of its orbits. The complexity of the pointwise dynamics suggests that we shouldconsider instead the action of the flow φ t on objects that are reminiscent of thesmooth structure of M . This is one of the motivations for the introduction of the Koopman operator defined, for t ≥
0, by L t : u (cid:55)−→ u ◦ φ t , (1)acting for instance on C ∞ ( M ) or on the space D (cid:48) ( M ) of distributions on M . Theadjoint of the Koopman operator is the well-known transfer operator f (cid:55)−→ f ◦ φ − t | det d φ t | ◦ φ − t . Our aim is now to understand the asymptotic of the operator L t when t tends to+ ∞ . The advantage of this approach is that if u ∈ C ∞ ( M ), we have:dd t ( L t u ) = X ( L t u ) , where X denotes the generator of the Anosov flow φ t (we identify X with its Liederivative). That is, we have replaced the non-linear dynamics of φ t on the finite-dimensional manifold M by a linear ODE on an infinite-dimensional space ( C ∞ ( M )for instance), which is presumably easier to understand. Indeed, we expect that,as in the case of finite-dimensional linear ODEs, the long-time behaviour of theoperator L t should be ruled by the spectrum of the differential operator X .Following this observation, much effort was invested in developing a spectraltheory for Anosov flows. The first decisive steps in this field were made by DavidRuelle [ Rue86 ] and Mark Pollicott [
Pol85 ], so that
Ruelle–Pollicott resonances –that will be defined below – now bear their names. The main difficulty is that thespectrum of X acting on L ( M ) is in general very wild, in sharp contrast with thecase of elliptic differential operators such as the Laplacian on a compact Riemannianmanifold. It is only with the introduction of the Banach spaces of anisotropic distributionsthat this conundrum was completely resolved. They are functional spaces tailoredto fit the dynamics of φ t . Using the spectral properties of X acting on such spaces,we may define a discrete spectrum for X called Ruelle–Pollicott spectrum (seethe introduction of Chapter 3 for more details). The article of Michael Blank,Gerhard Keller, and Carlangelo Liverani [
BKL02 ] is often considered the firstappearance of Banach spaces of anisotropic distributions in the dynamical literature.However the notion was already implicitly present in the works of Hans Henrik Rugh[
Rug92, Rug96 ] and David Fried [
Fri95 ] in the analytic category. A consequenceof the theory of Banach spaces of anisotropic distributions is that the resolvent of X , defined for Re z (cid:29) u ∈ C ∞ ( M ) by( z − X ) − u = (cid:90) + ∞ e − zt L t u d t, (2)admits, as an operator from C ∞ ( M ) to D (cid:48) ( M ), a meromorphic continuation to C with residues of finite ranks. The poles of ( z − X ) − are called the Ruelle–Pollicottresonances of X or φ t (the rank of the residue being the multiplicity of the resonance).These resonances are powerful tools in the study of the statistical properties of theflow φ t – for example see the article of Oliver Butterley [ But16 ].In some particular cases, mostly of algebraic nature, it is possible to obtainquite detailed informations on the distribution of these resonances (geodesic flows inconstant negative curvature or constant-time suspensions of cat maps for instance).However, in the general case, there is no reason that we should be able to obtainvery precise results, much less compute explicitly the values of the Ruelle–Pollicottresonances of X . We have to settle for qualitative counting estimates.It is well-known that much information about the global dynamics of φ t maybe retrieved from its periodic orbits – description of invariant measures, relationbetween the pressure and Poincar´e series, etc. It may not come as a surprise thenthat the Ruelle–Pollicott spectrum can be completely determined from the periodicorbits of φ t , via a sort of zeta function, as we explain now.For each periodic orbit γ , let T γ denote (resp. T γ ) its length (resp. primitivelength). We also denote by P γ its linearized Poincar´e map, i.e. P γ = d ϕ T γ ( x ) | Eu ⊕ Es for some x in γ (the definition of the stable and unstable directions E s and E u isrecalled in the introduction of Chapter 3). These data are gathered in the dynamicaldeterminant , defined for Re z (cid:29) ζ X ( z ) := exp (cid:32) − (cid:88) γ T γ T γ e − zT γ (cid:12)(cid:12) det( I − P γ ) (cid:12)(cid:12) (cid:33) . Since the flow φ t is C ∞ , it follows from the work of Paulo Giulietti, Liverani andPollicott [ GLP13 ] (see also the article of Semyon Dyatlov and Maciej Zworski[
DZ16 ]) that ζ X extends to a holomorphic function on C , whose zeros are theRuelle–Pollicott resonances of X . It is then of the utmost importance to understandthe complex analytic properties of ζ X in order to apprehend the distribution ofRuelle–Pollicott resonances. On a historical note, the Ruelle–Pollicott resonanceswere originally introduced by Ruelle and Pollicott as the zeros of some zeta functionclosely related to (3).The regularity of the flow φ t plays an important role in the approach describedabove. Indeed, if the flow φ t were only C k for some k >
1, then the meromorphiccontinuations of the resolvent (2) and of the dynamical determinant (3) would apriori only be defined on a half plane of the form { z ∈ C : Re z > − A } , with A that goes to + ∞ when k tends to + ∞ (see [ Bal92 ]). On the contrary, if theflow φ t is real-analytic then Rugh [ Rug92, Rug96 ] (in dimension 3) and Fried
NTRODUCTION 7 [ Fri95 ] (in higher dimension) are able to prove a bound on the growth of ζ X that implies that its order is less than the dimension of M . What has to beunderstood here is that the functional analytic tools available in the real-analyticcategory are usually stronger than those available in the C ∞ category. These toolshave also been used in hyperbolic dynamics to understand other systems thanAnosov flows: this is the case in the pioneering work of Ruelle [ Rue76 ] for instance.Functional analytic tools from the real-analytic category have been used a lotrecently, when studying for instance expanding maps or perturbations of cat maps(see for example[
FR06, Nau12, Ada17, SBJ17, BN19 ]).This situation suggests to study Anosov flows in regularity classes that areintermediate between C ∞ and real-analytic. The hope here is that certain propertiesthat fail for C ∞ Anosov flow but hold for real-analytic ones could in fact be truein larger classes of regularity. Following this idea, the second author was able toestablish a trace formula for certain ultradifferentiable Anosov flows in [
J´ez19 ]. Inthis paper, we focus on the Gevrey classes of regularity. We will give later in thisintroduction a precise definition, but let us just recall for now that these are classesof regularity indexed by a parameter s ≥ Gev18 ] in order to study the regularity of the solutions of certain PDEs.There are several reasons for which we decided to work with this hypothesis ofregularity in particular. First, it is convenient that when s = 1 the Gevrey regularityis in fact the real-analytic regularity: we will be able to confront our results withthose available in the literature. Moreover, the Gevrey classes of regularity arenotoriously very well-behaved, so that we could hope to develop practicable toolsin this setting. More importantly, we were interested in a very particular question:the finiteness of the order of the dynamical determinant (3), and we thought, forheuristic reasons explained below, that the Gevrey regularity was the good settingto understand this question.These expectations have been met, as our main result writes: Theorem . Let s ∈ [1 , + ∞ [ . Let M be an n -dimensional compact s -Gevreymanifold endowed with a s -Gevrey Anosov flow ( φ t ) t ∈ R generated by a vector field X . Then there is a constant C > such that for every z ∈ C , | ζ X ( z ) | ≤ C exp ( C | z | ns ) . In particular, ζ X has finite order less than ns . As already mentioned, when s = 1, the Gevrey regularity coincides with thereal-analytic regularity. In that case, Theorem 1 is a consequence of the results ofRugh [ Rug92, Rug96 ] and Fried [
Fri95 ] mentioned above. However, when s > R is O ( R ns ). This gives information on thecounting of resonances far from the L spectrum, or, so to speak, deep in the complexregion. It is to be contrasted with the much sharper local results obtained by Faureand Sj¨ostrand [ FS11 ] and more recently by Faure and Masato Tsujii [
FT17 ]. Theirresult hold only for counting in boxes near the L spectrum, but are much betterupper bounds. The only available general lower bound to our knowledge is almostlinear, and due to Long Jin, Zworski (and Fr´ed´eric Naud) [ JZ17 ]. In the contactflow case, there are much sharper results, see [
FT13, DDZ14 ].Regarding the optimality of Theorem 1, it is relevant to observe that accordingto Ruelle’s result [
Rue76 ], if the stable and unstable foliations of the flow arethemselves analytic, the bound on the order is much better than the dimension. Inparticular, for the geodesic flow of a hyperbolic manifold of dimension n , acting on INTRODUCTION the unit cosphere of dimension 2 n −
1, the order of the dynamical determinant isexactly n + 1 – instead of the dimension 2 n − FT17 ].The techniques employed here are quite different from those of Rugh and Fried.They enable us to study directly the resolvent ( z − X ) − on relevant spaces. Severalresults follow, that are gathered in the beginning of Chapter 3. We obtain: • a weighted version of Theorem 1 (see Theorem 8 and discussion below it); • a characterization of the resonant states by their Gevrey wavefront set(see Proposition 3.2); • a “global” version of the stability of Ruelle–Pollicott resonances understochastic perturbation proven by Dyatlov and Zworski in [ DZ15 ] (seeTheorem 9); • a Gevrey version of linear response, see § KKPW89 , Corollary 1] however for a related result).
Theorem . Let (cid:15) (cid:55)→ X (cid:15) be a real-analytic family of real-analytic vector fieldson a real-analytic manifold M , defined for (cid:15) near . We assume that X generatesan Anosov flow that admits a unique SRB measure µ . Let µ (cid:15) denote the uniqueSRB measure of the Anosov flow generated by X (cid:15) , for (cid:15) near zero. Then the map (cid:15) (cid:55)→ µ (cid:15) ∈ U ( M ) is real-analytic on a neighbourhood of zero. The space U ( M ) of analytic ultradistributions that appears in Theorem 2 isdefined in § C ∞ Anosov flows (mentioned in the beginning of this introduction).Let us describe here the strategy introduced by Fr´ed´eric Faure, Nicolas Royand Johannes Sj¨ostrand in [
FRS08 ] to study C ∞ Anosov diffeomorphisms, andadapted to flows in [
FS11 ]. The anisotropic spaces from [
FS11 ] are of the formOp( e G ) · L ( M ), where G is a symbol of logarithmic size called an escape functionand Op is a Weyl quantization. The action of X on Op( e G ) · L ( M ) is equivalentto the action on L of Op (cid:16) e G (cid:17) − X Op (cid:16) e G (cid:17) = X + A G . (4)Here, the operator A G is a pseudo-differential operator of logarithmic order, and G has been chosen so that the real part of the symbol of A G is negative. Thiscan be used with the sharp G˚arding inequality to invert the operator X − λ , upto a compact operator. The meromorphic continuation of (2) follows then fromFredholm analytic theory. However, when trying to invert the conjugated operatorOp( e G ) − ( X − λ ) Op( e G ) = X + A G − λ , the negative sign of the real part of thesymbol of A G competes with the possible positivity of − Re λ . Fortunately, we maychange G , and consequently make Re A G even more negative. Hence the strategy NTRODUCTION 9 above allows to continue ( X − λ ) − meromorphically to { λ ∈ C : Re λ > − N G } , forarbitrarily large N G . Notice in particular that the construction from [ FS11 ] requiresthe use of a scale of spaces, since the resolvent is meromorphically continued “stripby strip”. This is a consequence of the fact that the weight G is logarithmic, due tothe limitations of pseudo-differential calculus in the C ∞ category. Indeed, we knowthat the Fourier transform of a C ∞ function decays faster than the inverse of anypolynomials, but this is the best general bound available, hence the restriction onthe growth of G .In order to improve the situation that we just described, one could try to makethat operator A G (or rather its symbol) “larger”. However, this requires to take G with a growth faster than logarithmic, and we just explained that it cannot be donein the C ∞ category. It seems then natural to wonder what happens if we work in asmaller class of regularity. Indeed, Fourier transforms of functions that are moreregular than C ∞ should decay faster, and, morally, it should allow to soften theconstraint on the growth of G . We recall here that the question that we want toconsider is the finiteness of the order of the dynamical determinant (3). It is classicalto derive such a bound from Schatten estimates. Here, we would like the resolventof the operator (4) to be Schatten. A natural way to try to achieve that is to makethe operator (4) hypo-elliptic. To do so, we want the pseudo-differential operator A G to have positive order (cid:15) >
0, but this requires that G itself has order (cid:15) > G , we need to work in a class of regularity inwhich compactly supported functions have a Fourier transform that decays like astretched exponential exp( −(cid:104) ξ (cid:105) (cid:15) /C ). This is exactly the decay which characterizesthe Fourier transform of 1 /(cid:15) -Gevrey functions. This is why this paper is set in theframework of Gevrey classes of regularity.We used above the construction of anisotropic spaces from [ FS11 ] in order toexplain the motivation behind Theorem 1. However, this heuristic proof could havebeen formulated using any other construction of anisotropic spaces and the ideawould be the same: one wants to use the extra regularity to replace some weakweight by a stronger one. Unfortunately, we have not been able to use the availableconstructions of anisotropic Banach spaces to make the heuristic proof explainedabove rigorous. The main difficulty is that the weight e G is not a symbol in theusual sense when G is a symbol of order (cid:15) > J´ez19 ] give spacesfor which small time dynamics of the Koopman operator (1) is not sufficiently wellcontrolled. On the other hand, microlocal constructions, as the one from [
FS11 ]described above, could maybe be adapted, using results of Luisa Zanghirati [
Zan85 ]and Luigi Rodino [
Rod93 ] on spaces of infinite order Gevrey pseudo-differentialoperators. However, this would restrict our work to the case of large s , and it isnot clear in the first place that the spaces constructed in that way carry dynamicalinformation. It would also not yield a bound on the order as good as the one weobtain. FBI transform.
Thankfully, we were able to construct satisfying spaces usingan FBI – for Joseph Fourier, Jacques Bros and David Iagolnitzer – transform. Thisis a linear map taking functions of a space variable x ∈ M to functions on the phasespace α ∈ T ∗ M . It is an integral transform whose kernel has an oscillating Gaussianbehaviour.FBI Transforms are a sort of “localized” Fourier transform. Such transformsare common in the linear PDE literature, and we gathered some historical remarksto give a bit of context. Maybe the first popular occurrence is the one introduced byDennis Gabor [ Gab46 ] for use in signal processing. Later on, a continuous version of his construction was given, taking the form G x ( τ, ω ) = (cid:90) R x ( t ) e − π ( t − τ ) − iωt d t. To this day, the Gabor transform is a staple in signal analysis. It is a particularcase of a Short Time Fourier Transform whose general expression isSTFT { x } ( τ, ω ) = (cid:90) R n x ( t ) w ( t − τ ) e − iωt d t, where w is a window function.Another occurrence of generalized Fourier transform, more closely related toour main interest, is the so-called Bargmann–Segal transform. It was introduced byValentine Bargmann [ Bar61 ] and Irving Ezra Segal [
Seg63 ], as C n (cid:51) z (cid:55)→ Bf ( z ) = (cid:90) R n e − ( (cid:104) z,z (cid:105)− √ (cid:104) z,x (cid:105) + (cid:104) x,x (cid:105) ) f ( x )d x. It realizes an isometry between L ( R n ) and the Bargmann–Segal space of holomor-phic functions F on C n such that (cid:90) C n | F ( z ) | e −| z | d z < ∞ . Bros and Iagolnitzer introduced their “generalized Fourier transform” in [
BI75 ].Its form was(5) F u ( v, v , X ) = (cid:90) R n u ( x ) e i (cid:104) v,x (cid:105)− v ( x − X ) d x, (for u in some reasonable space, for example, u can be a tempered distribution).Their purpose was to define and study an analytic version of the wavefront set. Thiswas based upon previous works of Iagolnitzer and Henry Stapp [ IS69 ]. At thattime, another notion of analytic wavefront set had been proposed by Mikio Sato,Takahiro Kawai and Masaki Kashiwara [
SKK73 ]. Jean Michel Bony [
Bon77 ] thenproved that these definitions were equivalent.Nowadays, it is more common to use a semiclassical version of the transform,defined on R n by(6) T R n u ( x, ξ ; h ) := 1(2 πh ) n/ (cid:90) e ih ( (cid:104) x − x (cid:48) ,ξ (cid:105) + i | x − x (cid:48) | ) u ( x (cid:48) )d x (cid:48) . In the case of manifolds, to account for non-linear changes of variable, it is moreconvenient to take a slightly different scaling for the phase.In the second chapter of this article, given a compact real-analytic manifold M , we will construct an FBI transform T , which is an operator from D (cid:48) ( M ) to C ∞ ( T ∗ M ) given for α = ( α x , α ξ ) ∈ T ∗ M by T u ( α ) = (cid:90) M K T ( α, y )d y. Here K T is an analytic kernel, negligible away from the “diagonal” { y = α x } , andwhich near this diagonal has roughly the behaviour of e ih (cid:104) α x − y,α ξ (cid:105) + i (cid:104) α ξ (cid:105) ( α x − y ) a ( α, y ) , where a is an analytic symbol, elliptic in the relevant class. Contrary to (6), we usehere a phase that is a symbol of order 1 in α . The basic properties of this transformare investigated in Chapter 2.Let us explain now why FBI transforms are interesting objects. What motivatedthe works of Bros and Iagolnitzer is the following simultaneous observations. Let f NTRODUCTION 11 be a tempered distribution in R n . Then, ( x, ξ ) is not in the C ∞ wavefront set of f ,if and only if, for all ( x (cid:48) , ξ (cid:48) ) sufficiently close to ( x, ξ ), as h → T R n f ( x (cid:48) , ξ (cid:48) ; h ) = O ( h ∞ ) . On the other hand, if f is a real analytic function is some open set U ⊂ R n , thenfor x ∈ U , and ξ ∈ R n , there exists a constant C > T R n f ( x, ξ ; h ) = O (cid:18) exp (cid:18) − Ch (cid:19)(cid:19) . This suggests to define the analytic wavefront set of f as the set of ( x, ξ ) ∈ R n such that T R n f does not satisfy this last bound uniformly in any neighbourhood of( x, ξ ).The PDE specialist may wonder what is the purpose of introducing one moredefinition of the wavefront set, and the Dynamical Systems expert speculate whywe have to consider wavefront sets in the first place. The answer is twofold.The first point is (very) closely related to the Segal–Bargmann transform. Itis the observation that the FBI transform enables to represent (pseudo)differentialoperators as multiplication operators . This feature alone makes it useful for studyingelliptic regularity problems.Usually, one can chose the transform T such that T ∗ T is the identity, so that T is an isometry between L ( M ) and its image in L ( T ∗ M ). It turns out that, for asuitable class of functions p ∈ C ∞ ( T ∗ M ), u (cid:55)→ T ∗ pT u is a pseudo-differential operator with principal symbol p . Denoting Π = T T ∗ , it isthe orthogonal projector on the image of T . We then haveΠ p Π( T u ) = T ( T ∗ pT u ) . This formula relates a Toeplitz-like operator – Π p Π – with a pseudo-differentialoperator – T ∗ pT .On top of being a very practical tool, the fact that the FBI transform relatespseudo-differential operators with Toeplitz-like operators is thus a bridge betweenthe quantization of cotangent spaces via the algebra of pseudo-differential operators,and the quantization of compact symplectic manifolds.The second point is that for a function u , its transform T u ( x, ξ ) is a functionof parameters belonging to the classical phase space. In particular, one expectsthat (microlocal) propagation phenomenon can be observed directly on T u . Aparticularly striking consequence is the flexibility with which escape functions canbe used. This deserves a detailed explanation
Helffer–Sj¨ostrand theory.
Our use of the FBI transform is deeply inspiredby the work of Bernard Helffer and Sj¨ostrand [
HS86, Sj¨o96a ]. We will rely inparticular on the notion of complex Lagrangian deformation of the cotangent bundle.This concept is deeply related with the notion of escape function , which is maybemore popular nowadays. The idea of escape function has a long history. It is linkedto the technique of positive commutators, that appears in ´Eric Mourre’s estimates[
Mou81 ] for example. It was introduced by Helffer and Sj¨ostrand [
HS86 ] to studyquantum tunneling effects for some electric Schr¨odinger operators. The central ideais the following: given a (pseudo-)differential operator P with principal symbol p , ifone can build a function G that decreases along the Hamiltonian flow (Φ t ) t ∈ R of p in some region of phase space, then one can gain some sub-principal micro-ellipticityin that region. Helffer and Sj¨ostrand introduced a framework involving the FBItransform to implement this idea. Escape functions have since been popularizedand have become very much an independent technique. We will give first a (very) condensed presentation of the Helffer-Sj¨ostrand methoditself before coming back to historical considerations. If M is a real analytic manifold,one can choose a complexification (cid:102) M for M , and then T ∗ (cid:102) M is a complexificationfor the cotangent space T ∗ M of M . It is possible to construct an FBI transform T whose kernel is real analytic. This implies that if u ∈ D (cid:48) ( M ), then T u has aholomorphic extension to a complex neighbourhood of T ∗ M in T ∗ (cid:102) M . In particular,if Λ ⊆ T ∗ (cid:102) M is sufficiently close to T ∗ M , we can consider the restriction T u | Λ , butwe need to chose Λ wisely if we want this to be useful. In particular, there are somegeometric conditions that have to be fulfilled by Λ.Notice indeed that the complex manifold T ∗ (cid:102) M is endowed with a rich real symplectic structure, since both the real part ω R and imaginary part ω I of thecanonical complex symplectic form ω on T ∗ (cid:102) M are real symplectic forms on T ∗ (cid:102) M .Notions from the symplectic geometry of ω I (resp. ω R ) will be designed with anI (resp. R) – I-Lagrangian, I-symplectic... Since ω is an exact symplectic form( ω = d θ if one denotes by θ = ξ d x the canonical Liouville one-form), the 2-forms ω I and ω R also are.If G is a real valued symbol of order 1 on T ∗ (cid:102) M –that will play the role of anescape function in our context –, we consider for τ ≥ τ := exp( τ H ω I G )( T ∗ M ) ⊂ T ∗ (cid:102) M .
Since T ∗ M is I-Lagrangian, Λ τ also is: this is the complex Lagrangian deformationof T ∗ M that we announced. The one form Im θ is thus closed on Λ τ . The globaldescription as the image of T ∗ M implies the existence of a global solution H τ onΛ τ to the equation d H τ = − Im θ | Λ τ . Since G is a symbol, H τ also is. Since T ∗ M is R-symplectic, Λ τ also is for τ small enough, so that ω nR /n ! defines a volume formd α on Λ τ .Let P be a semi-classical differential operator with analytic coefficients, with(analytic) principal symbol p . Then for u analytic and τ small enough, we will seein Chapter 2 that under relevant hypotheses, we have(7) (cid:90) Λ τ T P u T u e − H τ /h d α = (cid:90) Λ τ ( p + O ( h )) | T u | e − H τ /h d α Here, we have identified p with its holomorphic extension. The idea of the methodis to consider (instead of L ( M )) the space(8) H Λ τ := (cid:110) u (cid:12)(cid:12)(cid:12) T u | Λ τ ∈ L ( e − H τ /h d α ) (cid:111) . (here u is assumed to be in a space of hyperfunctions we will define precisely lateron). On H Λ τ , the operator P has an effective principal symbol p | Λ τ , which may be(if G is suitably chosen) more elliptic than p | T ∗ M .This method also applies to analytic pseudo-differential operators. It can beextended to the case of Gevrey operators, after the necessary adjustments we presentat the end of the introduction. The rigorous statements and proofs of the resultsthat we just mentioned may be found in Chapter 2.Let us now give a bit of context and more explanations. We start by presentingwhat we call the “Martinez method”, and explain why it is not suited to our needs.As we have seen before, if σ is a function satisfying suitable estimates and T someFBI transform, T ∗ σT is a pseudo-differential operator with principal symbol σ .However, the operator T ∗ σT is a well defined bounded operator on L , even if σ is only a L ∞ function. This observation suggests to quantize a class of functionsmuch larger than the usual class of symbols. Instead of the usual construction ofmicrolocal spaces in the form (for some function G that will be called an escape NTRODUCTION 13 function ) Op( e G ) L ( M ) , defined with Op, the Weyl quantization, one can define a space H G with (cid:107) f (cid:107) G := (cid:90) T ∗ M e − G | T u | . Formally, this is the space ( T ∗ e G T ) L ( M ). It is considerably easier to define thanthe space H Λ introduced in (8). This approach was taken in Faure and Tsujii’swork [ FT17 ]. Similar spaces appear in the works of Andr´e Martinez [
Mar94 ], ShuNakamura [
Nak95 ] and Klaus Jung [
Jun00 ]. These authors study tunneling effectsfor the bottom of the spectrum of elliptic operators, and instead of exp( G ) considerstronger weights in the form exp( G/h ), with h > H G is bounded bysup x lim sup ξ →∞ exp( G ◦ Φ t − G )( x, ξ ) , where (Φ t ) t ∈ R denotes the symplectic lift of ( φ t ) t ∈ R . This is a sort of “integratedversion” of a G˚arding-type estimate on which rely Martinez et. al. Indeed, theyprove that(9) (cid:104) P f, f (cid:105) H G/h = (cid:90) T ∗ M p G ( x, ξ ) e Gh | T u | , where p G is the principal symbol of P , shifted by G according to some preciseformula. When p is real, the imaginary part of p G is given in first approximation,for G small enough, by ∇ ξ G · ∇ x p − ∇ x G · ∇ ξ p. The appearance of this Poisson bracket is revealing of some symplectic phenomenonat play here. Actually, if one were to take G to be the real part of an almostanalytic extension of G , one would obtain (in the relevant symbol class)(10) p (exp( τ H ω I G ( α ))) = p ( α ) + iτ ( ∇ ξ G · ∇ x p − ∇ x G · ∇ ξ p ) + O ( τ ) . In particular, if Λ is a I-Lagrangian corresponding to G , the spaces H G and H Λ have (at least approximately) the same microlocal behaviour. On the other hand,in the Helffer–Sjostrand method, one does not have to assume that the symbol G isconstructed as the extension of a symbol defined on T ∗ M . For our purpose, thispoint will be crucial, and justifies the use of I-symplectic geometry instead of the– a priori simpler – point of view of Martinez. Indeed, unlike the references citedabove, for the study of analytic flows, we need to assume that G is symbol of order1. In that case, the estimate (10) can be rewritten (for p of order 1) p (exp( τ H ω I G ( α ))) = p ( α ) + iτ ( ∇ ξ G · ∇ x p − ∇ x G · ∇ ξ p ) + O L ∞ ( (cid:104) α (cid:105) τ ) . The remainder is small as a symbol of order 1, but is not bounded. For this reasonthe Koopman operator may lose the property of semi-group on the spaces H G . Tocircumvent this, we have to build our escape function directly on T ∗ (cid:102) M .Let us come back to the Helffer–Sj¨ostrand method. Our presentation is moregeneral than what appeared in the original article [ HS86 ], so let us present thenovelty in our approach. In its first version, the method dealt with operators onthe flat space R n , and the weight G was assumed to be compactly supported in the ξ variable. The theory was then adapted to manifolds, by Sj¨ostrand [ Sj¨o96a ] andthen Sj¨ostrand and Zworski [
SZ99 ], to obtain asymptotics for the counting functionof resonances of an analytic convex obstacle, somehow closing the discussion openedby the Bardos–Lebeau–Rauch paper [
BLR87 ]. However, in these articles, there remained the restriction that G be compactlysupported in ξ . Additionally, the transform introduced did not have a globallyanalytic kernel, preventing the study of hyperfunctions. This was not a problemsince the main interest were solutions of the wave equation, which are highly regular a priori . In our study of analytic and, more generally, Gevrey Anosov flows, theeigenfunctions – or resonant states – are not regular, and this limitation had to belifted. We thus obtain a transform whose kernel is globally analytic.Additionally, we are able to deal with symbols G of order 1 – this is the naturallimit on arbitrary real analytic manifolds. Indeed, when G is of order 1, the manifoldΛ is contained in a Grauert tube of size ∼ T ∗ M . If one tries to take a symbolof order larger than 1, it would require to work with Grauert tubes of infinite radius,and only very particular real analytic manifolds possess such structure – for example,it is not the case of the real hyperbolic space.At the same time as we were elaborating this article, Jeffrey Galkowski andZworski [ GZ19b, GZ19a ] were studying a very similar extension in the analyticcategory. They obtain a version of the Helffer–Sj¨ostrand framework for symbols oforder 1, on tori. They have already found another application to the technique in[
GZ20 ].With [
LL97 ], Bernard and Richard Lascar extended the Helffer–Sj¨ostrandmethod to the case of Gevrey regularity on manifolds. Again, they only consideredsymbols with compact support in ξ , while we are able to consider symbols of order1 /s for s -Gevrey problems. We will see that this is the natural limit. Moreover, theranges of s ’s that are allowed in [ LL97 ] is quite restrictive, a constraint that we willalso lift.We close this short presentation of the Helffer-Sj¨ostrand method with thefollowing heuristic consideration. The gist of the technique is that given a linear(pseudo-)differential operator P , for the purpose of studying the regularity of thesolutions to some equation involving P , the regularity of P can be exhausted bytaking the right I-Lagrangian Λ. Replacing L ( M ) by H Λ is akin to replacing T ∗ M by Λ for all practical purposes. Instead of using analytic microlocal analysis on M –which may be complicated – one uses C ∞ microlocal analysis on Λ – which maybe considerably simpler. We hope that our application of the method to analyticAnosov flows in § WZ01 ], which deals with the C ∞ case. Our results on analytic FBI transform on compact manifold are detailedat the beginning of Chapter 2. Gevrey microlocal analysis.
Before we explain how the results in the analyticcase have to be adapted in the Gevrey case, let us recall some definitions. The classof Gevrey functions was introduced by Maurice Gevrey in [
Gev18 ] to study theregularity of solutions to certain linear PDEs, in particular the heat equation – itturns out that the heat kernel is 2-Gevrey with respect to the time variable.Let s ≥ U be an open subset of R n . A function f : U → C issaid to be s -Gevrey, or, for short, G s , if f is C ∞ and if, for every compact subset K of U , there are constants C, R >
0, such that, for all α ∈ N n and x ∈ K , we have | ∂ α f ( x ) | ≤ CR | α | α ! s . (11)The constant R in (11) may be interpreted as the inverse of a (Gevrey) radiusof convergence. Notice that when s = 1, this describes the class of real-analyticfunction on U . When s >
1, the class of s -Gevrey function is non-quasianalytic: itcontains compactly supported functions. NTRODUCTION 15
In [
Rou58 ], Charles Roumieu made the crucial observation that given a s -Gevrey function f , compactly supported in R n , for some constants c, C >
0, and ξ ∈ R n , | ˆ f ( ξ ) | ≤ Ce − c | ξ | /s . Conversely, a function whose Fourier transform satisfies such an estimate is s -Gevrey. He also initiated the study of Gevrey ultradistributions, i.e. the continuouslinear functionals on spaces of Gevrey functions. Later on, Hikosaburo Komatsu[ Kom73, Kom77, Kom82 ] gave a systematic treatment of such objects.Since Gevrey regularity can be characterized by the decay of Fourier transforms,it is only natural to expect that a specific version of microlocal analysis can bedeveloped for this class of regularity. Such a study was inaugurated less than adecade later by Louis Boutet de Monvel and Paul Kr´ee in [
BDMK67 ]. Theystudied an algebra of pseudo-differential operators on R n whose symbols σ satisfyestimates of the form | ∂ αx ∂ βξ σ | ≤ CR | α | + | β | α ! s β ! (cid:104) ξ (cid:105) m −| β | , for all multi-indices α, β ∈ N n . In other words, the symbols are s -Gevrey in the x variable, and real analytic in the ξ variable. We will denote this algebra by G sx G ξ Ψ.It contains the differential operators with s -Gevrey coefficients. They prove thatthe usual statement of C ∞ pseudo-differential calculus can be adapted, so that: • modulo some compactness of support assumption, G sx G ξ Ψ indeed forms analgebra; • the usual O ( (cid:104) ξ (cid:105) −∞ ) remainders, and C ∞ -smoothing properties can bereplaced by O (exp( − (cid:104) ξ (cid:105) /s /C )) remainders and G s smoothing properties.In particular, • given P ∈ G sx G ξ Ψ elliptic, there exists E another such operator, so that P E − I is G s smoothing ( E is called a parametrix for P ).With the development and popularization of microlocal analysis, their studywas furthered by many other authors. Notably, a group of Italian mathematicians,and among them Zanghirati [ Zan85 ] and Rodino [
Rod93 ]. When compared with[
BDMK67 ], the main innovation is the following: they study pseudo-differentialoperators whose order is not finite, such as exp t ( − ∆) (cid:15) , with (cid:15) > C ∞ regularity.The striking result of Bardos, Lebeau and Rauch [ BLR87 ] created new interestfor Gevrey microlocal analysis. The theory of propagation of singularities wasestablished by several authors, notably Yoshinori Morimoto with Kazuo Taniguchi[
MT85 ] and Lascar [
Las86 ]. In the latter, a different class of operators is considered,corresponding to symbols satisfying the estimates | ∂ αx ∂ βξ σ | ≤ CR | α | + | β | ( α ! β !) s (cid:104) ξ (cid:105) m −| β | . Now, the symbol is Gevrey in x and ξ , and we will refer to such operators as G sx G sξ or simply G s pseudors. When working with propagation of singularities,since Hamiltonian flows do not necessarily preserve the fibers in T ∗ M , they cannotpreserve the mixed G sx G ξ regularity from before. It is thus natural to consider suchoperators.Finally, we already mentioned [ LL97 ], where appears a semi-classical – i.e.with an h – version of the theory. In this situation, the remainders of size O (exp( −(cid:104) ξ (cid:105) /s /C )) from [ BDMK67 ] become remainders of size exp( − /Ch /s ).The purpose of Lascar & Lascar was to study the FBI transform in Gevrey classes,to improve upon the result of Sj¨ostrand [ Sj¨o96a ], itself a refinement of [
BLR87 ].In some sense, the present article completes [
LL97 ] in several directions.
We already mentioned several technical extensions we will provide with respectto the I-Lagrangian spaces method. In Chapter 1, the reader can also find a turnkeytheory of G s pseudors, which we did not find in the literature. Indeed, while algebraicstability of G s G pseudors was considered, it is not the case of the G s class. To ourunderstanding, when G s pseudors appeared in articles such as [ LL97, Jun00 ], theorigin of such an operator was either not explicitly mentioned or it was actuallya G x G sξ operator. Since there was only one G s operator, the authors were notconcerned with G s Ψ as a class of operators. We have thus taken the opportunityto prove that the class of G s pseudors is stable by composition, and several otherresults that enable one to practice microlocal analysis in that class.Let us come back to the I-Lagrangian spaces. When studying Gevrey PDEproblems instead of analytic ones, we have to rely on almost-analytic extensionsinstead of holomorphic extensions. A central fact is that if f is s -Gevrey, it admits s -Gevrey almost analytic extensions, that satisfy estimates in the form (cid:12)(cid:12) ∂f ( x + iy ) (cid:12)(cid:12) ≤ C exp (cid:32) − C | y | s − (cid:33) . For this to be a s -Gevrey remainder O (exp( − /Ch /s )), we have to impose that | y | ≤ Ch − /s . In the analytic case, for the I-Lagrangian spaces, we work in aneighbourhood on size 1 of the reals. In the s -Gevrey case, we have to considershrinking complex neighbourhoods of the reals as h →
0. For this reason, in the s -Gevrey case, instead of working with I-Lagrangians Λ = e τH G ( T ∗ M ) with G asymbol of order 1, we will have to considerΛ = exp( τ h − /s G )( T ∗ M ) , with G a symbol of order 1 /s . For a s -Gevrey Anosov flows, this means that wecan at most gain a (cid:104) ξ (cid:105) /s subprincipal term, leading to ns -Schatten estimates, andTheorem 1.Let us mention a direction in which our work can certainly be extended. Givena logarithmically convex sequence A = ( A k ) k ≥ of positive real numbers, theassociated Denjoy–Carleman class on R n is the set of smooth functions f such thatfor some constant C >
0, and for all α ∈ N n , | ∂ α f ( x ) | ≤ C | α | A | α | α ! . Denjoy–Carleman classes thus generalize Gevrey classes, since the s -Gevrey classis the ( k ! s − ) k ≥ –Denjoy–Carleman class. Since A is logarithmically convex, theassociated regularity can be characterized with the Fourier transform, as wasobserved in [ Rou58 ]. It follows that one can probably reformulate all our resultsin some of these more general classes. In order to extend our results to moregeneral Denjoy–Carleman classes, one would need to understand the almost analyticextension of functions in these classes. A discussion of this topic may be found in[
Dyn76 ].In another work of the second author [
J´ez19 ], it was observed that to obtainglobal trace formulae for flows, it was sufficient to work in Denjoy–Carlemanclasses much less regular than Gevrey. It is also probably the case that for Anosovdiffeomorphisms, the right scale of regularity necessary to obtain Schatten estimatesis much less than Gevrey, using I-Lagrangian spaces or another technique.
Structure of the paper.
Chapter 1 is devoted to recalling some basic factsabout the Gevrey regularity and describing the algebra of pseudo-differential op-erators with Gevrey symbols acting on manifolds. Chapter 2 of this monographwill be devoted to describing an FBI transform suited to our needs. To this end,
NTRODUCTION 17 we will revisit and expand the § Sj¨o96a ]. This is quite independent of Anosovflows, and we hope that it can be usefully applied to other situations. Finally, wewill apply this tool to the study of Gevrey Anosov flows in Chapter 3 and prove inparticular Theorem 1.Throughout the paper h >
Acknowledgements.
Thanks to M. Zworski’s suggestion to use H¨ormander’s ∂ trick, we were able to include the analytic case ( s = 1), we are in his debt.We would also like to thank Baptiste Morisse, San V˜u Ngo. c and JohannesSj¨ostrand for inspiring discussions.The second author is supported by by the European Research Council (ERC)under the European Union’s Horizon 2020 research and innovation programme(grant agreement No 787304). Part of this work was accomplished during the secondauthor’s residence at the Mathematical Sciences Research Institute in Berkeley,California, during the Fall 2019 Micro-local Analysis Program supported by theNational Science Foundation under Grant No. DMS-1440140.HAPTER 1 Gevrey microlocal analysis on manifolds
This first chapter is divided in three sections. In § § § G s pseudors announced in the introduction.Some classical microlocal questions that we do not deal with are the following: • the existence of a good parametrix in the G s class when s > • the related question of the functional calculus in that class; • propagation of singularities and radial estimates. First definitions.
We start by recalling the definition of Gevrey classesof regularity. Let s ≥ U be an open subset of R n . A function f : U → C is said to be s -Gevrey if f is C ∞ and if, for every compact subset K of U , there are constants C, R > α ∈ N n and x ∈ K , we have | ∂ α f ( x ) | ≤ CR | α | α ! s . (1.1)Notice that when s = 1, we retrieve the class of real-analytic functions on U . When s >
1, the class of s -Gevrey functions is non-quasianalytic: it contains compactlysupported functions. We will often write G s instead of s -Gevrey and denote by G s ( U )the space of G s functions on U and by G sc ( U ) the space of compactly supportedelements of G s ( U ). The non-quasianalyticity of G s when s > G s partitions of unity. As pointed out in the introduction, the inverseof the constant R in (1.1) will sometimes be called a G s radius of convergence.The definition above extends immediately to the case of Banach valued functions.A function f from U to some Banach space B is said to be G s if (1.1) holds with themodulus replaced by the norm of B . With this definition, the class of G s functionsis stable by composition, as was proved by Gevrey in his original paper [ Gev18 ]. Infact, the class of G s functions is very well-behaved and, when s >
1, quite flexible:for instance, it is Cartesian closed, stable by differentiation, solving ODEs, ImplicitFunction Theorem, etc, and there are versions of Borel’s and Whitney’s theorem for G s functions (see [ KMR09 ] for details).Since the class G s is stable by composition, we have a natural definition of a G s structure on a manifold: a G s manifold is a Hausdorff topological space withcountable basis endowed with a maximal G s atlas. Here, a G s atlas is defined tobe an atlas with G s change of charts (notice that we retrieve the usual notion ofreal-analytic manifold when s = 1). As usual, if M and N are two G s manifolds,then a map f : M → N is said to be G s if it is G s “in charts”. Since the ImplicitFunction Theorem holds in the class G s , most elementary results from differentialgeometry are easily checked to be true in the G s category. In particular, there is awell-defined notion of G s (vector-)bundle, and each usual bundle associated with
190 1. GEVREY MICROLOCAL ANALYSIS ON MANIFOLDS a G s manifold M (tangent, cotangent, etc) admits a natural G s structure. As aconsequence, it makes sense to say that a vector field over M is G s . Remark . Of course, a real-analytic manifold has a natural structure of G s manifold, since real-analytic maps are G s . As pointed out in [ LL97 ], all G s manifolds may be described in this way. Indeed, there is a Gevrey version of thefamous Whitney’s Embedding Theorem [ Whi36 ] : every G s compact manifold is G s -diffeomorphic to a real-analytic submanifold of an Euclidean space. The adaptationof the proof of Whitney’s Theorem to our setting is straightforward. Since theInverse Function Theorem holds in the G s category, it suffices to follow the linesof the proof of [ Hir94 , Theorem 7.1], replacing C ∞ by G s at every step. Thisembedding produces an analytic structure on any compact G s manifold, compatiblewith its G s structure.We want now to define ultradistributions on a G s manifold M . To do so, weneed to give a structure of topological vector space to the space G sc ( M ) of compactlysupported G s functions on M . If ( U, κ ) is a G s chart for M and K a relativelycompact subset of U , then we define for every R > f , infinitelydifferentiable on a neighbourhood of K , the semi-norm (cid:107) f (cid:107) s,R,K := sup x ∈ κ ( K ) α ∈ N n (cid:12)(cid:12)(cid:12) ∂ α (cid:16) f ◦ κ − (cid:17) ( x ) (cid:12)(cid:12)(cid:12) R | α | α ! s . (1.2)We extend this definition to any relatively compact subset K of M by covering K by a finite number K , . . . , K N of compact sets included in some domains of chartsand setting (cid:107) f (cid:107) s,R,K := N (cid:88) j =1 (cid:107) f (cid:107) s,R,K j . Then, if K is a relatively compact subset of M and R >
0, we define E s,R ( K )to be the Banach space of functions f ∈ C ∞ ( M ), supported in K , such that (cid:107) f (cid:107) s,R,K < + ∞ , endowed with the norm (cid:107)·(cid:107) s,R,K . For s = 1, consider thisdefinition as temporary, as we will give a more practical but equivalent scale ofspaces of real analytic functions in the next section § U is an open subsetof M , G sc ( U ) := lim −→ K (cid:98) U lim −→ R> E s,R ( K ) . Here, the first limit is taken over compact subsets K of U , and the inductivelimit is taken in the category of locally convex topological vector spaces. Noticethat the underlying set of this limit is indeed the set of compactly supported G s functions on U . In particular, when s = 1, if U is not compact itself then G sc ( U )is zero-dimensional. In the case that M itself is compact, we will write G s ( M ) for G sc ( M ).We define the space U s ( U ) of ultradistributions on U to be the strong dual of G sc ( U ). Then, by [ Gro50 ], we see that U s ( U ) identifies with the projective limit U s ( U ) = lim ←− K (cid:98) U lim ←− R> (cid:16) E s,R ( K ) (cid:17) (cid:48) . In particular, U s ( U ) is a Fr´echet space. It is a Roumieu-type space of ultradistribu-tions. More details on Roumieu and Beurling spaces can be found in [ BMT90 ]. .1. GEVREY SPACES OF FUNCTIONS AND SYMBOLS 21 Remark . The fact that the limit (1.1.1.1) does not depend on the choice ofcharts in the definition of the norms (cid:107)·(cid:107) s,R,K follows from the stability by compositionof the class G s . Indeed, if we denote by ( (cid:107)·(cid:107) (cid:48) s,R,K ) R> the family of norms that isobtained with another choice of charts, one sees (applying Faa di Bruno’s formulaor adapting the original proof of Gevrey [ Gev18 ]) that there is a constant
C >
R > (cid:107)·(cid:107) s,C max(1 ,R ) ,K ≤ C (cid:107)·(cid:107) (cid:48) s,R,K and (cid:107)·(cid:107) (cid:48) s,C max(1 ,R ) ,K ≤ C (cid:107)·(cid:107) s,R,K . (1.3)Consequently, when U is an open subset of R n , we may and will assume that thechart κ used to define the norms (cid:107)·(cid:107) s,R,K is the inclusion of U in R n . Remark . Let us explain how we will use Landau notations. We assumethat M is a compact manifold. If w : R ∗ + → R ∗ + is a function and ( u h ) h> a familyof function on M depending on a small parameter h >
0, we will oftentimes writethat u h is an O ( w ( h )) in G s ( M ) (or even in G s ), or that u = O G s ( w ( h )). It has tobe understood in the following way: there is an R > u h is an O ( w ( h ))in E s,R ( M ).If U is a subset of M , we will also write that u h is an O ( w ( h )) in G s outside of U . It means that there is an R > (cid:107) u h (cid:107) s,R,M \ U = O ( w ( h )) . Notice that we retrieve the previous case by taking U = ∅ . We will not be concernedwith regularity with respect to the parameter h . However, when there are otherparameters, in a function f of ( x, y ; h ) for example, more care is required. Wewill thus use the notation f h ( x, y ) = O G sx ( w ( h, y )) to indicate that for every y thefunction x (cid:55)→ f ( x, y ) is an O G s ( w ( h, y )), with constants uniform in y . It is not tobe confused with f h ( x, y ) = O G s ( w ( h )) which indicates Gevrey regularity in both x and y . Remark . Let s ≥
1. In order to discuss perturbations of G s Anosov flows,we need to define a topology on the space of G s sections of a G s vector bundle. Let p : F → M denote a real G s vector bundle on M (the case of complex vector bundleis similar). Let ( U, κ ) be a G s chart for M and ( κ, Ψ) : p − ( U ) → κ ( U ) × R d be atrivialization for p : F → M . Then, if K is a compact subset of U and R >
0, wedefine for every C ∞ section f of F the semi-norm (cid:107) f (cid:107) s,R,K = sup x ∈ κ ( K ) α ∈ N n (cid:13)(cid:13)(cid:13) ∂ α Ψ (cid:16) f ◦ κ − (cid:17) ( x ) (cid:13)(cid:13)(cid:13) R | α | α ! s , where (cid:107)·(cid:107) denotes any norm on R d . Using this semi-norm to replace (1.2), the caseof sections of the vector bundle F is dealt with as the case of the trivial line bundleover M . In particular, when M is a compact manifold, we have a definition thespaces E s,R ( M ; F ), for R >
0, and a topology on the space G s ( M ; F ) of G s sectionsof F .1.1.1.2. The particular case of real-analytic functions.
We want now to rewritethe definitions of the previous paragraph in a way that may be more intuitive,in the case s = 1. Indeed, it may seem more natural to describe real analyticfunctions as restrictions of holomorphic functions. If K is a compact subset of R n ,then we see that a smooth function f defined on a neighbourhood of K such that (cid:107) f (cid:107) ,R,K < + ∞ admits a holomorphic extension to a complex neighbourhood of K of size ( CR ) − (for some C > R ), the L ∞ norm ofthis extension being bounded by C (cid:107) f (cid:107) ,R,K . Reciprocally, if f admits a bounded holomorphic extension to a complex neighbourhood of size CR − , then (cid:107) f (cid:107) ,R,K isfinite and controlled by the L ∞ norm of this extension (independently on R ). Wewill now explain how this remark generalizes to the case of compact manifold.Let then M be a compact real-analytic manifold of dimension n . By a resultof Fran¸cois Bruhat and Hassler Whitney [ WB59 ], the manifold M admits a com-plexification (cid:102) M . That is, (cid:102) M is a holomorphic manifold of complex dimension n endowed with a real-analytic embedding M ⊆ (cid:102) M , such that M is a totally realsubmanifold of (cid:102) M . This means that at each p ∈ M , we have T p M ∩ iT p M = { } . Itfollows then that if N is a complex manifold and f : M → N is a real-analytic map,then f extends to a holomorphic map from a neighbourhood of M in (cid:102) M to N . Inparticular, if (cid:102) M (cid:48) is another complexification for M then the identity of M extendsto a biholomorphism between a neighbourhood of M in (cid:102) M and a neighbourhood of M in (cid:102) M (cid:48) . Remark . If (cid:102) M is a complexification for M , let B ( (cid:102) M ) denote the space ofbounded holomorphic functions on (cid:102) M . Then, we may give a new definition of thespace of real-analytic functions on M by G ( M ) = lim −→ (cid:102) M B (cid:0) (cid:102) M (cid:1) . Here, the inductive limit (in the category of locally convex topological vector spaces)is over all the complexifications (cid:102) M of M . This coincides with (1.1.1.1) when s = 1.This definition may be quite appealing because it is very intrinsic. However, wewill rather use a more concrete description of G ( M ), that boils down to choosing aparticular basis of complex neighbourhoods for M .We will use particular complexifications of M called Grauert tubes. The notionof Grauert tube first appeared in [ Gra58 ], but we will rely on the exposition from[
GS91 ]. First, according to [
Mor58 ], there is a real-analytic embedding of M intoan Euclidean space. Hence, we may choose a real-analytic Riemannian metric g on M . According to [ WB59 ], there exists a complexification (cid:102) M of M endowed withan anti-holomorphic involution z (cid:55)→ ¯ z such that M is the set of fixed point of z (cid:55)→ ¯ z .Then, since the square of the distance induced by g on M is real-analytic near thediagonal, it extends to a holomorphic function on a neighbourhood of the diagonalof M in (cid:102) M × (cid:102) M . Following [ GS91 ], we define ρ on (cid:102) M (up to taking (cid:102) M smaller) by ρ ( z ) = − d ( z, ¯ z ) . From [
GS91 ], we know that ρ defines a strictly plurisubharmonic function on (cid:102) M such that M = { z ∈ (cid:102) M : ρ ( z ) = 0 } . Then, if (cid:15) > M ) (cid:15) as the sub-level of ρ :( M ) (cid:15) := (cid:110) z ∈ (cid:102) M : ρ ( z ) < (cid:15) (cid:111) . (1.4)Notice that, since ρ is strictly plurisubharmonic, the Grauert tube ( M ) (cid:15) is strictlypseudo-convex. Moreover, the real (1 , i∂ ¯ ∂ρ is K¨ahler and the associatedhermitian form coincides with g on M . We will consequently still denote thishermitian form by g .Using the notion of Grauert tube, we can replace the spaces E ,R ( M ) thatappeared in § R ≥ E ,R ( M ) the space of bounded holomorphic functions on ( M ) /R (endowedwith the sup norm). Here, we only work with R large enough so that ( M ) /R iswell-defined and E ,R ( M ) is non-trivial. The spaces E ,R ( M ) defined in this way do .1. GEVREY SPACES OF FUNCTIONS AND SYMBOLS 23 not need to coincide with those of § G ( M ) = lim −→ R → + ∞ E ,R ( M ) . Hence, we will always assume that we use this definition of the spaces E s,R ( M )in the case s = 1. To see that we get the same topology on G ( M ) as in § G ( M ) is not very important,since we will have to work directly with the spaces E ,R ( M ) in order to get thebest order for the dynamical determinant in Theorem 1.Let us point out that there is another way to describe Grauert tubes. Indeed,since the Riemannian metric g is real-analytic, so is its exponential map. Conse-quently, if x is a point in M , then exp x : T x M → M extends to a holomorphic map,still denoted by exp x , from a neighbourhood of 0 in T x M ⊗ C to (cid:102) M . Then, the map( x, v ) (cid:55)→ exp x ( iv )(1.5)defines a real analytic diffeomorphism between a neighbourhood of the zero sectionin T M and a neighbourhood of M in (cid:102) M . For (cid:15) > M ) (cid:15) is the image of (cid:110) ( x, v ) ∈ T M : g ( v, v ) < (cid:15) (cid:111) by the map (1.5). With this description of the Grauert tube ( M ) (cid:15) , we see that theprojection T M → M induces a real-analytic projection from ( M ) (cid:15) to M , that wewill denote by Re. We define also the function | Im | : ( M ) (cid:15) → R + as the square rootof ρ . We will sometimes, slightly abusively, write | Im z | instead of | Im | ( z ).Since M is compact, we could have chosen any decreasing basis of neighbour-hoods for M in (cid:102) M to define the spaces E ,R ( M ). However, we will need to considerreal-analytic functions defined on T ∗ M (for instance symbols) or more generally onproducts of the type M N × (cid:0) T ∗ M (cid:1) N . Since these manifolds are non-compact, thechoice of a complex neighbourhood for T ∗ M becomes non-trivial. As we want toconsider symbols on T ∗ M , it seems natural to introduce Grauert tubes using theKohn–Nirenberg metric g KN . Recall that g KN is defined in the following way. TheLevi–Civita connexion associated with g gives a splitting T ( T ∗ M ) = V ⊕ H intovertical and horizontal bundles, where both subbundles are identified with T M , sothat we can define g KN ( x, ξ ) := g H ( x, ξ ) + 11 + | ξ | x g V ( x, ξ ) . In charts, it is uniformly equivalent to its flat version g flat KN = d x + 11 + ξ d ξ . The curvature of g KN is bounded, and so are all its covariant derivatives, and onecan check that it admits Grauert tubes, which look like conical neighbourhoods of T ∗ M at infinity.More precisely, the cotangent space T ∗ (cid:102) M of (cid:102) M is a complexification of T ∗ M .Notice that there is a natural inclusion of T ∗ M ⊗ C into T ∗ (cid:102) M and that the anti-holomorphic involution that fixes T ∗ M is given on T ∗ M ⊗ C by ( x, ξ ) (cid:55)→ ( x, ¯ ξ ). Asabove, we find a strictly plurisubharmonic function ρ KN defined on a neighbourhoodof T ∗ M in T ∗ (cid:102) M . Then, mimicking (1.4), we set for (cid:15) > (cid:0) T ∗ M (cid:1) (cid:15) := (cid:110) α ∈ T ∗ (cid:102) M : ρ KN ( α ) < (cid:15) (cid:111) . As in the compact case, | Im α | will denote the square root of ρ KN ( α ). To describethese tubes in more concrete terms, we may examine them in local coordinates.Given a real-analytic chart for M , it extends holomorphically to a chart for (cid:102) M . Ithence defines a holomorphic trivialization for T ∗ (cid:102) M , mapping T ∗ M on T ∗ R n . If wedenote these coordinates by ˜ x = x + iy, ˜ ξ = ξ + iη , then for a point α ∈ T ∗ (cid:102) M thatwrites (˜ x, ˜ ξ ) in local coordinates, the quantity | Im α | is uniformly equivalent to | Im α | (cid:16) | y | + | η |(cid:104) ξ (cid:105) . This gives a rough but tractable idea of the shape of Grauert tubes in local coordi-nates.Let us discuss some others notations. Since x (cid:55)→ g x is a K¨ahler metric, themap α (cid:55)→ g α x ( α ξ , α ξ ) is real analytic and non-negative. On the other hand, we canconsider the holomorphic extension ˜ g of g , so that α (cid:55)→ ˜ g α x ( α ξ , α ξ ) is a holomorphicmap. With the determination of the square root positive on the reals, we define for α in T ∗ (cid:102) M the Japanese brackets (cid:104) α (cid:105) = (cid:113) g α x ( α ξ , α ξ ) and (cid:104)| α |(cid:105) = (cid:113) g α x ( α ξ , α ξ ) . (1.6)Hence, (cid:104) α (cid:105) is holomorphic in α , while (cid:104)| α |(cid:105) is not. However, notice that on a Grauerttube (cid:0) T ∗ M (cid:1) (cid:15) , for (cid:15) > (cid:104)| α |(cid:105) , |(cid:104) α (cid:105)| and Re (cid:104) α (cid:105) are uniformly equivalent. Notice also that we may define a Kohn–Nirenberg metricon T ∗ (cid:102) M (since T ∗ (cid:102) M identifies with the cotangent bundle of (cid:102) M seen as a real-analyticmanifold).Finally, let us mention that we will use simpler definitions of the notions abovewhen working on R n . If U is an open subset of R n and (cid:15) > (cid:0) T ∗ U (cid:1) (cid:15) = { ( x, ξ ) ∈ C n : d ( x, U ) < (cid:15) and | Im ξ | < (cid:15) (cid:104) Re ξ (cid:105)} (1.7)of T ∗ U . We think of (cid:0) T ∗ U (cid:1) (cid:15) as (an approximation of) the Grauert tube of T ∗ U for the Kohn–Nirenberg metric, even if this case is not covered by our discussionabove (since U is not a compact manifold). Similarly, for ξ ∈ C n with Re (cid:104) ξ, ξ (cid:105) ≥ R ∗ + ) (cid:104) ξ (cid:105) = (cid:112) (cid:104) ξ, ξ (cid:105) and (cid:104)| ξ |(cid:105) = (cid:113) (cid:107) ξ (cid:107) . Here, the norm in the definition of (cid:104)| ξ |(cid:105) is the one given by the identification C n (cid:39) R n , while (cid:104) ξ, ξ (cid:105) = (cid:80) nj =1 ξ j if ξ = ( ξ , . . . , ξ n ). Hence (cid:104) ξ (cid:105) is holomorphic but (cid:104)| ξ |(cid:105) is not. As in the manifold case, for U an open subset of R n and ( x, ξ ) ∈ ( T ∗ U ) (cid:15) ,the quantities (cid:104)| ξ |(cid:105) , (cid:104) Re ξ (cid:105) , |(cid:104) ξ (cid:105)| and Re (cid:104) ξ (cid:105) are equivalent.1.1.1.3. Almost analytic extensions.
Let (
M, g ) be a compact real-analytic Rie-mannian manifold. To study deformations in the Grauert tube of M (or of T ∗ M ),we will make extensive use of almost-analytic extensions of smooth functions on M .The notion of almost analytic extension was introduced by Lars H¨ormander [ H¨or69 ]and then by Louis Nirenberg [
Nir71 ]. It has become a very common notion inmicrolocal analysis, and are essential in [
MS75 ] for instance.Recall that if f is a C ∞ function on M then an almost-analytic extension for f is a compactly supported C ∞ functions ˜ f on some ( M ) (cid:15) that coincides with f on M and such that ¯ ∂ ˜ f vanishes to all orders on M . It is classical that such a ˜ f exists [ Zwo12 , Theorem 3.6]. While this is hardly surprising, it will be crucial inour analysis that if f is G s then ˜ f may be chosen G s as well. This will allow us tomake the flatness of ¯ ∂ ˜ f near M quantitative using Lemma 1.3 below. .1. GEVREY SPACES OF FUNCTIONS AND SYMBOLS 25 Observe that in [
FNRS19 ] (for example), weaker forms of almost analyticextensions were obtained with the adequate decay. However, they are only C ,which is not sufficient for our purposes. One can also find in [ Jun00 ] another notionof almost analytic extension: instead of working with a single function ˜ f such that¯ ∂ ˜ f vanishes to all orders on M , a family of function ( ˜ f h ) h> , where f h is defined ona tube whose size depends on h and such that ¯ ∂f h vanishes when h tends to 0, isused.We start by proving that Gevrey functions admit Gevrey almost analyticextensions. Lemma . Given a Grauert tube ( M ) (cid:15) ⊃ M , for each s > , there exists aconstant C s and a compact subset K ⊆ ( M ) (cid:15) so that for all R ≥ , there exists abounded map f (cid:55)→ ˜ f from E s,R ( M ) to E s,C s R ( K ) , such that ˜ f is an almost-analyticextension for f . Covering M by real-analytic charts and then using a G s partition of unity, wereduce to the Euclidean case, that is, we only need to prove: Lemma . Let s > . Let K be a compact subset of R n and K (cid:48) be a compactneighbourhood of K in C n . Then, there is A > , and for each R > thereis a continuous map f (cid:55)→ ˜ f from E s,R ( K ) to E s,AR (cid:0) K (cid:48) (cid:1) such that for every f ∈ E s,R ( K ) , the function ˜ f is an almost analytic extension for f . It seems to be folklore that this may be deduced from results of Lennart Carlesonon universal moment problems [
Car61 ], but we are not aware of any referencecontaining a proof, and thus we provide one.
Proof.
First of all, we need to extend to higher dimensions the one dimensionalresults of Carleson. This is quite straightforward, since most difficulties are alreadypresent in the one dimensional case. For x ∈ R we define w ( x ) = (cid:16) x (cid:17) − exp (cid:32) − s | x | s e (cid:33) . Let us denote by H the Hilbert space of measurable functions (up to modificationon zero measure sets) u from R n to C that satisfy (cid:107) u (cid:107) H := (cid:90) R n | u ( x ) | ¯ w ( x )d x < + ∞ , where ¯ w ( x ) = (cid:81) nj =1 w ( x j ). Let ( P m ) m ∈ N be the sequence (depending on s ) oforthogonal polynomials, defined, up to a sign, by ∀ m, p ∈ N : (cid:90) R P m ( t ) P p ( t ) w ( t )d t = (cid:40) m = p, , and deg P m = m for every m ∈ N . Then define for α ∈ N n the polynomial P α ( x ) = n (cid:89) j =1 P α j ( x j ) , and notice that the P α ’s form an orthogonal family in H . According to [ Car61 ,(2.6)], there are constants
C, r > k ∈ N we have + ∞ (cid:88) (cid:96) =0 (cid:12)(cid:12)(cid:12) P ( k ) (cid:96) (0) (cid:12)(cid:12)(cid:12) ≤ Cr k k ! − s ) . For some new constants C and r , and for every β ∈ N n , it follows that (cid:88) α ∈ N n (cid:12)(cid:12)(cid:12) ∂ β P α (0) (cid:12)(cid:12)(cid:12) ≤ Cr | β | β ! − s ) . Now, let S denotes the Hilbert space of sequences s = ( s α ) α ∈ N n of complex numberssuch that (cid:107) s (cid:107) S := (cid:88) α ∈ N n (cid:32) | s α | (2 r ) | α | α ! s (cid:33) < + ∞ . If s ∈ S , define the sequence ( b α ) α ∈ N n by b α = (cid:88) β ∈ N n ∂ β P α (0) β ! s β , for α ∈ N n (notice that this sum is finite). Then we have (cid:88) α ∈ N n | b α | = (cid:88) α ∈ N n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) β ∈ N n ∂ β P α (0) β ! s β (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , ≤ (cid:107) s (cid:107) S (cid:88) α ∈ N n (cid:88) β ∈ N n (cid:12)(cid:12)(cid:12) ∂ β P α (0) (cid:12)(cid:12)(cid:12) β ! β ! s r ) | β | , ≤ (cid:107) s (cid:107) S (cid:88) β ∈ N n β ! s − r ) | β | (cid:88) α ∈ N n (cid:12)(cid:12)(cid:12) ∂ β P α (0) (cid:12)(cid:12)(cid:12) , ≤ C (cid:107) s (cid:107) S (cid:88) β ∈ N n − | β | = C (cid:18) (cid:19) n (cid:107) s (cid:107) S . Thus, if we set L ( s ) = (cid:88) α ∈ N n b α P α , then L is a bounded operator from S to H . The main point about L is that forevery s ∈ S and α ∈ N n an elementary computation ensures that(1.8) (cid:90) R n L ( s )( t ) t α ¯ w ( t )d t = s α . Now, if f ∈ E s,R ( K ), we define for every x ∈ R n the sequence s ( x ) =( s α ( x )) α ∈ N n by s α ( x ) = ∂ α f ( x ) b | α | , where b = 3 × s Rr . Notice that s ( x ) ∈ S and define ˜ f for x, y ∈ R n by(1.9) ˜ f ( x + iy ) = χ ( y ) (cid:90) R n e ib (cid:104) t,y (cid:105) L ( s ( x ))( t ) ¯ w ( t )d t = χ ( y ) (cid:104) e ib (cid:104) y, ·(cid:105) , L ( s ( x )) (cid:105) H , where χ is a compactly supported s (cid:48) -Gevrey function for some 1 < s (cid:48) < s , identicallyequal to 1 on a neighbourhood of 0. One easily checks that the map x (cid:55)→ s ( x ) ∈ S is C ∞ and supported in K . Moreover, if x ∈ K and β ∈ N n then we have(1.10) ∂ β s ( x ) = (cid:32) ∂ α + β f ( x ) b | α | (cid:33) α ∈ N n . Thus, thanks to our choice of b we have for all β ∈ N n and x ∈ K (1.11) (cid:13)(cid:13)(cid:13) ∂ β s ( x ) (cid:13)(cid:13)(cid:13) S ≤ n (cid:107) f (cid:107) s,R,K (2 s R ) | β | β ! s . .1. GEVREY SPACES OF FUNCTIONS AND SYMBOLS 27 Thus x (cid:55)→ s ( x ) is G s with inverse radius 2 s R . By dominated convergence, we seethat the map G : y (cid:55)→ ( t → e ib (cid:104) t,y (cid:105) ) is C ∞ from R n to H . Moreover, if α ∈ N n thederivative ∂ α G is y (cid:55)→ (cid:16) t (cid:55)→ ( ibt ) | α | e ib (cid:104) t,y (cid:105) (cid:17) and its norm at y ∈ R n is (cid:107) ∂ α G ( y ) (cid:107) H = b | α | (cid:90) R n | t α | ¯ w ( t )d t = b | α | n (cid:89) j =1 (cid:90) R | t | α j w ( t )d t , ≤ b | α | (cid:16) e s (cid:17) s | α | (cid:18) se s s s (cid:19) n n (cid:89) j =1 Γ (cid:0) s (2 α j + 1) (cid:1) . Then, applying Stirling’s formula, we see that there are constants
C, M > α ∈ N n and y ∈ R n we have(1.12) (cid:107) ∂ α G ( y ) (cid:107) H ≤ C ( M R ) | α | α ! s . Consequently, the map F : x + iy (cid:55)→ (cid:104) e ib (cid:104) y, ·(cid:105) , L ( s ( x )) (cid:105) H is C ∞ on C n and if α, β ∈ N n and x, y ∈ R n then we have(1.13) ∂ αx ∂ βy F ( x + iy ) = (cid:104) ∂ β G ( y ) , L ( ∂ α s ( x )) (cid:105) H . Thus, from (1.11) and (1.12), we see that the map f (cid:55)→ ˜ f is continuous from E s,R ( K ) to E s,MR (cid:0) K (cid:48) (cid:1) for K (cid:48) a compact complex neighbourhood of K (we mayand do assume that M ≥ s ).It remains to see that ˜ f is indeed an almost analytic extension for f . Since˜ f and F coincide near R n , we only need to study F to do so. Notice that for all x ∈ R n and α, β ∈ N n , according to (1.13), (1.8) and (1.10), we have ∂ αx ∂ βy F ( x ) = (cid:90) R n ( ibt ) β L ( ∂ α s ( x )) ( t ) ¯ w ( t )d t = i | β | ∂ α + β f ( x ) . In particular, F and f coincide on R n . Now, if j ∈ { , . . . , n } , we have ∂F∂ ¯ z j = (cid:16) ∂F∂x j + i ∂F∂y j (cid:17) , and thus if x ∈ R n and α, β ∈ N n we have ∂∂ ¯ z j (cid:16) ∂ αx ∂ βy F (cid:17) ( x ) = 12 (cid:16) ∂ α + e j x ∂ βy F + i∂ αx ∂ β + e j y F (cid:17) = 12 (cid:16) i | β | ∂ α + β + e j f ( x ) + i | β | +2 ∂ α + β + e j f ( x ) (cid:17) = 0 . Consequently, ¯ ∂F vanishes to all orders on R n , and ˜ f is indeed an almost analyticextension of f . (cid:3) In order to apply Lemma 1.1, we need to investigate the way a Gevrey functioncan be flat. To do so, we will apply the “sommation au plus petit terme”, a methodfor regularizing certain divergent series that is particularly well suited for Taylorseries of Gevrey functions. The interested reader may refer to [
Ram93 ] for detailsand historical references.
Lemma . Let U be an open subset of R n and K a compact and convex subsetof U . Then for every s > , there are constants C, C > such that for every R > and f ∈ C ∞ ( U ) such that the quantity (cid:107) f (cid:107) s,R,K defined by (1.2) is finite, if x, y ∈ K then we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( y ) − (cid:88) | α |≤ ( R | x − y | ) − s − C ∂ α f ( x ) α ! ( y − x ) α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:107) f (cid:107) s,R,K exp (cid:32) − C ( R | x − y | ) s − (cid:33) . (1.14) We explain in Remark 1.7 below how Lemma 1.3 allows to control the size ofthe Cauchy–Riemann operator applied to an almost analytic extension of a Gevreyfunction.
Proof of Lemma 1.3.
Taylor’s formula gives for every non-negative integer kf ( y ) = (cid:88) | α |
1) ln 24 ( nR | x − y | ) s − (cid:33) , and the lemma is proved. (cid:3) Remark . We will mostly use Lemma 1.3 with an f that vanishes to infiniteorder on x . In that case, we get a control on the size of f ( y ) for y near x . However,it will sometimes be useful to have the general result at our disposal. Concerningthe general result, notice that the constant C in (1.14) may be chosen arbitrarilylarge (up to taking C larger). Indeed, the terms that we add by taking C largerare controlled by the right hand side of (1.14). Remark . Let us explain how Lemma 1.3 allows to control the size of analmost analytic extension of a Gevrey function. Let s, K, K (cid:48) , M and R be as inLemma 1.2. We may assume that K (cid:48) is convex. Then, if f ∈ E s,R ( K ) is a G s function, we know that it admits a G s almost analytic extension ˜ f ∈ E s,MR (cid:0) K (cid:48) (cid:1) .Then if R > M R , we see that the components of ¯ ∂ ˜ f belongs to E s,R (cid:0) K (cid:48) (cid:1) . Byassumption, these components vanish at infinite order on R n . Hence, it follows fromLemma 1.3 that there are constants C > K and K (cid:48) ) and C R (that may also depend on R ) such that for z ∈ C n we have (cid:12)(cid:12)(cid:12) ¯ ∂ ˜ f ( z ) (cid:12)(cid:12)(cid:12) ≤ C R (cid:107) f (cid:107) s,R,K exp (cid:32) − C ( R | Im z | ) s − (cid:33) . (1.15)Here, we used the fact that the norms of the coordinates of ¯ ∂ ˜ f in E s,R (cid:0) K (cid:48) (cid:1) arecontrolled by (cid:107) f (cid:107) s,R,K .It will be useful to control also the derivatives of ¯ ∂ ˜ f . In fact, we can improve(1.15) into a Gevrey estimates. Indeed, if α, β ∈ N n then the components of ∂ αx ∂ βy ¯ ∂ ˜ f are G s (we write z = x + iy for the coordinates in C n ). To see so, just notice thatfor α (cid:48) , β (cid:48) ∈ N n , j ∈ { , . . . , n } and z ∈ K (cid:48) we have ( C > K and .1. GEVREY SPACES OF FUNCTIONS AND SYMBOLS 29 K (cid:48) and may vary from one line to another, C R may also depend on R ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ α + α (cid:48) x ∂ β + β (cid:48) y ∂ ˜ f∂ ¯ z j ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:13)(cid:13)(cid:13) ˜ f (cid:13)(cid:13)(cid:13) s,MR,K (cid:48) ( M R ) | α | + | α (cid:48) | + | β | + | β (cid:48) | (cid:0) α + α (cid:48) + e j (cid:1) ! s (cid:0) β + β (cid:48) + e j (cid:1) ! s ≤ C R (cid:107) f (cid:107) s,R,K ( CR ) | α | + | β | α ! s β ! s ( CR ) | α (cid:48) | + | β (cid:48) | (cid:0) α (cid:48) (cid:1) ! s (cid:0) β (cid:48) (cid:1) ! s . This estimate can be rewritten as (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∂ αx ∂ βy ∂ ˜ f∂ ¯ z j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) s,CR,K (cid:48) ≤ C R (cid:107) f (cid:107) s,R,K ( CR ) | α | + | β | α ! s β ! s . Since ∂ αx ∂ βy ∂ ˜ f∂ ¯ z j vanishes at all orders on R n , it follows from Lemma 1.3 that thereare constants C >
0, that only depends on K and K (cid:48) , and C R > R , such that for every z ∈ C n and α, β ∈ N n we have (cid:12)(cid:12)(cid:12) ∂ αx ∂ βy ¯ ∂ ˜ f ( z ) (cid:12)(cid:12)(cid:12) ≤ C R (cid:107) f (cid:107) s,R,K ( CR ) | α | + | β | α ! s β ! s exp (cid:32) − C ( R | Im z | ) s − (cid:33) . To close this section, we present here a trick that will be useful to deriveGevrey bounds from L ∞ bounds on almost analytic extensions. This trick is a slightrefinement of the proof of [ FNRS19 , Proposition 3.8] in the Gevrey case (which isin some sense the reciprocal of Lemma 1.2).
Lemma . Let s ≥ . Let D be a ball in R n . Thereis a constant (cid:101) C > such that, for every R ≥ , there is a constant C R such that thefollowing holds. For every λ ≥ , let f λ be a function from R n to C . We assumethat for all λ ≥ the map f λ admits a C extension F λ to a neighbourhood of D in C n , and there exist R > and C > such that(i) for every λ > and x ∈ C n if x is at distance less than R − s λ s − of D then | F λ ( x ) | ≤ C exp (cid:32) − (cid:18) λR (cid:19) s (cid:33) ; (ii) for every λ > and x ∈ C n if x is at distance less than R − /s λ s − of D then (cid:12)(cid:12) ¯ ∂F λ ( x ) (cid:12)(cid:12) (cid:40) ≤ C exp (cid:16) − ( R | Im x | ) − s − (cid:17) if s >
1= 0 if s = 1 . Then for R (cid:48) = (cid:101) CR and every λ ≥ we have (cid:107) f λ (cid:107) s,R (cid:48) ,D ≤ C R C exp (cid:32) − (cid:18) λ (cid:101) CR (cid:19) s (cid:33) . Proof.
The case s = 1 is just an application of Cauchy’s formula. Consequently,we focus on the case s > s > λ ≥ D λ the R − /s λ s − / D . Then choose x ∈ D and write the Bochner–Martinelli formulafor F λ at x to find f λ ( x ) = (cid:90) ∂D λ F λ ( z ) ω ( z, x ) − (cid:90) D λ ¯ ∂F λ ( z ) ∧ ω ( z, x ) , (1.16) where ω ( · , x ) is the ( n, n −
1) form (the hat means that the corresponding factor isomitted) ω ( z, x ) = ( n − iπ ) n | z − x | n n (cid:88) j =1 (cid:0) ¯ z j − ¯ x j (cid:1) d¯ z ∧ d z ∧ · · · ∧ (cid:99) d¯ z j ∧ · · · ∧ d¯ z n ∧ d z n . By Fa`a di Bruno’s formula and the Leibniz rule, we get as in [
FNRS19 , Proposition3.8] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ αx (cid:32) ¯ z j − ¯ x j | z − x | n (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( n ) | α | α ! | x − z | n + | α |− , where the constant C ( n ) only depends on the dimension n . Hence, by differentiationunder the integral, we see that f λ is C ∞ with the estimate (for λ ≥
1, the constant
C > C does not depend on C ) | ∂ α f λ ( x ) | ≤ CC | α | C ( n ) | α | α ! (cid:16) R s λ − s (cid:17) n + | α |− exp (cid:32) − (cid:18) λR (cid:19) s (cid:33) + CC C ( n ) | α | α ! sup t ∈ (cid:20) , R − s λ s − (cid:21) exp (cid:16) − ( Rt ) − s − (cid:17) t n + | α |− . (1.17)Then, notice that sup r ∈ R ∗ + e − r r ( s − n + | α |− ≤ C ( s, n ) | α | α ! s − , so thatexp (cid:32) − (cid:18) λR (cid:19) s (cid:33) (cid:16) R s − λ − s (cid:17) n + | α |− ≤ sup t ∈ R ∗ + exp (cid:32) − t /s (cid:33) t (1 − /s )(2 n + | α |− ≤ ( s − n + | α |− sup r ∈ R ∗ + e − r r ( s − n + | α |− ≤ C | α | α ! s − , where the constant C > n and s . We also have for t ∈ (cid:104) , R − s λ s − (cid:105) thatexp (cid:16) − ( Rt ) − s − (cid:17) t n + | α |− ≤ exp (cid:32) − ss − (cid:18) λR (cid:19) s (cid:33) sup τ ∈ R + exp (cid:16) − ( Rτ ) − s − (cid:17) τ n + | α |− ≤ (cid:16) s − R (cid:17) n + | α |− exp (cid:32) − ss − (cid:18) λR (cid:19) s (cid:33) sup r ∈ R + e − r r ( s − n + | α |− ≤ C | α | α ! s − R n + | α |− exp (cid:32) − ss − (cid:18) λR (cid:19) s (cid:33) . The announced result follows by plugging the last two estimates in (1.17). (cid:3)
The Bochner–Martinelli trick will be quite useful when studying oscillatoryintegrals, for the following reason: .1. GEVREY SPACES OF FUNCTIONS AND SYMBOLS 31
Lemma . Let U ⊂ R n be a bounded open set, and Φ : U → C be a G s function. Let K ⊂ U be a compact set. Then there exists C, R , h > such thatfor R ≥ R , there exists a constant C R > , so that for h ∈ ]0 , h ] , (cid:13)(cid:13)(cid:13) x (cid:55)→ e ih Φ( x ) (cid:13)(cid:13)(cid:13) s,R,K ≤ C R exp (cid:32) − h inf K Im Φ + 1 C ( hR ) /s (cid:33) . In particular, taking
R > h − /s term. Proof.
Up to replacing Φ by Φ − i inf K Im Φ, we may assume that inf K Im Φ =0. Then, we pick ˜Φ an almost analytic extension for Φ, and we observe that for x at distance at most R − /s h − /s from K , we haveIm ˜Φ ≥ − C h − s R s . On the other hand, (cid:12)(cid:12)(cid:12) ∂ x (cid:16) e ih (cid:101) Φ( x ) (cid:17)(cid:12)(cid:12)(cid:12) ≤ C exp (cid:32) C h − /s R s − R | Im x | ) s − (cid:33) , ≤ C exp (cid:32) C h − /s R s − R | Im x | ) s − (cid:33) , provided that R ≥ R . We can then apply the Bochner–Martinelli trick toexp( − CR − /s h − /s ) e i Φ( x ) h and the result follows. (cid:3) Definitions.
In anticipation of the study of G s pseudors in § BDMK67, Zan85, Rod93 ]. Letus fix an open subset U of R n for the remainder of this section. Definition . Let m ∈ R , s ≥
1. A C ∞ function a : T ∗ U (cid:39) U × R n → C belongs to the symbol class S s,m (cid:0) T ∗ U (cid:1) if, for every compact subset K of U , thereare constants C, R > α, β ∈ N n and ( x, ξ ) ∈ K × R n , we have (cid:12)(cid:12)(cid:12) ∂ αx ∂ βξ a ( x, ξ ) (cid:12)(cid:12)(cid:12) ≤ CR | α | + | β | ( α ! β !) s (cid:104) ξ (cid:105) m −| β | . (1.18)In a more transparent way, we will also write that a is a G s symbol (of order m ) . Remark . Symbols are allowed to depend on the small implicit parameter h > h > Remark . For N , N ∈ N , we define mutatis mutandis classes of symbols S s,m (cid:16)(cid:0) T ∗ U (cid:1) N × U N (cid:17) . It is understood that the results and definitions below extend naturally to thosemore general classes of symbols. Later on, we will mostly consider symbols a ( α, x )in S s,m (cid:0) T ∗ M × M (cid:1) that are in fact only defined when x and α x are close to eachother.As we did for functions in § R n introduced in (1.7). Using Taylor’s and Cauchy’sFormula we can prove (the proof is left as an exercise to the reader): Lemma . Let m ∈ R . Let a : T ∗ U → C be a function. Then a belongs to thesymbol class S ,m (cid:0) T ∗ U (cid:1) if and only if, for every relatively compact open subset Ω of U , there is (cid:15) > such that a admits a holomorphic extension to (cid:0) T ∗ Ω (cid:1) (cid:15) , whichsatisfies for some C > and every ( x, ξ ) ∈ (cid:0) T ∗ Ω (cid:1) (cid:15) | a ( x, ξ ) | ≤ C (cid:104) Re ξ (cid:105) m . Remark . In the absence of holomorphic extensions in the case s > a ∈ S s,m (cid:0) T ∗ U (cid:1) and every relatively compact open subset Ω of U , thesymbol a admits an almost analytic extension on (cid:0) T ∗ Ω (cid:1) (cid:15) for some (cid:15) >
0. Thisextension satisfies G s symbolic estimates as in Definition 1.1. In particular, byrescaling Lemma 1.3, we find that there is a constant C > x, ξ ) ∈ (cid:0) T ∗ Ω (cid:1) (cid:15) we have (cid:12)(cid:12) ¯ ∂a ( x, ξ ) (cid:12)(cid:12) ≤ C (cid:104) Re ξ (cid:105) m exp (cid:32) − C (cid:18) | Im x | + | Im ξ |(cid:104) Re ξ (cid:105) (cid:19) − s − (cid:33) . (1.19)The proof of the existence of such an extension for a is based on Lemma 1.1 anda standard rescaling argument, using for instance a partition of unity of Paley–Littlewood type. Observe that the quantity in the exponential is comparable to (cid:107) Im( x, ξ ) (cid:107) − / ( s − g KN . As in Remark 1.7, the estimate (1.19) can be upgraded to aGevrey estimates: there are constants C, R > x, ξ ) ∈ (cid:0) T ∗ Ω (cid:1) (cid:15) and α, β ∈ N n we have – under the identification R n (cid:39) C n , (cid:12)(cid:12)(cid:12) ∂ αx ∂ βξ ¯ ∂a ( x, ξ ) (cid:12)(cid:12)(cid:12) ≤ CR | α | + | β | α ! s β ! s (cid:104) Re ξ (cid:105) m −| β | exp (cid:32) − C (cid:18) | Im x | + | Im ξ |(cid:104) Re ξ (cid:105) (cid:19) − s − (cid:33) . In the case s = 1, we will construct in § Definition . Let m ∈ R . We define an element of the class of formal symbols F S ,m (cid:0) T ∗ U (cid:1) to be a formal series (cid:80) k ≥ h k a k where the a k ’s are C ∞ functions from T ∗ U to C such that, for every open relatively compact subset Ω of U , there are (cid:15) > k ≥
0, a holomorphic extension of a k to (cid:0) T ∗ Ω (cid:1) (cid:15) , and constants C, R > k ∈ N and ( x, ξ ) ∈ (cid:0) T ∗ Ω (cid:1) (cid:15) we have | a k ( x, ξ ) | ≤ CR k k ! (cid:104) Re ξ (cid:105) m − k . (1.20) Remark . If V is an open subset of (cid:0) T ∗ U (cid:1) (cid:15) and ( a k ) k ≥ is a family ofholomorphic functions on V that satisfies (1.20) then we will also say that (cid:80) k ≥ h k a k is a formal symbol on V . Definition . Let m ∈ R and a = (cid:80) k ≥ h k a k ∈ F S ,m (cid:0) T ∗ U (cid:1) . Let Ω be anopen subset of U . A realization of a on Ω is an element a of S ,m (cid:0) T ∗ Ω (cid:1) such that,for every open relatively compact subset Ω (cid:48) of Ω, there is (cid:15) > a andthe a k ’s have holomorphic extensions to (cid:0) T ∗ Ω (cid:48) (cid:1) (cid:15) such that, for every C > C >
0, such that, for every ( x, ξ ) ∈ (cid:0) T ∗ Ω (cid:48) (cid:1) (cid:15) , we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a ( x, ξ ) − (cid:88) ≤ k ≤ (cid:104) Re ξ (cid:105) C h h k a k ( x, ξ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C exp (cid:18) − (cid:104) Re ξ (cid:105) Ch (cid:19) . (1.21) .1. GEVREY SPACES OF FUNCTIONS AND SYMBOLS 33 We will prove in § Lemma . Let m ∈ R . Let a ∈ F S ,m (cid:0) T ∗ U (cid:1) and let Ω be a relatively compactopen subset of U . Then a admits a realization a ∈ S ,m (cid:0) T ∗ Ω (cid:1) on Ω . Remark . Let us explain why we do not introduce formal G s symbols. Thenatural notion of such a symbol would be a formal sum (cid:88) h k a k , with an estimate of the form (cid:12)(cid:12)(cid:12) ∂ αx ∂ βξ a k (cid:12)(cid:12)(cid:12) ≤ CR | α | + | β | ( α + β + 2 k )! s (cid:104) ξ (cid:105) m −| β | . To build an actual symbol from a formal symbol, the only procedure we have is theso-called sommation au plus petit terme . The problem is that the result is a symbolthat is well-defined modulo an error in the class O G sx,ξ (cid:32) exp (cid:32) − (cid:18) (cid:104) ξ (cid:105) Ch (cid:19) s − (cid:33)(cid:33) . While it has the right regularity , it does not decay fast enough to produce G s smoothing remainder. In the class of G sx G ξ symbols, we could consider formalsymbols, and this is what is done in [ BDMK67, Zan85, Rod93 ].We will also need Gevrey symbols defined on manifolds. There is a naturaldefinition that is valid for any s ≥
1: we ask for Definition 1.1 to be satisfied inany charts. This definition is not empty. Indeed, it follows from Fa`a di Bruno’sformula, that the class of symbols presented in Definition 1.1 is stable by G s changeof coordinates. Definition . Let M be a G s manifold and m ∈ R . We say that a : T ∗ M → C belongs to the class of symbol S s,m (cid:0) T ∗ M (cid:1) if for every G s chart ( U, κ ) for M thesymbol a κ ( x, ξ ) := a (cid:16) κ − ( x ) , T d κ − ( x ) κ · ξ (cid:17) belongs to S s,m (cid:0) T ∗ ( κ ( U )) (cid:1) . Similarly, we say that a = (cid:80) k ≥ h k a k is a formalanalytic symbol in F S ,m (cid:0) T ∗ M (cid:1) if, for every analytic chart ( U, κ ) for M , the formalsymbol (cid:80) k ≥ h k a κk belongs to F S ,m (cid:0) T ∗ ( κ ( U )) (cid:1) . Remark . As in the Euclidean case, we extend these definitions in thenatural way to symbols defined on products of the form (cid:0) T ∗ M (cid:1) N × M N . Moreover,if M is compact and s >
1, we see by taking a partition of unity that if a is in S s,m (cid:0) T ∗ M (cid:1) then it admits an almost analytic extension to (cid:0) T ∗ M (cid:1) (cid:15) (for some (cid:15) > s = 1, we do not need to use a partition of unity (bythe principle of analytic continuation) to prove that a admits a holomorphic extensionto some (cid:0) T ∗ M (cid:1) (cid:15) with growth at most polynomial. The following characterizationof analytic symbol follows. Lemma . Let ( M, g ) be a compact real-analytic Riemannian manifold and m ∈ R . Then a : T ∗ M → C belongs to S ,m (cid:0) T ∗ M (cid:1) if and only if there are C, (cid:15) > such that a admits a holomorphic extension to (cid:0) T ∗ M (cid:1) (cid:15) that satisfies for every α ∈ (cid:0) T ∗ M (cid:1) (cid:15) | a ( α ) | ≤ C (cid:104)| α |(cid:105) m . Similarly, a = (cid:80) k ≥ h k a k belongs to F S ,m (cid:0) T ∗ M (cid:1) if and only if there are constants C, R, (cid:15) > such that the a k ’s admit holomorphic extensions to (cid:0) T ∗ M (cid:1) (cid:15) that satisfyfor every k ∈ N and α ∈ (cid:0) T ∗ M (cid:1) (cid:15) : | a k ( α ) | ≤ CR k k ! (cid:104)| α |(cid:105) m − k . Remark . As in Definition 1.4, the notion of realization for symbols onmanifolds is deduced immediately from the notion on Euclidean spaces by takingcharts. In the case of analytic symbols on a compact manifold, we may use Lemma1.8 to reformulate this notion: a ∈ S ,m (cid:0) T ∗ M (cid:1) is a realization for (cid:80) k ≥ h k a k ifand only if for every C large enough, there are constants C, (cid:15) > α ∈ (cid:0) T ∗ M (cid:1) (cid:15) we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a ( α ) − (cid:88) ≤ k ≤ (cid:104)| α |(cid:105) C h h k a k ( α ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C exp (cid:18) − (cid:104)| α |(cid:105) Ch (cid:19) . As stated below, realizations of formal analytic symbols on manifold do exist. Wewill indeed use H¨ormander’s solution to the ¯ ∂ -equation to prove the following lemma(the proof is delayed to § Lemma . Let M be a compact G manifold and m ∈ R . Let a = (cid:80) k ≥ h k a k be a formal analytic symbol in F S ,m (cid:0) T ∗ M (cid:1) . Then a admits a realization on M . H¨ormander’s problem on Grauert tube and analytic approximations.
Inorder to define the kernel of the FBI transform T and to find realizations of formalanalytic symbols, we need a tool to construct globally defined real analytic functions.This will be performed using the version of H¨ormander’s estimates [ H¨or65, H¨or66 ]exposed in [
Dem12 ]. Notice that it is common to apply H¨ormander’s solution tothe ¯ ∂ equation to solve such problems, see for instance the appendix of [ Sj¨o96b ] or[
Sj¨o82 , Proposition 1.2]. We will prove the following lemma.
Lemma . Let ( M, g ) be a compact real analytic Riemannian manifold. Let (cid:15) > be small enough. Let f be a (0 , -form on (cid:0) T ∗ M (cid:1) (cid:15) (that may depend on theimplicit parameter h ) with C ∞ coefficients and such that ¯ ∂f = 0 . Assume that thereis a constant C > such that for every α ∈ (cid:0) T ∗ M (cid:1) (cid:15) we have | f ( α ) | ≤ C exp (cid:18) − (cid:104)| α |(cid:105) Ch (cid:19) . (1.22) Then, there are (cid:15) ∈ ]0 , (cid:15) [ and a constant C (cid:48) > such that, for h small enough, thereis a C ∞ function r on (cid:0) T ∗ M (cid:1) (cid:15) such that ¯ ∂r = f and r ( α ) = O (cid:18) exp (cid:18) − (cid:104)| α |(cid:105) C (cid:48) h (cid:19)(cid:19) on ( T ∗ M ) (cid:15) . Moreover, the same result holds with the natural modifications if wereplace (cid:0) T ∗ M (cid:1) (cid:15) by a manifold of the form (cid:0) T ∗ M (cid:1) N (cid:15) × ( M ) N (cid:15) . Remark . Most of the time, we will apply Lemma 1.10 with f = − ¯ ∂F where F is a C ∞ function on (cid:0) T ∗ M (cid:1) (cid:15) . Then, if we define (cid:101) F = F + r on (cid:0) T ∗ M (cid:1) (cid:15) ,we see that (cid:101) F is holomorphic and satisfies (cid:101) F ( α ) = F ( α ) + O (cid:18) exp (cid:18) − (cid:104)| α |(cid:105) C (cid:48) h (cid:19)(cid:19) on (cid:0) T ∗ M (cid:1) (cid:15) . This is the method that we will use to construct a FBI transform withanalytic kernel for instance (see Lemma 2.3). .1. GEVREY SPACES OF FUNCTIONS AND SYMBOLS 35 Remark . In order to make sense of the hypothesis (1.22) in Lemma 1.10,we need to specify which metric we use to measure the (0 , f . To do so,recall from § g KN , thereis a non-negative plurisubharmonic function ρ KN on the complexification of T ∗ M .The (1 , ω KN = i∂ ¯ ∂ρ defines a K¨ahler metric on (cid:0) T ∗ M (cid:1) (cid:15) that coincideswith g KN on T ∗ M . Then ω KN induces a hermitian metric on T (cid:0) T ∗ M (cid:1) (cid:15) , hence on T ∗ (cid:0) T ∗ M (cid:1) (cid:15) and following on T ∗ (cid:0) T ∗ M (cid:1) (cid:15) ⊗ C . This is the metric that appears in(1.22).We will also need the following version of Lemma 1.10 on R n , whose proofis slightly easier (one may rely directly on the original version of H¨ormander’sargument [ H¨or65 ]) and will consequently be omitted.
Lemma . Let U be an open subset of R n . Let Ω be a relatively compactopen subset of U . Let (cid:15) > be small enough. Let f be a (0 , -form on (cid:0) T ∗ U (cid:1) (cid:15) (thatmay depend on the implicit parameter h ) with C ∞ coefficients and such that ¯ ∂f = 0 .Assume that there is a constant C > such that for every α ∈ (cid:0) T ∗ U (cid:1) (cid:15) we have | f ( α ) | ≤ C exp (cid:18) − (cid:104)| α |(cid:105) Ch (cid:19) . Then, there are (cid:15) ∈ ]0 , (cid:15) [ and a constant C (cid:48) > such that, for h small enough, thereis a C ∞ function r on (cid:0) T ∗ Ω (cid:1) (cid:15) such that ¯ ∂r = f and r ( α ) = O (cid:18) exp (cid:18) − (cid:104)| α |(cid:105) C (cid:48) h (cid:19)(cid:19) on ( T ∗ Ω) (cid:15) . Moreover, the same result holds with the natural modifications if wereplace (cid:0) T ∗ U (cid:1) (cid:15) by a manifold of the form (cid:0) T ∗ U (cid:1) N (cid:15) × ( U ) N (cid:15) . Let us mention that there are no particular difficulties to deal with a manifold ofthe form (cid:0) T ∗ M (cid:1) N (cid:15) × ( M ) N (cid:15) rather than with (cid:0) T ∗ M (cid:1) (cid:15) in Lemma 1.10 and thus wewill only write the proof in the latter case. The proof of Lemma 1.10 is based on anapplication of the following result, Theorem 3, from [ Dem12 ]. This is just [
Dem12 ,Theorem VIII.6.5] applied to the holomorphic line bundle ( X × C ) ⊗ Λ n T X where X × C denotes the trivial line bundle on X (see the remark below Theorem VIII.6.5in [ Dem12 ]). Notice that we apply here a result which is true in a very generalcontext (namely weakly pseudo-convex K¨ahler manifold) to a much more favorablecase (Grauert tubes are strictly pseudo-convex). In particular, one could probablyretrieve “by hand” Lemma 1.10 by adapting H¨ormander’s argument [
H¨or65, H¨or66 ].However, we thought that it was easier to rely on an available powerful technology,since it avoids to rediscuss boundary and non-compactness issues. Moreover, itseemed to us that a more elementary proof of Lemma 1.10 would have been mostlya repetition of some technical estimates that are very well detailed in [
Dem12 ]. Theorem
Dem12 ]) . Let ( X, ω ) be a weakly pseudo-convex K¨ahler manifoldof complex dimension d . Let ϕ be a C ∞ function from X to R . For every x ∈ X denotes by λ ( x ) ≤ · · · ≤ λ d ( x ) the eigenvalues of Ricci( ω ) + i∂ ¯ ∂ϕ and assumethat λ ( x ) > . Then, if f is a form of type (0 , on X , with C ∞ (resp. L )coefficients, that satisfies ¯ ∂f = 0 and (cid:90) X λ + · · · + λ d | f | e − ϕ dVol < + ∞ . (1.23) Then, there is a C ∞ (resp. L ) function r on X such that ¯ ∂r = f and (cid:90) X | r | e − ϕ dVol ≤ (cid:90) X λ + · · · + λ d | f | e − ϕ dVol . (1.24) Remark . We refer to [
Dem12 ] for precise definitions and extensivediscussions of the complex analytic notions that appear in the statement of Theorem3. Let us mention however that dVol denotes the Lebesgue measure associated withthe Hermitian form associated with ω (this measure is identified with the volumeform ω d /d !) and that Ricci( ω ) denotes the Ricci curvature of ω .We want to apply Theorem 3 with X = (cid:0) T ∗ M (cid:1) (cid:15) and ω = ω KN (as defined inRemark 1.16). To do so, we will start by checking that: Lemma . (cid:0)(cid:0) T ∗ M (cid:1) (cid:15) , ω KN (cid:1) is a weakly pseudo-convex K¨ahler manifold. Proof.
We already saw in § (cid:0)(cid:0) T ∗ M (cid:1) (cid:15) , ω KN (cid:1) is K¨ahler. Recallthat a complex manifold is said to be pseudoconvex if it admits a plurisubharmonicexhaustion function ψ . In the case of (cid:0) T ∗ M (cid:1) (cid:15) , we can take for instance ψ : α (cid:55)→ − log (cid:16) (cid:15) − ρ ( α ) (cid:17) + Re g α x ( α ξ , α ξ ) . (1.25)The first term in (1.25) is plurisubharmonic because it is a convex increasing functionof a plurisubharmonic function. The second term is plurisubharmonic because itis pluriharmonic, as the real part of a holomorphic function ( g denotes here theholomorphic extension of the Riemannian metric on M ). To see that ψ is indeedan exhaustion function, notice that both terms in (1.25) are bounded from below.Thus, the first term ensures that the sublevel sets { ψ < c } remain away from theboundary of (cid:0) T ∗ M (cid:1) (cid:15) while the second term ensures that they remain bounded inthe fibers (recall that in (cid:0) T ∗ M (cid:1) (cid:15) the quantity Re g α x ( α ξ , α ξ ) is equivalent to thesize of α ξ ). (cid:3) We need then to construct the function ϕ that appears in Theorem 3. Beforedoing so, recall that if u is a real (1 , V there is a unique hermitian form h u on V that satisfies ∀ ξ, η ∈ V : u ( ξ, η ) = − h u ( ξ, η ) . Moreover,if we choose coordinates ( z , . . . , z d ) on V , the map u (cid:55)→ h u writes i (cid:88) ≤ j,k ≤ n h jk d z j ∧ d¯ z k (cid:55)→ (cid:88) ≤ j,k ≤ n h jk d z j ⊗ d¯ z k . Now, if V is the tangent space of (cid:0) T ∗ M (cid:1) (cid:15) at some point α , we define the eigenvaluesof u as the diagonal elements of the matrix of h u in an orthonormal basis for h ω KN that diagonalizes h u (these are the eigenvalues that appear in the statement ofTheorem 3). We can now construct the weight ϕ that we are going to use. Lemma . Assume that (cid:15) > is small enough. Then there is a strictlyplurisubharmonic function ϕ on (cid:0) T ∗ M (cid:1) (cid:15) with the following properties:(i) there is C > such that for all α ∈ (cid:0) T ∗ M (cid:1) (cid:15) we have − C (cid:104)| α |(cid:105) ≤ ϕ ( α ) ≤ − (cid:104)| α |(cid:105) C ; (ii) there is C > such that for all α ∈ (cid:0) T ∗ M (cid:1) (cid:15) we have h i∂ ¯ ∂ϕ ≥ (cid:104)| α |(cid:105) C h ω KN . Proof.
Choose
A >
0. We will see that if A is large enough and (cid:15) is smallenough then the function ϕ = − (cid:104)| α |(cid:105) (1 − Aρ ) .1. GEVREY SPACES OF FUNCTIONS AND SYMBOLS 37 satisfies the desired properties. To do so, we compute i∂ ¯ ∂ϕ = − i∂ ¯ ∂ ( (cid:104)| α |(cid:105) ) + A ( (cid:104)| α |(cid:105) ω KN + u )(1.26)where u = i∂ρ ∧ ¯ ∂ ( (cid:104)| α |(cid:105) ) + i∂ ( (cid:104)| α |(cid:105) ) ∧ ¯ ∂ρ. In terms of Hermitian forms, (1.26) rewrites h i∂ ¯ ∂ϕ = h − i∂ ¯ ∂ ( (cid:104)| α |(cid:105) ) + A (cid:0) (cid:104)| α |(cid:105) h ω KN + h u (cid:1) . Working in local coordinates, we see that the regularity of g KN (it has boundedderivatives with respect to itself, i.e. it satisfies symbolic estimates) implies that h ω KN is uniformly equivalent to the Kohn–Nirenberg metric on T ∗ C n . For the samereason, ρ is a symbol of order 0. Using that the differential of ρ vanishes on the realand that (cid:104)| α |(cid:105) is a symbol of order 1, we see that h u = O ( (cid:15) (cid:104)| α |(cid:105) ) h ω KN . Hence, if (cid:15) is small enough, we have (cid:104)| α |(cid:105) h ω KN + h u ≥ (cid:104)| α |(cid:105) h ω KN . From the symbolic behavior of (cid:104)| α |(cid:105) , we deduce that h − i∂ ¯ ∂ ( (cid:104)| α |(cid:105) ) = O ( (cid:104)| α |(cid:105) ) h ω KN , and thus (ii) holds if A is large enough (we may even take C arbitrarily small).Finally, if we impose that (cid:15) < A , it is immediate that (i) holds. (cid:3) We are now ready to prove Lemma 1.10.
Proof of Lemma 1.10.
We want to apply Theorem 3 with ϕ replaced by νϕh ,where ν is a small positive constant and ϕ is from Lemma 1.13. To do so, we needto check that the eigenvalues λ ≤ · · · ≤ λ n ofRicci( ω KN ) + iν∂ ¯ ∂ϕh are positive. Using again the regularity of the Kohn–Nirenberg metric, we see that h Ricci( ω KN ) = O (cid:0) h ω KN (cid:1) . Hence, with (ii) in Lemma 1.13, we have for some constant
C > h Ricci( ω KN )+ iν∂ ¯ ∂ϕh ≥ (cid:18) ν (cid:104)| α |(cid:105) Ch − C (cid:19) h ω KN . Hence, if h is small enough (depending on ν ), we have λ ≥ ν (cid:104)| α |(cid:105) Ch > , (1.27)and we may apply Theorem 3 to the (0 , f .Indeed, thanks to (1.22), (i) in Lemma 1.13 and (1.27), we see that for ν and h small enough we have (cid:90) ( T ∗ M ) (cid:15) λ + · · · + λ n | f | e − νϕh dVol ≤ C < + ∞ , (1.28)where C does not depend on h . Hence, according to Theorem 3, there is a C ∞ function r on (cid:0) T ∗ M (cid:1) (cid:15) such that ¯ ∂r = f and (cid:90) ( T ∗ M ) (cid:15) | r | e − νϕh dVol ≤ C. (1.29) It remains to show that r is an O (exp( − (cid:104)| α |(cid:105) /C (cid:48) h )) in ( T ∗ M ) (cid:15) for some (cid:15) ∈ ]0 , (cid:15) [.However, this fact follows easily from (1.28) and (1.29) (recall that ¯ ∂r = f ) byaveraging Bochner–Martinelli’s Formula. (cid:3) Remark . It will be useful to know the dependence of r on f in Lemma 1.10.We will see that the solution to the ¯ ∂ equation in Theorem 3 may be obtained by theapplication of a continuous linear operator. In particular, if f depends smoothly ona parameter in the space of forms that satisfy (1.22) then r has the same dependenceon this parameter (in a space of very rapidly decaying functions). We will need thisfact to check that the measurability assumption in Fubini’s Theorem is satisfied inthe proof of Theorem 5 below.To see that in Theorem 3, the function r may be deduced from f by theapplication of a continuous linear operator, denote by H the Hilbert space of forms f of type (0 ,
1) with L coefficients that satisfy (1.23) and ¯ ∂f = 0 (in the sense ofdistributions). Define also the space H of L functions r on X such that ¯ ∂r ∈ H and (cid:90) X | r | e − ϕ dVol < + ∞ . The space H is made a Hilbert space when endowed with the norm (cid:107) r (cid:107) H := (cid:115)(cid:90) X | r | e − ϕ dVol + (cid:13)(cid:13) ¯ ∂r (cid:13)(cid:13) H . Then the operator ¯ ∂ is clearly a bounded operator between H and H . Denote by p the orthogonal projection in H on the closed subspace (cid:0) ker ¯ ∂ (cid:1) ⊥ . Now, if f ∈ H ,we know by Theorem 3 that there is an r ∈ H such that ¯ ∂r = f with the estimates(1.24). Then p ( r ) is the unique element of (cid:0) ker ¯ ∂ (cid:1) ⊥ such that ¯ ∂ ( p ( r )) = f , andconsequently we may define a linear operator L from H to H by Lf = p ( r ). Forevery f ∈ H , we have ¯ ∂Lf = r by definition. Moreover we have (cid:90) X | Lf | e − ϕ dVol = (cid:107) Lf (cid:107) H − (cid:107) f (cid:107) H ≤ (cid:107) r (cid:107) H − (cid:107) f (cid:107) H = (cid:90) X | r | e − ϕ dVol ≤ (cid:107) f (cid:107) H . Hence, Lf also satisfies (1.24). Finally, when r has C ∞ coefficients then so does Lf by ellipticity of the ¯ ∂ operator.We are now ready to prove Lemma 1.7. The proof of Lemma 1.9 is very similar,applying Lemma 1.10 instead of Lemma 1.11, and we will omit it. Proof of Lemma 1.7.
Let us choose a relatively compact open subset Ω (cid:48) of U such that Ω ⊆ Ω (cid:48) . Let also χ : R → [0 ,
1] be a C ∞ function such that χ ( t ) = 0 if t ≤ and χ ( t ) = 1 if t ≥
1. Then define for k ≥ χ k ( ξ ) = χ (cid:18) (cid:104)| ξ |(cid:105) C kh (cid:19) for ξ ∈ C n . Here C is a large constant to be fixed later. We also define χ to beidentically equal to 1. Then for ( x, ξ ) ∈ (cid:0) T ∗ Ω (cid:48) (cid:1) (cid:15) we define b ( x, ξ ) = (cid:88) k ≥ χ k ( ξ ) h k a k ( x, ξ ) . .2. GEVREY OSCILLATORY INTEGRALS 39 Notice that if ( x, ξ ) ∈ (cid:0) T ∗ Ω (cid:48) (cid:1) we have for any C ≥ C (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) b ( x, ξ ) − (cid:88) ≤ k ≤ (cid:104)| ξ |(cid:105) C h h k a k ( x, ξ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:88) k> (cid:104)| ξ |(cid:105) C h | χ k ( ξ ) | h k | a k ( x, ξ ) | . (1.30)By assumption there are constants C, R > h k | a k ( x, ξ ) | ≤ CR k k k h k (cid:104)| ξ |(cid:105) m − k . Notice then that if χ k ( ξ ) (cid:54) = 0 then kh (cid:104)| ξ |(cid:105) − < /C and consequently h k | a k ( x, ξ ) | ≤ C (cid:104)| ξ |(cid:105) m exp (cid:18) k ln (cid:18) RC (cid:19)(cid:19) . By taking C large enough, we ensure that ln (2 R/C ) < (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) b ( x, ξ ) − (cid:88) ≤ k ≤ (cid:104)| ξ |(cid:105) C h h k a k ( x, ξ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:104)| ξ |(cid:105) m (cid:88) k> (cid:104)| ξ |(cid:105) C h exp (cid:18) k ln (cid:18) RC (cid:19)(cid:19) ≤ C (cid:104)| ξ |(cid:105) m exp (cid:16) ln (cid:16) RC (cid:17) (cid:104)| ξ |(cid:105) C h (cid:17) − exp (cid:16) ln (cid:16) RC (cid:17)(cid:17) . Hence, b satisfies the bound (1.21), but b is not analytic in ξ . We will correct thisusing H¨ormander’s solution to the ¯ ∂ equation. Notice indeed that¯ ∂b ( x, ξ ) = (cid:88) k ≥ (cid:104)| ξ |(cid:105) C h h k a k ( x, ξ ) ¯ ∂χ k ( ξ ) . The same computation than above shows then that, provided that C is largeenough, ¯ ∂b ( x, ξ ) is a O (exp( − (cid:104)| ξ |(cid:105) /Ch ). Thus, we end the proof by applyingLemma 1.11. (cid:3) Behind most results of this exposition, lies a careful analysis of oscillatoryintegrals with parameter, of the form(1.31) ( x, h ) (cid:55)→ (cid:90) Ω e ih Φ( x,y ) a ( x, y )d y. We will either want to prove that they are small, in some functional space, or givea precise expansion as h → x tends to ∞ in some sense). Suchestimates will fall in the usual category of non-stationary and stationary phaseestimates. We expect the reader to be accustomed with these techniques, at least inthe case that Φ is a real valued function, and the integrands are C ∞ . However, sincewe will be working in Gevrey regularity, there are some particularities, and tricksthat we gathered here. We hope that this can help in understanding the details ofthe proofs.To study oscillatory integrals, it is usual to use many integration by parts.This can be a useful technique in the Gevrey setting. However, it imposes tocount derivatives precisely, and introduce some formal norms. It is the approach of[ BDMK67, Gra87, LL97 ], and many others. On the other hand, in the analyticcase, it is often more convenient to use Cauchy estimates. Instead of having to find C k estimates for real values of the parameter, only L ∞ estimates are necessary, butfor values of the parameter in a complex neighbourhood of the reals. Using almost analytic extensions, one can mimic this tactic in the Gevrey case.It is the point of view that we have adopted as much as possible throughout thepaper. Naturally, this can also be used in the C ∞ case, and we leave this to thereader’s imagination. Integrations by parts still appear, but they only serve onepurpose: showing that some non-absolutely converging oscillatory integrals are welldefined. The assumptions on the functions Φ, a , andthe set Ω in (1.31) will depend on the context. In the general case, Im Φ will beallowed to be slightly negative. We will have to distinguish between the stationary and non-stationary regions for Φ. We deal with the latter case in this section. Letus start with the easy case: the phase is non-stationary of the whole of U × Ω. Proposition . Let s ≥ , and U × V be an open subset of R n × R k . Let Φ : U × V → C be a G s function and Ω be a relatively compact open subset of V .For λ ≥ and a ∈ G s ( U × V ) let I λ ( a ) : x (cid:55)→ (cid:90) Ω e iλ Φ( x,y ) a ( x, y )d y. (1.32) We make the following assumptions:(i) the imaginary part of Φ is non-negative on U × Ω ;(ii) there exists (cid:15) > such that Im Φ( x, y ) > (cid:15) for x ∈ U and y ∈ ∂ Ω ;(iii) the differential d y Φ does not vanish on U × V .Let K be a compact subset of U . Denote respectively by K and K a compactneighbourhood of K in U and of Ω in V . Then, there exists C > such that, forevery R ≥ , there is a constant C R > such that for all λ ≥ and a ∈ G s ( U ) wehave (cid:107) I λ ( a ) (cid:107) s,CR,K ≤ C R (cid:107) a (cid:107) s,R,K × K exp (cid:32) − (cid:18) λCR (cid:19) s (cid:33) . (1.33)We will give a proof of Proposition 1.1 based on an application of the Bochner–Martinelli Trick 1.4. Thus, we will give a bound on an almost analytic extension of I λ ( a ). However, when a and Φ are analytic in x then I λ has a natural holomorphicextension that satisfies the same bound as the almost analytic extension of I λ ( a ). Itwill be useful when studying FBI transforms to have this bound isolated, and this isthe point of the following proposition. Proposition . Under the hypothesis of Proposition 1.1, assume in additionthat Φ is analytic and that a is of the form a ( x, y ) = b ( x, y ) u ( y ) with b ∈ G ( U × V ) fixed and u ∈ G s ( V ) (we write then I λ ( a ) = I λ ( u ) ).Let K be a compact subset of U and K (cid:48) be a compact neighbourhood of Ω in V .Then, there exists a complex neighbourhood W of K such that for every u ∈ G s ( U ) ,the function I λ ( u ) has a holomorphic extension to W . Moreover, there is C > such that, for every R ≥ , there is C R > such that, if λ ≥ and z ∈ W are suchthat | Im z | ≤ ( CR ) − /s λ − /s , then for every u ∈ G s ( U ) , we have (1.34) | I λ ( u )( z ) | ≤ C R (cid:107) u (cid:107) s,R,K (cid:48) exp (cid:32) − (cid:18) λCR (cid:19) s (cid:33) . Proof of Proposition 1.1 and 1.2.
We start with the proof of Proposition1.1 and then explain how the same argument also gives Proposition 1.2.Let (cid:101)
Φ denote either the holomorphic extension of Φ (if s = 1) or a G s almostanalytic extension of Φ (if s > s > K and K are fixed). .2. GEVREY OSCILLATORY INTEGRALS 41 Then, we fix R ≥
1. In the following,
C > , K , K but not on R , while C R > , K , K and R . The value of C and C R may changefrom one line to another.Let a ∈ G s ( U × V ) be such that (cid:107) a (cid:107) s,R,K × K < + ∞ . We denote by ˜ a eitherthe holomorphic extension of a if s = 1 (which is defined for x, y in C n at distanceat most C − R − from K × K ) or an almost analytic extension for a given byLemma 1.2 if s > a is supported in K × K ). We can then extend the definition of I λ ( a ) to a smoothfunction on a complex neighbourhood of K in C n just by replacing Φ and a in (1.32)respectively by (cid:101) Φ and (cid:101) a .It will be useful to pick a cutoff χ ∈ C ∞ ( R + ), such that χ equals 1 in [0 , / , + ∞ [. We want to perform a contour deformation in the definition of I λ ( a ) in order to make the imaginary part of the phase positive. Since near theboundary of Ω, the imaginary part of the phase is positive, we will only have toconsider deformations in the complex domain for y ∈ Ω that are away from ∂ Ω. Forsome small (cid:15) > t ∈ R , we define the contourΓ t : ( x, y ) (cid:55)→ y + t (cid:18) − χ (cid:18) d ( y, ∂ Ω) (cid:15) (cid:19)(cid:19) ∇ y Im (cid:101) Φ( x, y ) . Here, the gradient is defined using the identification C n (cid:39) R n .Since the differential of Φ does not vanish on K × K and (cid:101) Φ satisfy the Cauchy–Riemann equations on R n , we see that ∇ y Im (cid:101) Φ does not vanish on K × K , andhence on a complex neighbourhood W × W of K × K . Thus, applying Taylor’sformula, we see that there is t > t ∈ [0 , t ] , x ∈ W and y ∈ K we have Im (cid:101) Φ ( x, Γ t ( x, y )) ≥ tC (cid:18) − χ (cid:18) d ( y, ∂ Ω) (cid:15) (cid:19)(cid:19) + Im (cid:101) Φ( x, y ) . For x ∈ K , the imaginary part of the phase is strictly positive when y is near theboundary of Ω , so that, for x ∈ W , this remains true, and we may assume thatIm (cid:101) Φ( x, y ) > (cid:15)/ d ( y, ∂ Ω) < (cid:15) / x ∈ W . Away from the boundary, wehave Im (cid:101) Φ( x, y ) ≥ − C | Im x | , for some constant C >
0, when x ∈ W and y ∈ K (this is because we assumed that the imaginary part of the phase is non-negative forreals x and y ). Summing up, we have for t ∈ [0 , t ] , x ∈ W and y ∈ Ω,(1.35) Im (cid:101) Φ( x, Γ t ( x, y )) ≥ tC − C | Im x | .∂ Ω ΩΓ t Im ˜Φ > (cid:15)
Im ˜Φ > t/C(cid:15)(cid:15)
Figure 1.
Sketch of the contour deformation.Now, choose t ∈ [0 , t ] and x ∈ W and let H be the map from M t = { x } × Ω × [0 , t ] to C n defined by H ( x, y, t ) = Γ t ( x, y ). If s = 1, we assumein addition that t ≤ ( CR ) − and d ( x, K ) ≤ ( CR ) − in order to ensure that ˜ a ( x, Γ t ( x, y )) is well-defined for every y ∈ Ω and t ∈ [0 , t ]. We apply Stokes’formula to the n -form H ∗ ( e iλ ˜Φ( z ) ˜ a ( z )d z ) to get I λ ( a )( x ) = (cid:90) Ω e iλ (cid:101) Φ ( x, Γ t ( x,y ) )˜ a ( x, Γ t ( x, y )) J Γ t ( y )d y + (cid:90) M t d H ∗ (cid:16) e iλ (cid:101) Φ( x,z ) ˜ a ( x, z )d z (cid:17) , (1.36)where J Γ t denotes the Jacobian of Γ t . We did not make any regularity assumptionon Ω, but this is not a problem when applying Stokes’ Formula, since we only deformaway from the boundary of Ω. Notice that there is a uniform bound on the Jacobian J Γ t for t ∈ [0 , t ] and that the supremum of ˜ a is controlled by C R (cid:107) a (cid:107) s,R,K × K (provided that d ( x, K ) ≤ ( CR ) − in the case s = 1). Consequently, using (1.35), wesee that (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) Ω e iλ (cid:101) Φ ( x, Γ t ( x,y ) )˜ a (cid:0) x, Γ t ( x, y ) (cid:1) J Γ t ( x )d y (cid:12)(cid:12)(cid:12)(cid:12) ≤ C R (cid:107) a (cid:107) s,R,K × K exp (cid:18) λ (cid:18) C | Im x | − t C (cid:19)(cid:19) . (1.37)This only gets better when t increases. However, this will not be the case of thesecond term if s >
1. To bound the second term in (1.36), notice thatd( e iλ ˜Φ( x,z ) ˜ a ( x, z )d z ) = ¯ ∂ z (cid:16) e iλ ˜Φ( x,z ) ˜ a ( x, z ) (cid:17) ∧ d z = iλe iλ ˜Φ( x,z ) ˜ a ( x, z ) ¯ ∂ z (cid:101) Φ( x, z ) ∧ d z + e iλ ˜Φ( x,z ) ¯ ∂ z ˜ a ( x, z ) ∧ d z. Hence, the second term in (1.36) is null if s = 1, since ˜ a is holomorphic. Thus, inthe case s = 1, if x ∈ C n is at distance at most ( AR ) − of K (for some large A > R ), we find taking t = min (cid:16) t , ( CR ) − (cid:17) that, | I λ ( a )( x ) | ≤ C R (cid:107) a (cid:107) ,R,K × K exp (cid:18) λ (cid:18) CAR − CR (cid:19)(cid:19) ≤ C R (cid:107) a (cid:107) ,R,K × K exp (cid:18) − λCR (cid:19) . On the second line, we assumed that A was large enough and changed the valueof C . Proposition 1.1 in the case s = 1 follows, since I λ ( a ) is holomorphic bydifferentiation under the integral.We turn now to the case s >
1. According to Remark 1.7, the integrand in thesecond term in (1.36) is controlled by(1.38) C R (cid:107) a (cid:107) s,R,K × K exp (cid:32) C | Im x | − C | R ( t + | Im x | ) | s − (cid:33) . Here, we recall that t ∈ [0 , t ] for some t > − t/C in (1.35), this term was needed only to bound the first termin (1.36). If we take t = A − R − /s λ /s − and assume that | Im x | ≤ A − R − /s λ s − (for some large A , A > C R (cid:107) a (cid:107) s,R,K × K exp CA − A s − C λ s R s ≤ C R (cid:107) a (cid:107) s,R,K × K exp (cid:32) − (cid:18) λCR (cid:19) s (cid:33) . .2. GEVREY OSCILLATORY INTEGRALS 43 We assumed that A and A were large enough on the second line. Togetherwith (1.36) and (1.37), this proves that for | Im x | < A − λ /s − R − /s (recall that t = A − R − /s λ /s − and assume that A (cid:29) A ),(1.39) | I λ ( a )( x ) | ≤ C R (cid:107) a (cid:107) s,R,K × K exp (cid:32) − (cid:18) λCR (cid:19) s (cid:33) . On the other hand, to estimate ∂ x I λ ( a ), we have to give a bound for (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) Ω e iλ (cid:101) Φ( x,y ) ( ∂ x ˜ a ( x, y ) + iλ ˜ a ( x, y ) ∂ x (cid:101) Φ( x, y ))d y (cid:12)(cid:12)(cid:12)(cid:12) . For this, we do not use a contour shift but a direct L bound. Recalling Remark1.7 and that the imaginary part of (cid:101) Φ( x, y ) is positive when x and y are real, we findthat this integral is bounded by C R (cid:107) a (cid:107) s,R,K × K λ exp (cid:32) Cλ | Im x | − C ( R | Im x | ) s − (cid:33) . But if | Im x | < A − λ /s − R − /s then λ ≤ A − ss − | Im x | − ss − R − s − and thus the estimate above gives if A is large enough (cid:12)(cid:12) ∂I λ ( a ) ( x ) (cid:12)(cid:12) ≤ C R (cid:107) a (cid:107) s,R,K × K exp (cid:32) − C ( R | Im x | ) s − (cid:33) . Proposition 1.1 follows then from the Bochner–Martinelli Trick, recalling (1.39).To prove Proposition 1.2, just notice that in that case, one may take (cid:101)
Φ = Φand ˜ a ( x, y ) = b ( x, y )˜ u ( y ) where ˜ u is an almost analytic extension for u . Then, I λ ( u )has an holomorphic extension by differentiation under the integral and the estimatethat we want to prove is just (1.39) (there is no difference in the proof). (cid:3) Remark . The same result holds if instead of U × Ω ⊂ R n + k , we assumethat U × Ω ⊂ M × N , with M, N two compact Riemannian manifolds. The proof isessentially the same. The main difference is that we have to give a more “geometric”expression for the contour shift. For example, using the integral lines of ∇ y Im (cid:101) Φ, orusing the exponential map.
It is a classical observation that the Sta-tionary Phase method in Gevrey regularity suffers from a loss of regularity. Thiscomes essentially from the fact that in the usual expansion (cid:90) R e − λw f ( w )d w ∼ (cid:114) πλ (cid:88) k ≥ λ ) k k ! f (2 k ) (0) , to gain a power of λ , we have to differentiate f twice . Let us be a bit more precise.We consider f a G s function, defined in a neighbourhood of 0 in R n × R k , and for U a small enough neighbourhood of 0 in R k ,we define g λ : x (cid:55)→ λ k/ π k/ (cid:90) U e − λw f ( x, w )d w. By differentiation under the integral, we find that g λ is uniformly G s as λ → + ∞ .Integrating by parts, we also find (for C, C > g λ ( x ) = (cid:88) ≤ (cid:96) ≤ λ s − C (cid:96) ! (cid:18) ∆ w λ (cid:19) (cid:96) f ( x,
0) + O G s (cid:32) exp (cid:32) − λ s − C (cid:33)(cid:33) . Here, notice that while the remainder is measured in G s , it has the size of anerror term in the G s − category (compare for instance with the bound given inProposition 1.1). In some sense, the reason behind this is that I λ is (if only formally) G s in the small parameter 1 /λ ; we already encountered a similar problem in Remark1.12. The case of more general phases is considerably worse: if instead of w weallow a general phase function Φ( x, w ), we will only recover a G s − estimate in x .One can find a general bound in [ Gra87 , Theorem 3.1], for example.Before we discuss this further, we recall the analytic version of Stationary Phase.This is a well-known tool (see for instance [
Sj¨o82 , Th´eor`eme 2.8 and Remarque2.10]. We will use the following version of it (which is stated on a real contour in[
Sj¨o82 ], the generalization that we state here is straightforward).
Proposition . Let U × V be an open subsetof C n × C k . Let Γ be a k -dimensional compact real submanifold with boundary of V .Let a, Φ : U × V → C be holomorphic functions. Let x ∈ U and assume that(i) the imaginary part of Φ( x , y ) is non-negative for every y ∈ Γ ;(ii) the imaginary part of Φ( x , y ) is strictly positive for every y ∈ ∂ Γ ;(iii) the function y (cid:55)→ Φ( x , y ) has a unique critical point y in Γ , which is inthe interior of Γ and is non-degenerate;(iv) Φ( x , y ) = 0 .Then there is a neighbourhood W of x in U such that for every x ∈ W , thefunction y (cid:55)→ Φ( x, y ) has a unique critical point y c ( x ) near Γ . We denote by Ψ( x ) = Φ( x, y c ( x )) the associated critical value. Then, there is a family ( b m ) m ≥ of holomorphic functions on W such that there are constants C, C such that, forevery m ∈ N , sup x ∈ W | b m ( x ) | ≤ C m m ! and, for every λ ≥ , sup x ∈ W (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e − iλ Ψ( x ) λ k/ (cid:90) Γ e iλ Φ( x,y ) a ( x, y )d y − (cid:88) ≤ m ≤ λ/C λ − m b m ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C exp (cid:18) − λC (cid:19) . Let us come back to the non-analytic case s >
1. In some cases, the loss ofregularity can be mitigated, and we can obtain an estimate as sharp as (1.40). Forexample, we obtain this statement, used in the proof of Lemma 1.21:
Lemma . Let s > . Let U × V be an open subset of R n × R k . Let K = K × K be a compact subset of U × V . Let Φ : U × V → R be a G s functionon U × V . Assume that for every x ∈ U the function y (cid:55)→ Φ( x, y ) has a uniquecritical point y c ( x ) in V that satisfies Φ( x, y c ( x )) = 0 . Let a ∈ E s,R ( K ) for some R > . Define then for λ ≥ the function I λ ( a ) : x (cid:55)→ λ k (cid:90) V e iλ Φ( x,y ) a ( x, y )d y (1.41) on U . Then the function I λ ( a ) is G s uniformly in λ . More precisely, there is R > ,a family ( b m ) m ≥ of functions in E s,R ( K ) and constants C, C > such that, for .2. GEVREY OSCILLATORY INTEGRALS 45 every m ∈ N , (cid:107) b m (cid:107) s,R ,K ≤ C m m ! s − and, for every λ ≥ , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) I λ ( a ) − (cid:88) ≤ m ≤ λ s − C λ − m b m (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) s,R ,K ≤ C exp (cid:32) − λ s − C (cid:33) . (1.42) Proof.
We can work locally in x . By the Morse Lemma, there is a changeof coordinates ρ x near y c ( x ) that changes the phase in Φ( x, y ) = y − y − for somedecomposition y = ( y + , y − ) of R k . Moreover, since the Implicit Function Theoremis true in the Gevrey category so is the Morse Lemma and hence ρ x is G s (with a G s dependence on x ). Since y c ( x ) is the only critical point of y (cid:55)→ Φ( x, y ), we mayintroduce a cut off function to split the integral I λ ( a )( x ) into an integral over y ’sthat are close to y c ( x ) and y ’s that are away from y c ( x ). The latter is negligiblethanks to Proposition 1.1. Hence, we may assume that a ( x, y ) vanishes when y istoo far from y c ( x ), in particular if it does not belong to the domain of definition of ρ x . We can then use ρ x as a change of variable in I λ ( a )( x ). We just reduced to thecase where Φ( x, y ) = Φ( y ) = y − y − .From now, on the proof is just the Gevrey version of the argument from theproof of [ Zwo12 , Theorem 3.13] in the C ∞ version. We introduce the new parameter h = λ − and take Fourier Transform in y in order to write (for some constant c (cid:54) = 0) I λ ( a )( x ) = f ( a, h )( x ) = c (cid:90) R k e ih ξ − ξ − F y a ( x, ξ )d ξ. In this form, it is easy to prove that the function h → f ( a, h ) is a smooth functionfrom ]0 ,
1] to E s,R ( K ) (we may differentiate under the integral in x ). To obtain amore precise statement, we introduce the differential operator P = − i (∆ y + − ∆ y − )and notice that d m d h m f ( a, h ) = f ( P m a, h ) . However, reasoning as in Remark 1.7, we see that for some R >
0, there is aconstant
C > m ∈ N , we have (cid:107) P m a (cid:107) s,R ,K ≤ C m (2 m !) s . It follows that h → f ( a, h ) is in fact a G s function from ]0 ,
1] to E s,R ( K ). Sincethe estimates on the derivatives are uniform, it extends to a G s function on [0 , b m ( x ) = P m a ( x, m ! . (cid:3) Another situation in which we are able to get a G s remainder in the stationaryphase method is when the phase Φ( x, y ) in (1.41) is purely imaginary and satisfiessome sort of positivity condition. More generally, if Φ is a real analytic phase, withIm Φ ≥ x is itself a non-degenerate critical point ofthe critical value Φ c ( x ), then one gets a G s remainder; we will not need this result.In the key Lemma 2.10 in the next chapter, a somewhat different phenomenonwill take place. The amplitude will be analytic in x , and G s in w , so that we can consider the holomorphic extension of I λ ( a ) in the x variable. We will be able tocontrol the stationary integral in a region in the complex bigger than a “ G s − ”region {| Im x | (cid:28) λ / (2 s − − } , but smaller than a “ G s ” region {| Im x | (cid:28) λ /s − } . Remark . All of the avatars of the Stationary Phase Method that we useare proved using the same global strategy: we reduce to the case of a quadraticphase using some version of Morse Lemma. Consequently, the coefficients in theexpansion e − iλ Ψ( x ) λ k (cid:90) e iλ Φ( x,y ) a ( x, y )d y ∼ (cid:88) m ≥ λ − m b m ( x )that appear in Proposition 1.3 or Lemma 1.14, are given by the same expression b m ( x ) = (2 π ) k m ! P m a ( x, y c ( x )) . Here, y c ( x ) denotes the critical point of y (cid:55)→ Φ( x, y ), and P m is the differentialoperator acting on the y variable defined using Morse coordinates ρ x , such thatΦ( x, y ) = Φ( x, y c ( x )) + i ρ x ( y ) , (1.43)by the expression P m u = (cid:20)(cid:18) ∆2 (cid:19) m (cid:16) u ◦ ρ − x Jρ − x (cid:17)(cid:21) ◦ ρ x . (1.44)Here, Jρ − x denotes the Jacobian of ρ − x . Concerning (1.43), let us point out that if z = ( z , . . . , z k ) ∈ C k , we write z = (cid:104) z, z (cid:105) = (cid:80) nj =1 z j . The Laplacian in (1.44) isdefined on C k using holomorphic derivatives: ∆ = (cid:80) nj =1 ∂ ∂z j (we retrieve the usualLaplacian when restricting this operator to R k ).One can check that these are the same operators that appear in the proof ofLemma 1.14. Of particular importance is the first term in the expansion b ( x ) = (2 π ) k a ( x, y c ( x )) (cid:114) det (cid:16) − i d y Φ( x, y c ( x )) (cid:17) . The choice of the determination of the square root here depends on the choiceof orientation on the contour of integration. We will always arrange so that theargument of the square root has non-negative real part and choose the correspondingholomorphic determination of the square root (see [
Sj¨o82 ] for details).Let us now state the version of the Morse Lemma that will be needed in theStationary Phase Argument in the proof of Proposition 2.10. The phase there willbe analytic rather than Gevrey, so that we will rely on the following version of theHolomorphic Morse Lemma. (This is also the version of Morse Lemma that is usedin the proof of Proposition 1.3).
Lemma . Let U and V be open subsetsrespectively of C n and C k . Let Φ : U × V → C be a holomorphic function. Assumethat there is ( x , y ) ∈ U × V such that y (cid:55)→ Φ( x , y ) has a non-degenerate criticalpoint at y . Then there exist open neighbourhoods U , V and W of x , y and respectively in C n , C k and C k and holomorphic maps ρ : U × V (cid:55)→ C n and y c : U → V such that(i) for every x ∈ U , the point y c ( x ) is the unique critical point of y (cid:55)→ Φ( x, y ) in V ; .2. GEVREY OSCILLATORY INTEGRALS 47 (ii) for every x ∈ U , the map V (cid:51) y (cid:55)→ ρ ( x, y ) is a diffeomorphism onto itsimage, which contains W ;(iii) for every x ∈ U and y ∈ V , we have Φ( x, y ) = Φ( x, y c ( x )) + i ρ ( x, y ) . The proof of this classical result may be found for instance in [
Sj¨o82 ] (seeLemme 2.7 there and the remark just after it). In several proofs, in order to applya method of steepest descent, it will be important to control the imaginary partof the critical value of the phase. The tool for this is the so-called “fundamental”lemma, presented in an elaborated version in [
Sj¨o82 , Lemme 3.1]. We give here amore elementary version.
Lemma . Let Φ be a phase as in Lemma 1.15. Assume in addition that x and y are real points, that Im Φ( x , y ) = 0 and that Im Φ( x, y ) ≥ for real x, y .Then for x real close to x , Im Φ( x, y c ( x )) ≥ . If additionally, we assume that
Im d Φ( x , y ) is non-degenerate, then for someconstant C > , and x real close to x , Im Φ( x, y c ( x )) ≥ C ( x − x ) . Proof.
Let ρ be as in Lemma 1.15 and write ρ x = ρ ( x, · ) for x ∈ U . We startby decomposing the Jacobian matrix of ρ x at y into real and imaginary part Dρ x ( y ) = A + iB. According to the definition of Morse coordinates, we obtain i t ( A + iB )( A + iB ) = D y,y Φ( x , y ) . Identifying the imaginary part, it follows that t AA − t BB = Im D y,y Φ( x , y ) ≥ A + iB has to be invertible, this implies that A is actually invertible. Wededuce that, up to taking V smaller, ρ x ( V ∩ R k ) is a graph over the reals, andthis remains true by perturbation for x close to x . We can thus write ρ x ( V ∩ R k ) = { w + iF x ( w ) | w ∈ Re ρ x ( V ∩ R k ) } . Here, F x is a real analytic map in all its variables. In particular, if x is real, at thepoint y = ρ − x ( iF x (0)), we haveΦ( x, y ) = Φ( x, y c ( x )) − i | F x (0) | , so that Im Φ( x, y c ( x )) ≥ | F x (0) | / ≥ x, y ) ≥ Φ( x , y ) is non-degenerate. SinceIm Φ( x, y ) ≥ x, y , the quantity Im Φ( x, y c ( x )) has to be critical at x = x .Hence it suffices for our purpose to show that the imaginary part of the Hessian ofthe critical value is invertible at x = x . For this, it is easier to directly work withΦ. According to the Inverse function theorem, Dy c = − ( D y,y Φ) − D y,x Φ . Here, our convention regarding matrices of second differential is that the lines of D y,x Φ have the size of x and its columns the size of y . It comes that ∇ x Φ c = ∇ x Ψ( x, y c ( x )), and D Φ c = D x,x Φ( x, y c ( x )) − D x,y Φ( D y,y Φ) − D y,x Φ . This is the Schur complement of D y,y Φ in D Φ. We deduce that D Φ is invertibleif and only if D Φ c is invertible. Since it is the case, with x = x and y = y , theinverse of the Hessian D Φ is given by (cid:32) − (cid:16) D y,y Φ (cid:17) − D y,x Φ 1 (cid:33) (cid:16) D Φ c (cid:17) − (cid:16) D y,y Φ (cid:17) − × (cid:32) − D x,y Φ (cid:16) D y,y Φ (cid:17) − (cid:33) . For a vector u ∈ R n , we deduce (cid:28)(cid:16) D Φ c (cid:17) − u, u (cid:29) = (cid:28)(cid:16) D Φ (cid:17) − (cid:18) u (cid:19) , (cid:18) u (cid:19)(cid:29) . Since Im D Φ >
0, we deduce that Im( (cid:16) D Φ (cid:17) − ) <
0, so that Im (cid:16) D Φ c (cid:17) − < D Φ c >
0. This ends the proof. (cid:3)
So far, we have considered integrals of the form (cid:90) Ω e iλ Φ( x,y ) a ( x, y ) dy, where Ω ⊂ R k is relatively compact and x is assumed to be varying in a compactset. However, sometimes, we will need to replace the parameter x by a parameter( x, ξ ) where x varies in a compact set but ξ ∈ R n . In that case, the phase Φ and theamplitude a will have symbolic behavior with respect to ξ and the large parameter λ will be replaced by (cid:104) ξ (cid:105) /h where h > Proposition . Let s ≥ , m ∈ R and U × V be an open subset of R n × R k .Let Φ be an element of S s, (cid:0) T ∗ U × V (cid:1) and Ω be a relatively compact open subsetof V . Let a ∈ S s,m (cid:0) T ∗ U × V (cid:1) and, for < h ≤ , define I h ( a ) : ( x, ξ ) (cid:55)→ (cid:90) Ω e ih Φ( x,ξ,y ) a ( x, ξ, y )d y. (1.45) We make the following assumptions:(i)
Im Φ ≥ on T ∗ U × Ω ;(ii) there exists (cid:15) > such that Im Φ( x, ξ, y ) > (cid:15) (cid:104) ξ (cid:105) for ( x, ξ ) ∈ T ∗ U and y ∈ ∂ Ω ;(iii) there is a C > such that for every ( x, ξ, y ) ∈ T ∗ U × V such that | ξ | ≥ C we have (cid:12)(cid:12) d y Φ( x, ξ, y ) (cid:12)(cid:12) ≥ (cid:104) ξ (cid:105) C .
Let K be a compact subset of U . Then, there are constants C, R > such that, forevery ( x, ξ ) ∈ K × R n with | ξ | ≥ C and every < h ≤ , we have (1.46) sup α,β ∈ N n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:104) ξ (cid:105) | β | ∂ αx ∂ βξ I h ( a )( x, ξ ) R | α | + | β | α ! s β ! s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C exp (cid:32) − (cid:18) (cid:104) ξ (cid:105) Ch (cid:19) s (cid:33) . Proposition . Let s ≥ , m ∈ R and U × V be an open subset of R n × R k .Let Φ be an element of S , (cid:0) T ∗ U × V (cid:1) , a be an element of S ,m (cid:0) T ∗ U × V (cid:1) and .2. GEVREY OSCILLATORY INTEGRALS 49 Ω be a relatively compact open subset of V . Then, for u ∈ G s ( V ) and < h ≤ ,define I h ( u ) : ( x, ξ ) (cid:55)→ (cid:90) Ω e ih Φ( x,ξ,y ) a ( x, ξ, y ) u ( y )d y. (1.47) We make the following assumptions:(i)
Im Φ ≥ on T ∗ U × Ω ;(ii) there exists (cid:15) > such that Im Φ( x, ξ, y ) > (cid:15) (cid:104) ξ (cid:105) for ( x, ξ ) ∈ T ∗ U and y ∈ ∂ Ω ;(iii) there is a C > such that for every ( x, ξ, y ) ∈ T ∗ U × V such that | ξ | ≥ C we have (cid:12)(cid:12) d y Φ( x, ξ, y ) (cid:12)(cid:12) ≥ (cid:104) ξ (cid:105) C .
Let K be a compact subset of U and K (cid:48) be a compact neighbourhood of Ω in V .Then, there is an (cid:15) > such that for every u ∈ G s ( U ) , the function I λ ( u ) has aholomorphic extension to (cid:0) T ∗ K (cid:1) (cid:15) . Moreover, there is C > such that, for every R ≥ , there is C R > , such that if < h ≤ and ( x, ξ ) ∈ (cid:0) T ∗ K (cid:1) (cid:15) are such that | ξ | ≥ C and | Im x | + | Im ξ |(cid:104) Re ξ (cid:105) ≤ CR ) s (cid:18) h (cid:104) Re ξ (cid:105) (cid:19) − s and u ∈ G s ( V ) then | I h ( u )( x, ξ ) | ≤ C R (cid:107) u (cid:107) s,R,K (cid:48) exp (cid:32) − (cid:18) (cid:104) Re ξ (cid:105) CRh (cid:19) s (cid:33) . Proofs of Proposition 1.4 and 1.5 may be deduced easily from the proofs ofProposition 1.1 and 1.2, replacing the estimate from Remark 1.7 by the bound fromRemark 1.10 on the ¯ ∂ of almost analytic extension of Gevrey symbols.However, there is also a standard rescaling argument that allows to deducePropositions 1.4 and 1.5 from Propositions 1.1 and 1.2. It also applies to stationaryphase estimates. Let us recall this argument, for instance in the case of Proposition1.5 . To deal with | ξ | ∼ M (cid:29)
1, we want to use the rescaling η = M − ξ and thusdefine the phase and amplitudeΦ M ( x, η, y ) = Φ ( x, M η, y ) M and a M ( x, η, y ) = M − m a ( x, M η, y )that satisfy uniform Gevrey estimates when ( x, y ) is in a compact subset of U × V and C − ≤ | η | ≤ C for C > I h ( a ) ( x, ξ ) = M m (cid:90) Ω e iλ Φ M ( x,η,y ) a M ( x, η, y ) d y, (1.48)where λ = h − M and η = M − ξ . Notice then that when | ξ | ∼ M we have λ ∼ h − (cid:104) ξ (cid:105) and the point η remains in the domain C − ≤ | η | ≤ C where we haveuniform Gevrey estimates on a M and Φ M . Hence, the integral in the right handside of (1.48) falls into the domain of application of Proposition 1.1, and (1.46)is deduced from (1.33) (taking into account the scaling ξ = M η and neglectingthe factor M m in (1.48) whose growth is annihilated by the decay of the stretchedexponential).The same rescaling argument also gives a symbolic version of Proposition 1.3. Proposition . Let U × V × W be an open subset of R n × R k × R d . Let r ∈ R . Let a ∈ S ,r ( T ∗ U × V × W ) and Φ ∈ S , ( T ∗ U × V × W ) . Let Γ be a d -dimensional compact real submanifold with boundary of C d . Assume that Γ is contained in a small enough complex neighbourhood of W so that a and Φ are well-defined (and satisfy symbolic estimates) on a complex neighbourhood of T ∗ U × V × Γ .Let ( x , z ) ∈ U × V , y be in the interior of Γ and F ⊆ R n . Assume that(i) the imaginary part of Φ( x , ξ, z , y ) is non-negative for every y ∈ Γ and ξ ∈ F ;(ii) there is (cid:15) > such that the imaginary part of Φ( x , ξ, z , y ) is greater than (cid:15) for every y ∈ ∂ Γ and ξ ∈ F ;(iii) for every ξ ∈ F , the function y (cid:55)→ Φ( x , ξ, z , y ) has a unique critical pointin Γ , which is y and is non-degenerate, with symbolic estimates in ξ ;(iv) for every ξ ∈ F we have Φ( x , ξ, z , y ) = 0 .Then there is a neighbourhood U × V of ( x , z ) in C n + k and η > such that,if we denote by G the set of ξ ∈ C n at distance less than η of F for the Kohn–Nirenberg metric, then, for every ( x, z ) ∈ ( U , V ) and ξ ∈ G , the function y (cid:55)→ Φ( x, ξ, z, y ) has a unique critical point y c ( x, ξ, z ) near Γ . We denote by Ψ( x, ξ, z ) =Φ( x, ξ, z, y c ( x, ξ, z )) the associated critical value. Then, there is a formal analyticsymbol (cid:80) m ≥ h m b m of order r on U × G × V such that for C > large enough,there is a constant C > such that, for every < h ≤ , ( x, z ) ∈ U × V and ξ ∈ G ,we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e − i (cid:104) ξ (cid:105) h Ψ( x,ξ,z ) (cid:18) (cid:104) ξ (cid:105) πh (cid:19) k/ (cid:90) Γ e i (cid:104) ξ (cid:105) h Φ( x,ξ,z,y ) a ( x, ξ, z, y )d y − (cid:88) ≤ m ≤ λ/C λ − m b m ( x, ξ, z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C exp (cid:18) − (cid:104)| ξ |(cid:105) Ch (cid:19) . (1.49) Remark . In most of the applications, we will have G = R n so that (cid:80) m ≥ h m b m is a proper formal analytic symbol on T ∗ U × V in the sense ofDefinition 1.2 (otherwise, see Remark 1.11). Then, we see that the integral e − i (cid:104) ξ (cid:105) h Ψ( x,ξ,z ) (cid:18) (cid:104) ξ (cid:105) πh (cid:19) k/ (cid:90) Γ e i (cid:104) ξ (cid:105) h Φ( x,ξ,z,y ) a ( x, ξ, z, y )d y (1.50)is a realization of (cid:80) m ≥ h m b m in the sense of Definition 1.3. Remark . It will sometimes happen when applying Proposition 1.6 thatthe contour Γ depends on x or ξ so that the integral (1.50) will not necessarily bea holomorphic functions of the corresponding variables. However, Γ will alwaysremain uniformly smooth and the hypothesis of Proposition 1.6 will hold uniformly.Thus, the estimate (1.49) will still hold.In particular, in the case when G = R n , then the integral (1.50) is approximatedwith a precision exp ( − (cid:104)| ξ |(cid:105) /h ) by any realization of the formal symbol (cid:80) m ≥ h m b m .We end this section by stating a rescaled version of Lemma 1.14. Lemma . Let s > , r ∈ R . Let U × V be an open subset of R n × R k .Let K be a compact subset of V . Let Φ : V → R be a G s function on V and a ∈ S s,r (cid:0) T ∗ U × V (cid:1) . Assume that(i) if ( x, ξ ) ∈ T ∗ U and y ∈ V \ K then a ( x, ξ, y ) = 0 ;(ii) the function Φ has a unique critical point y ∈ V , which is non-degenerateand satisfies Φ( y ) = 0 .Then b ( x, ξ ) = (cid:18) (cid:104) ξ (cid:105) (2 πh ) (cid:19) k (cid:90) V e i (cid:104) ξ (cid:105) h Φ( y ) a ( x, ξ, y )d y .2. GEVREY OSCILLATORY INTEGRALS 51 defines a G s symbol b ∈ S s,r (cid:0) T ∗ U (cid:1) given at first order by b ( x, ξ ) = e i π q (cid:12)(cid:12)(cid:12) det (cid:16) d Φ( y ) (cid:17)(cid:12)(cid:12)(cid:12) a ( x, ξ, y ) mod hS s,r − (cid:0) T ∗ U (cid:1) , where q denotes the signature of d Φ( y ) . Remark . We do not give here the most general version possible of Lemma1.17. In particular, we could make Φ depend on x and ξ (assuming that Φ is a G s symbol of order 0).Notice also that we can give a full asymptotic expansion b ∼ (cid:88) m ≥ h m b m , where b m ∈ S s,r − m (cid:0) T ∗ U (cid:1) . Here, the asymptotic expansion holds in quite a strongsense (not as satisfactory as in the case s = 1 though). We will not need the fullstrength of this expansion, and, consequently, let us just mention that, for every N ≥
0, we have b − N (cid:88) m =0 h m b m ∈ h N +1 S s,r − N − (cid:0) T ∗ U (cid:1) . For the expression of the b m ’s we may use either the one that is given in the proofof Lemma 1.14 or in Remark 1.20. Remark . Taking into account the rescaling, we may deduce from Remark1.20 an expression for the coefficients in the expansion e − i (cid:104) ξ (cid:105) h Ψ( x,ξ,z ) (cid:18) (cid:104) ξ (cid:105) πh (cid:19) k (cid:90) e i (cid:104) ξ (cid:105) h Φ( x,ξ,z,y ) a ( x, ξ, z, y )d y ∼ (cid:88) m ≥ h m b m from Proposition 1.6 (or Remark 1.23). The coefficients b m ’s are given by theexpression b m ( x, ξ, z ) = P m a ( x, ξ, z, y c ( x, ξ, z )) m ! . Here, the differential operator P m acts on the y variable and is defined by theformula P m u = (cid:104) ξ (cid:105) − k (cid:20)(cid:18) ∆2 (cid:19) m (cid:16) u ◦ ρ − x,ξ,z Jρ − x,ξ,z (cid:17)(cid:21) ◦ ρ x,ξ,z , where ρ x,ξ,z denotes Morse coordinate such thatΦ( x, ξ, z, y ) = Ψ( x, ξ, z, y c ( x, ξ, z )) + i ρ x,ξ,z ( y ) . For later use, notice that the first term in the expansion is given by b ( x, ξ, z ) = a ( x, ξ, z, y c ( x, ξ, z )) (cid:114) det (cid:16) − i d y Φ( x, ξ, z, y c ( x, ξ )) (cid:17) , the choice of the determination of the square root will be dealt with as we explainedin Remark 1.20. Now that we have introduced classes of G s symbols, we are ready todefine and study the basic properties of G s pseudo-differential operators (or fromnow on, pseudors). Thus, let s ≥ M be a compact G s manifold. Accordingto Remark 1.1, we may endow M with a coherent structure of real-analytic manifoldand choose a real-analytic metric g on M . Then, we endow M with the Lebesguemeasure associated with g , so that we may consider Schwartz kernels of operatorsfrom C ∞ ( M ) to D (cid:48) ( M ).We will work in the semi-classical setting. Hence, throughout the paper, h > Zwo12 ] foran introduction to semi-classical pseudo-differential operators (or from now on, h -pseudors) in the C ∞ category. These pseudors are defined by two properties:pseudo-locality and a fixed local model for their kernels. Definition . Let s ≥
1, and P = ( P h ) h> a family ofoperators from C ∞ ( M ) to D (cid:48) ( M ), with Schwartz kernel K P . We say that P is G s pseudo-local if its kernel is small in G s outside of the diagonal. More precisely, P is G s pseudo-local, if, for every compact subset K of M × M that does not intersectthe diagonal of M × M , there are constants C, R > (cid:107) K P (cid:107) s,R,K ≤ C exp (cid:18) − Ch s (cid:19) . (1.51) Definition . Let m ∈ R , s ≥
1, and P = ( P h ) h> a family of operators from C ∞ ( M ) to D (cid:48) ( M ), with Schwartz kernel K P . Let us assume that:(i) P is G s pseudo-local;(ii) near the diagonal, the kernel of P is given (in any relatively compact opensubset V of a local G s coordinate patch U ) by the oscillatory integral K P ( x, y ) = 1(2 πh ) n (cid:90) R n e i (cid:104) x − y,ξ (cid:105) h p ( x, ξ )d ξ + O G s (cid:18) exp (cid:18) − Ch s (cid:19)(cid:19) , (1.52) where p ∈ S s,m (cid:0) T ∗ V (cid:1) (in the sense of Definition 1.1).Then we say that P is a G s h -pseudor. The set of G s pseudors (resp. of order m )on M will be denoted by G s Ψ ( M ) (resp. G s Ψ m ( M )). Remark . As we explained before, one could define a smaller class ofpseudo by requiring in (ii) that the symbol p is a G sx G ξ symbol as defined in theintroduction. Of course, when s = 1, this is the same class of symbol. Most resultsfrom the current section (Propositions 1.7, 1.8, 1.11 and 1.9) remains true in theclass of G sx G ξ pseudors (and are in fact more standard).Concerning the technical results from the following sections, let us list here themain differences for G sx G ξ pseudors (these results were present in an earlier versionof this paper). In Lemma 1.18, we have the better bound O G s (exp( − /Ch )). InLemma 1.21, we have an efficient asymptotic expansion for the symbol b . TheKuranishi trick Lemma 1.24 is also true also for s > G sx G ξ pseudors (since it only relies on a contour shift in the ξ variable), and there isa weaker version of Proposition 1.12 valid when s >
1. We also expect that theparametrix construction Theorem 4 can be adapted to the case s > G sx G ξ pseudors. The reader interested with the pseudo-differential calculus associatedto G sx G ξ symbols may refer to [ BDMK67, Zan85, Rod93 ].We state now the main properties of G s pseudors. These results will be provedin § § .3. GEVREY PSEUDO-DIFFERENTIAL OPERATORS 53 that G s pseudors are in particular pseudors in the usual (i.e. C ∞ ) sense. To startwith, a G s pseudor sends C ∞ into C ∞ . Hence, it makes sense to compose G s pseudorand without surprise we obtain that: Proposition G s pseudors) . Let P and Q be G s pseudors on M . Then P Q is a G s pseudor. Consequently, G s Ψ is a subalgebra of C ∞ Ψ . This algebra is stable by taking adjoints:
Proposition G s pseudor) . Let A be a G s pseudor on M .Then the L adjoint A ∗ of A is also a G s pseudor. As one could expect, G s pseudors have better mapping properties than C ∞ pseudors. Proposition G s pseudors) . Let P be a G s pseudoron M . Then P is continuous from G s ( M ) to itself and extends continuously from U s ( M ) to itself. We will also need to discuss the principal symbol of a G s pseudor. We will usethe following result. The proof in the case s > G s pseudors have been established. The proof in the analytic case will be abit more involved due to the lack of partition of unity. Proposition G s pseudors) . Let A be a G s pseudorof order m ∈ R on M . Then, there is a symbol a ∈ S s,m (cid:0) T ∗ M (cid:1) such that, forevery small enough coordinate patch U , if p denotes the symbol such that thekernel of A has the form (1.52) on U , then, after change of coordinates, we have p = a | T ∗ U mod hS s,m − .We say that a is a G s principal symbol for A . Here, it is important to notice that being a G s principal symbol for A is aslightly stronger property than being a representative of the principal symbol of A (defined in the C ∞ sense) of regularity G s .Finally, one may wonder if there are any G s pseudors on a given compactmanifold M . It is straightforward to check that differential operators with G s coefficients are indeed G s pseudors, but we can do better than that. Indeed, we canquantize any G s symbol on M . Proposition G s symbols) . There is a linear map Op from ∪ m ∈ R S s,m to the space of continuous operators from C ∞ ( M ) to D (cid:48) ( M ) suchthat, for every p ∈ ∪ m ∈ R S s,m , the operator Op ( p ) is a G s pseudor on M such that p is a principal symbol for P . To close this section, we state some results in the analytic class that will beproved in § G h -pseudor. We say that a G h -pseudor P of order m is (semi-classically) elliptic if there are c, (cid:15) > G principal symbol p ∈ S ,m (cid:0) T ∗ M (cid:1) for P such that for every α ∈ (cid:0) T ∗ M (cid:1) (cid:15) we have | p ( α ) | ≥ c (cid:104)| α |(cid:105) m . (1.53)We say that P is classically elliptic if (1.53) only holds when (cid:104)| α |(cid:105) is large enough.With this definition, we have the following result, whose proof is given in § Theorem . Assume that P is a a semi-classically elliptic G h -pseudor of order m ∈ R on M . Then, for h small enough, there is a G h -pseudor Q of order − m such that P Q = QP = I . Remark . We cannot find precisely this statement in [
BDMK67 ], sincethis reference only deals with pseudors on open sets of Euclidean spaces and discusspolyhomogeneous rather than semi-classical symbolic calculus. However, the proofthat we give of Theorem 4 relies in a fundamental way on Sj¨ostrand’s version [
Sj¨o82 ,Theorem 1.5] of the argument of [
BDMK67 ]. Notice however that the parametrixconstruction from [
BDMK67 ] is valid for any s ≥
1, but in a G sx G ξ class.Finally, in § G h -pseudors. Theorem . Let P be a self-adjoint G h -pseudor, classically elliptic of order m > . Assume that the spectrum of P is contained in [ s, + ∞ [ for some s ∈ R and h small enough. Let (cid:15) > and f be a holomorphic function on U s,(cid:15) = { z ∈ C : Re z > s − (cid:15) and | Im z | < (cid:15) (cid:104) Re z (cid:105)} . Assume in addition that there is N ∈ R and a constant C > such that for every z ∈ U s,(cid:15) we have | f ( z ) | ≤ C (cid:104)| z |(cid:105) N . Then, for h small enough, the operator f ( P ) is a G h -pseudor of order mN .Moreover, if p is a G principal symbol for P , then f ( p ) is a G principal symbolfor f ( P ) . Arguably, microlocal analysis isan application of the study of oscillatory integrals, as exposed in § G s h -pseudors. Hence, let U be an open subsetof R n , let m ∈ R and let a a G s symbol of order m in T ∗ U × U . We want to establishthe basic properties of the distributional kernel K a ( x, y ) = 1(2 πh ) n (cid:90) R n e i (cid:104) x − y,ξ (cid:105) h a ( x, ξ, y )d ξ. (1.54)The integral in (1.54) must be interpreted as an oscillating integral as usual. Fortechnical purposes, we allowed here the symbol a to depend on an additional variable y (that does not appear in (1.52)), but we will see in Lemma 1.21 that it does notintroduce more generality. This will be the main technical result of this section.Since we required G s h -pseudors to have a negligible kernel away from thediagonal, we need to prove that the local model (1.52) for their kernels has thisproperty. Lemma . Let m ∈ R , s ≥ . Let a be a G s symbol of order m in T ∗ U × U .Let K be a compact subset of U × U that does not intersect the diagonal. Then thekernel K a defined by (1.54) is C ∞ on a neighbourhood of K , and there are C, R > such that (cid:107) K a (cid:107) E s,R ( K ) ≤ C exp (cid:18) − Ch s (cid:19) . Proof.
Notice that we want to prove a local statement in U × U . Hence, wemay cut K in small pieces and then assume that K is contained in the interiorof K × K , where K and K are two disjoint balls, and that there is v ∈ R n ,such that for every ( x, y ) ∈ K × K we have ( x − y ) v ≥
1. A priori, K a is not .3. GEVREY PSEUDO-DIFFERENTIAL OPERATORS 55 a converging integral. However, when x (cid:54) = y , we can perform a finite number ofintegration by parts to reduce to that case, and find K a ( x, y ) = ( − N h N (2 πh ) n | x − y | N (cid:90) R n e i (cid:104) x − y,ξ (cid:105) h ∆ Nξ a ( x, ξ, y )d ξ, (1.55)with N > m + n . The integral is now convergent, and we can drop the prefactor.We denote by a N the new symbol. We observe that in the ξ variable, the phase isnon-stationary; however, we cannot apply Proposition 1.1 because the domain ofintegration is non-compact. We will have to employ a slightly different method. Weconsider δ > t := (cid:110) ξ + iδth − s (cid:104) ξ (cid:105) s v | ξ ∈ R n (cid:111) . Then, we denote by ˜ a N an almost analytic extension for a N in the sense of Remark1.10 (or the holomorphic extension of a N if s = 1) and obtain (cid:90) R n e i (cid:104) x − y,ξ (cid:105) h a N ( x, ξ, y )d ξ = (cid:90) Γ e i (cid:104) x − y,ξ (cid:105) h ˜ a N ( x, ξ, y )d ξ + (cid:90) [0 , × R n H ∗ (cid:16) e i (cid:104) x − y,ξ (cid:105) h ∂ ξ ˜ a N ( x, ξ, y ) ∧ d ξ (cid:17) , (1.56)where H denotes the homotopy ( t, ξ ) (cid:55)→ ξ + iδth − /s (cid:104) ξ (cid:105) /s v . Along Γ t , the imaginarypart of the phase is(1.57) Im (cid:104) x − y, ξ (cid:105) = δth − s (cid:104) ξ (cid:105) s (cid:104) x − y, v (cid:105) ≥ δth − s (cid:104) ξ (cid:105) s . Using the estimate on ∂ ξ ˜ a N given by Remark 1.7, we obtain, for some C > (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R n e i (cid:104) x − y,ξ (cid:105) h a N ( x, ξ, y )d ξ (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:90) Γ exp (cid:18) − Im (cid:104) x − y, ξ (cid:105) h (cid:19) d | ξ | + C sup t ∈ [0 , (cid:90) Γ t exp (cid:32) − Im (cid:104) x − y, ξ (cid:105) h − C (cid:12)(cid:12)(cid:12)(cid:12) Im ξ (cid:104) ξ (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) − s − (cid:33) d | ξ |≤ C exp (cid:18) Ch s (cid:19) , provided that δ is small enough . Here d | ξ | denotes the total variation of themeasure associated to the n -form d ξ restricted to the contour of integration. Now,we have to obtain a similar bound in G s instead of L ∞ . Actually, this is not muchfurther effort.It follows from Lemma 1.5 and (1.57) that, for some C, R >
0, the (cid:107)·(cid:107) s,R,K × K norm of ( x, y ) (cid:55)→ exp ( i (cid:104) x − y, ξ (cid:105) /h ) is an O (exp( − C − ( (cid:104)| ξ |(cid:105) /h ) /s )) when ξ ∈ Γ .Thus, differentiating under the integral and applying Leibniz formula, we find thatthe first term in (1.56) is an O (exp( − C − h − /s )) in G s .To deal with the second term in (1.56), we use Lemma 1.5 again: given (cid:15) >
0, wecan find
R > (cid:107)·(cid:107) s,R,K × K norm of ( x, y ) (cid:55)→ exp ( i (cid:104) x − y, ξ (cid:105) /h ) isan O (exp( (cid:15) ( (cid:104)| ξ |(cid:105) /h ) /s )) for ξ ∈ ∪ t ∈ [0 , Γ t . Recalling the bound on the derivativesof the coordinates of ¯ ∂ ξ ˜ a N given in Remark 1.10, we see that for some C, R > (cid:107)·(cid:107) s,R,K × K norm of ( x, y ) (cid:55)→ ¯ ∂ ξ ˜ a N ( x, ξ, y ) is an O (exp( − C − ( (cid:104)| ξ |(cid:105) /h ) /s ))for ξ ∈ ∪ t ∈ [0 , Γ t . Since we may assume that (cid:15) < C − , we can differentiate underthe integral in the second term of (1.56) and apply the Leibniz formula to find that the second term in (1.56) is also an O (exp( − C − h − /s )) in G s (for some other C > (cid:3)
In order to control the action of G s pseudors acting on G s functions, we will usethe following lemma. Lemma . Let m ∈ R , s ≥ and a ∈ S s,m (cid:0) T ∗ U (cid:1) . Let D, D (cid:48) and D (cid:48)(cid:48) be ballssuch that D ⊆ D (cid:48) , D (cid:48) ⊆ D (cid:48)(cid:48) and D (cid:48)(cid:48) ⊆ U . Then, for every R > and δ > , thereexists a constant C R > and R (cid:48) > such that for all u ∈ E s,R (cid:0) D (cid:48)(cid:48) (cid:1) the function x (cid:55)→ (cid:90) D (cid:48) K a ( x, y ) u ( y )d y (1.58) is G s on D , with (cid:107)·(cid:107) s,R (cid:48) ,D (cid:48) norm less than C R (cid:107) u (cid:107) E s,R ( D (cid:48)(cid:48) ) exp (cid:18) δh s (cid:19) . Proof.
Start by noticing that the integral in (1.58) is well-defined for x ∈ D since the singular support of K a is contained in the diagonal of U × U . We denotethis integral by Au ( x ). Let χ : R n → [0 ,
1] be a C ∞ function supported in D (cid:48) andidentically equals to 1 on a neighbourhood of D . Then, we may write Au ( x ) = (cid:90) D (cid:48) K a ( x, y ) χ ( y ) u ( y )d y (cid:124) (cid:123)(cid:122) (cid:125) := A u ( x ) + (cid:90) D (cid:48) K a ( x, y )(1 − χ ( y )) u ( y )d y (cid:124) (cid:123)(cid:122) (cid:125) := A u ( x ) . The term A u ( x ) may be ignored because K a is G s -small outside the diagonal(Lemma 1.18). For this we do not need u to be G s . Then, notice that, for N largeenough, we have A u ( x ) = 1(2 πh ) n (cid:90) D (cid:48) × R n e i (cid:104) x − y,ξ (cid:105) h a ( x, ξ, y ) (cid:16) ξ (cid:17) N (cid:16) − h ∆ y + 1 (cid:17) N ( χ ( y ) u ( y )) d y d ξ. Then, let ˜ u be an almost analytic extension for u given by Lemma 1.1 if s >
1, orthe holomorphic extension of u if s = 1. We also define ˜ χ on C n by ˜ χ ( z ) = χ (Re z ).Then, we let Γ ξ be the contourΓ ξ = (cid:40) y − i(cid:15) (cid:18) h (cid:104) ξ (cid:105) (cid:19) − s ξ (cid:104) ξ (cid:105) : y ∈ D (cid:48) (cid:41) , where (cid:15) > A u ( x ) = 1(2 πh ) n (cid:90) R n (cid:90) Γ ξ e i (cid:104) x − y,ξ (cid:105) h a ( x, ξ, y ) (cid:16) ξ (cid:17) N F u ( y )d y d ξ (cid:124) (cid:123)(cid:122) (cid:125) := R u ( x ) + R u ( x ) + R u ( x ) . (1.59)Here, F u ( y ) comes from the development of (cid:16) − h ∆ y + 1 (cid:17) N ( ˜ χ ( y )˜ u ( y )) , and wesplit the error term coming from Stokes’ Formula into the integral R that impliesapplication of the Cauchy–Riemann operator ∂ y to ˜ χ and its derivatives, and theintegral R that implies application of ∂ y to ˜ u and its derivatives.We start by dealing with R u ( x ). Our definition of ˜ χ implies that if Re z isnear D (in particular near x ) then ˜ χ is locally constant near z . Consequently, in thedefinition of R u ( x ), we only integrate over point y such that y is uniformly awayfrom D (and hence x ). Thus, the argument from the proof of Lemma 1.18 applies: .3. GEVREY PSEUDO-DIFFERENTIAL OPERATORS 57 we may shift the contour (in ξ ) in the integral defining R u ( x ) to prove that R u decays like exp( − C − h − /s ) in G s , and this does not require G s regularity on ˜ u nor˜ χ . We deal now with R u ( x ). Notice that if y ∈ Γ ξ thenIm (cid:18) − yξh (cid:19) = (cid:15) (cid:18) (cid:104) ξ (cid:105) h (cid:19) /s ξ (cid:104) ξ (cid:105) . (1.60)Then, we apply Lemma 1.5, and observe that for some R (cid:48) >
1, the (cid:107)·(cid:107) s,R (cid:48) ,D normof x (cid:55)→ exp ( i (cid:104) x, ξ (cid:105) /h ) a ( x, ξ, y ) grows at most like O (exp( δ (cid:48) ( (cid:104) ξ (cid:105) /h ) /s )) for δ (cid:48) > R is G s with the announced estimates.It remains to deal with R . This is very similar to the case of R , but insteadof using the positivity of the imaginary part of the phase to prove the decay of theintegrand, we apply Lemma 1.3 to control the Cauchy–Riemann operator applied to˜ u and its derivatives: see Remark 1.7. Since we consider points with imaginary partsless than (cid:15) ( h/ (cid:104) ξ (cid:105) ) − /s , we get from application of the Cauchy–Riemann operatorto ˜ u and its derivatives a decay likeexp (cid:32) − C (cid:18) (cid:104) ξ (cid:105) h (cid:19) s (cid:33) in G s . Then, we may work as in the case of R (or in the proof of Lemma 1.18),differentiating under the integral to prove that R u decays like O (exp( − C/h /s ))in G s , ending the proof of the lemma. Notice that the growth Au is only due to theterm R u in (1.59), and more precisely to small frequencies in the integral defining R . (cid:3) Another preliminary that will come in handy is the following
Lemma . Let U ⊂ R n be open, and Ω ⊂ U a relatively compact open set.Let a be a G symbol on T ∗ U × U . Then we can find C, δ > and a distribution (cid:101) K a ( x, u ) , defined for x ∈ Ω , u ∈ R n , such that (cid:101) K a has a holomorphic extension to (1.61) { ( x, u ) ∈ C | d ( x, Ω) < δ, u ∈ C n \ { } , | Im u | < δ | Re u | } , with (cid:101) K a ( x, u ) = K a ( x, x + u ) + O G (cid:18) exp (cid:18) − Ch (cid:19)(cid:19) for | u | < δ , | (cid:101) K a ( x, u ) | ≤ C exp (cid:18) − | u | Ch (cid:19) for | u | > δ/ . (1.62) Region of holomorphy of K a Region where χ is not constantConical neighbourhood V u
Figure 2.
The complex neighbourhoods involved in the extensionof the kernel for s = 1. Proof.
For this, we will use the ∂ inversion trick Lemma 1.10. First of all, itfollows from the proof of Lemma 1.18 that, for (cid:15) > x, u ) (cid:55)→ K a ( x, x + u ) has a holomorphic extension to (cid:110) ( x, u ) ∈ C n : d ( x, Ω) < (cid:15), < | u | < (cid:15) and | Im u | < (cid:15) | Re u | (cid:111) . Let us choose a C ∞ function χ : C n → [0 ,
1] such that χ ( u ) = 0 if | u | ≥ (cid:15)/ χ ( u ) = 1 if | u | ≤ (cid:15)/
4. Then, we use the bump function χ to define a form f of type(1 ,
0) by f ( x, u ) = − K a ( x, x + u ) ¯ ∂χ ( u ) . The 1-form f is defined for ( x, u ) ∈ (cid:0) T ∗ Ω (cid:1) (cid:15) with (cid:15) (cid:28) (cid:15) and it follows from Lemma1.18 that f ( x, u ) is an O (exp( − | u | / ( Ch ))). Hence, it follows from Lemma 1.10 thatthere is a smooth function r : ( T ∗ Ω) (cid:15) → C , for some small (cid:15) >
0, such that ¯ ∂r = f and r is an O (exp( − | u | / ( Ch ))). We can then define (cid:101) K a by (cid:101) K a ( x, u ) = χ ( u ) K a ( x, x + u ) + r ( x, u ) . For δ > ∂ (cid:16) χ ( u ) (cid:101) K a ( x, u ) (cid:17) + ¯ ∂r ( x, u ) = 0 , and is consequently holomorphic. We may assume that δ < (cid:15)/
4. For | u | < δ , wehave then ¯ ∂r ( x, u ) = 0 and hence r is an O G (exp( − / ( Ch ))) there. The first line of(1.62) follows. The second line of (1.62) follows from Lemma 1.18, the assumptionon the support of χ and the estimate on r . (cid:3) After these preliminary steps, we turn to the most basic trick: elimination ofvariables in symbols. We will see that we can eliminate the dependence on y of a in(1.54). We will then state the usual consequences of Lemma 1.21 (namely Lemmas1.22 and 1.23). As the knowledgeable reader will observe, the proof is more subtlethan in the C ∞ case, or even the G sx G ξ case. For the latter setting, one can findproofs for the h = 1, s > Zan85 , Theorem 2.25] (see also the proof of[
Rod93 , Theorem 3.2.24]).
Lemma . Let m ∈ R , s ≥ . Let a be a G s symbol of order m in T ∗ U × U .Let Ω be a relatively compact open subset of U . Then there exists a G s symbol b oforder m in T ∗ Ω such that for every compact subset K of Ω × Ω , there are constants C, R > such that (cid:107) K a − K b (cid:107) s,R,K ≤ C exp (cid:18) − Ch s (cid:19) , (1.63) where K a and K b are defined by (1.55) . Proof.
The first observation is that if b ∈ S s,m (cid:0) T ∗ R n (cid:1) and K b ( x, y ) = 1(2 πh ) n (cid:90) R n e ih (cid:104) x − y,ξ (cid:105) b ( x, ξ )d ξ, then it follows from the Fourier Inversion Formula that b may be recovered fromthe kernel K b by b ( x, ξ ) = (cid:90) R n e ih (cid:104) u,ξ (cid:105) K b ( x, x + u )d u. We would like to replace K b by the kernel K a in this formula and take this asa definition for b , but there are several issues with this formula, and we needconsequently to make a few corrections in this expression. .3. GEVREY PSEUDO-DIFFERENTIAL OPERATORS 59 The first problem is that for x ∈ Ω, the distribution K a ( x, x + u ) is not definedfor all u ∈ R n , and hence we cannot use this formula directly. When s >
1, we canintroduce a G s cutoff χ , equal to 1 on [0 , δ/ , δ ] for δ > δ -neighbourhood of Ω is included in U . Then, we replace a by a = a ( x, ξ, y ) χ ( | x − y | ) and set (cid:101) K a ( x, u ) := K a ( x, x + u ) χ ( | u | ) = K a ( x, x + u ) . Without loss of generality, we can assume that a = a (the error is negligibleconsidering Lemma 1.18). When s = 1, we change the definition of (cid:101) K a ( x, u ), anduse instead the kernel given by Lemma 1.20. In both cases, we let(1.64) b ( x, ξ ) = (cid:90) R n e ih (cid:104) u,ξ (cid:105) (cid:101) K a ( x, u )d u. One can see that (cid:101) K a ( x, y − x ) is the kernel of a pseudor, and it follows consequentlyfrom the usual theory of C ∞ pseudor that b is a symbol in the usual Kohn–Nirenbergclasses. Hence, the kernel K b is well-defined and from the Fourier Inversion Formulawe have (cid:101) K a ( x, u ) = K b ( x, x + u ). Would we know that b ∈ S s,m , we could take b = b and the bound (1.63) would be satisfied. However, it is not obvious that b is a G s symbol.In order to bypass this difficulty, we will provide a decomposition:(1.65) b ( x, ξ ) = b ( x, ξ ) (cid:124) (cid:123)(cid:122) (cid:125) a G s symbol + O G sx (cid:32) exp (cid:32) − (cid:104) ξ (cid:105) /s Ch /s (cid:33)(cid:33) . Here, the important point is that we are not requiring any regularity beyond L ∞ inthe ξ variable for the error. Indeed, it follows from Lemma 1.5 that for any r > e i (cid:104) x − y,ξ (cid:105) h = O G sx,y (cid:32) exp (cid:32) r (cid:18) (cid:104) ξ (cid:105) h (cid:19) s (cid:33)(cid:33) . Consequently, by taking r >
C > e i (cid:104) x − y,ξ (cid:105) h × O G sx (cid:32) exp (cid:32) − (cid:104) ξ (cid:105) /s Ch /s (cid:33)(cid:33) = O G sx,y (cid:32) exp (cid:32)(cid:18) r − C (cid:19) (cid:18) (cid:104) ξ (cid:105) h (cid:19) s (cid:33)(cid:33) = O G sx,y (cid:32) exp (cid:32) − C (cid:18) (cid:104) ξ (cid:105) h (cid:19) s (cid:33)(cid:33) . Hence, multiplying (1.65) by exp( i (cid:104) x − y, ξ (cid:105) /h ) and integrating over ξ , we find that K b ( x, y ) = K b ( x, y ) + O G sx,y (cid:32) exp (cid:32) − C (cid:18) (cid:104) ξ (cid:105) h (cid:19) s (cid:33)(cid:33) , and the estimate (1.63) is then satisfied. Indeed, near the diagonal K b and K a agrees (up to an O G (exp( − / ( Ch ))) in the case s = 1) and away from the diagonalboth kernels are negligible due to Lemma 1.18.Before going into the core of the proof, let us make another reduction. Wechoose a large integer N , then we perform integration by parts in the oscillatingintegral defining the kernel K a to write K a = (cid:96) (cid:88) j =1 h | α j | ∂ α j y K a j , where (cid:96) ∈ N and, for j ∈ { , . . . , (cid:96) } , we have a j ∈ S m − N,s (cid:0) T ∗ U × U (cid:1) and α j ∈ N n satisfies (cid:12)(cid:12) α j (cid:12)(cid:12) ≤ N . Hence, we may assume in the following that the order m of a isless than − n −
1, for if K a j − K b j satisfies the bound (1.63) for j = 1 , . . . , N , then(1.63) also holds for a if we set b ( x, ξ ) = (cid:96) (cid:88) j =1 ( − iξ ) α j b j ( x, ξ ) . Analytic case.
We establish (1.65) first in the case s = 1. In the case s = 1,it seems natural to work using holomorphic extension, thus we should rather prove(1.66) b ( x, ξ ) = b ( x, ξ ) (cid:124) (cid:123)(cid:122) (cid:125) a G symbol + O L ∞ (cid:32) exp (cid:32) − (cid:104) ξ (cid:105) s Ch s (cid:33)(cid:33) for ξ ∈ R n and x ∈ C n at distance less than C − of Ω (for some large C > b into a small remainder, and an oscillatory integral, to which we will be able toapply Proposition 1.6.We want to get rid of the non-diagonal terms first. For this, we can introducethe contour shift ( χ is the same cutoff function as above)(1.67) Γ := (cid:26) u + i(cid:15) (cid:18) − χ (cid:18) | u | (cid:19)(cid:19) ξ (cid:104) ξ (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) u ∈ R n (cid:27) . This deformation is not trivial only for | u | > δ , so that deforming from R n to Γ ,we find(1.68) b ( x, ξ ) = (cid:90) Γ e ih (cid:104) u,ξ (cid:105) (cid:101) K a ( x, u )d u. We only deform R n away from the singularity u = 0 of (cid:101) K a and thus there is noerror in Stokes’ Formula.Along Γ , when | Re u | > δ , the imaginary part of the phase is (cid:15) (cid:104) ξ (cid:105) − | ξ | , andthe kernel (cid:101) K a is O (exp( − /Ch )). As a consequence, we can remove the part ofthe integral corresponding to | u | > δ , since this is an O (exp( − C − (cid:104)| ξ |(cid:105) /h )) when | Im x | ≤ C − .In the remaining region | u | < δ , we can use the explicit expression for (cid:101) K a . Wemust consequently deal with the error term in (1.62). We want to see that that aterm of the form (cid:90) Γ , | Re u |≤ δ e i (cid:104) u,ξ (cid:105) h O G x,u (cid:18) exp (cid:18) − Ch (cid:19)(cid:19) d u (1.69)is negligible. When | ξ | is smaller than some given constant, the decay of the factor O G x,u (exp( − / ( Ch ))) is sufficient. When ξ is large, the phase in (1.69) is non-stationary, has non-negative imaginary part and is positive on the boundary of thecontour of integration. Hence, by a non-stationary phase argument, we find thatthe term (1.69) is in fact an O (cid:16) exp (cid:16) − C − (cid:104) ξ (cid:105) h (cid:17)(cid:17) . Here, we cannot apply directlyProposition 1.5 because we integrate on a contour rather than on a compact subsetof R n . However, the same contour shift strategy may be applied, and the readershould have no difficulty completing the argument (adapting for instance the proofof Proposition 1.1). .3. GEVREY PSEUDO-DIFFERENTIAL OPERATORS 61 We may consequently neglect the error term in (1.62). Thus, we may assumethat when | u | ≤ δ is real the kernel (cid:101) K a ( x, u ) is given by the formula (cid:101) K a ( x, u ) = 1(2 πh ) n (cid:90) R n e − i (cid:104) u,η (cid:105) h a ( x, η, x + u )d η. (1.70)In general, the integral in the right hand side of (1.70) would be oscillating, but wereduced to the case when the order of a is less than − n − u is away from 0 before plugging (1.70) into (1.68). To do so, we introduce for u ∈ R n such that | u | ≤ δ the contour ( (cid:15) > ( u ) := (cid:26) η − i(cid:15) (cid:18) − χ (cid:18) | u | (cid:19)(cid:19) (cid:104) η (cid:105) u : η ∈ R n (cid:27) . Notice that when | u | ≤ δ then Γ ( u ) is in fact R n . Then, we may shift contour in(1.70) in order to replace R n by Γ ( u ). We find consequently that for u ∈ C n suchthat | Re u | ≥ δ and | Im u | ≤ (cid:15) δ , the holomorphic extension of (cid:101) K a is given by (cid:101) K a ( x, u ) = 1(2 πh ) n (cid:90) Γ (Re u ) e − i (cid:104) u,η (cid:105) h a ( x, η, x + u )d η. (1.71)It may not be clear at first sight that the expression given by (1.71) is indeedholomorphic (indeed, the dependence of the contour on u is not). However, due tothe invariance of the integral under contour shift, we may replace Γ (Re u ) by alocally constant contour when proving holomorphicity. Now, we assume that theconstant (cid:15) > satisfies (cid:15) (cid:28) (cid:15) so that the equality(1.71) holds for u ∈ Γ . Thus, if we form the contourΓ := { ( u, η ) : u ∈ Γ , η ∈ Γ (Re u ) } , then b is given up to negligible terms by the integral b ( x, ξ ) (cid:39) πh ) n (cid:90) ( u,η ) ∈ Γ | Re u |≤ δ e i (cid:104) u,ξ − η (cid:105) h a ( x, η, x + u )d η d u. (1.72)We need a further contour deformation, to get rid of the lack of compactness in the η variable. To this end, we observe that the phase from (1.70) phase is non-stationaryin the u variable when η (cid:54) = ξ . This suggest to consider the contourΓ := (cid:26)(cid:18) u + i(cid:15) ξ − η (cid:104) ξ (cid:105) , η (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) ( u, η ) ∈ Γ (cid:27) . Replacing R n × Γ by Γ in (1.72), the only error comes from the boundary of thecontour of integration in (1.72). However, we see that for ( u, η ) ∈ Γ such that | Re u | = 4 δ the imaginary part of the phase at ( u, η ) is given by (provided that (cid:15) (cid:28) (cid:15) (cid:28)
1) Im ( u, ξ − η ) = (cid:15) (cid:32) (cid:104) ξ (cid:105) (cid:104) ξ (cid:105) − (cid:104) ξ, Re η (cid:105)(cid:104) ξ (cid:105) (cid:33) + (cid:15) δ (cid:104) Re η (cid:105)≥ ( (cid:104) ξ (cid:105) + (cid:104)| η |(cid:105) ) C .
Since the imaginary part of the phase increases along the homotopy, we see thatthe boundary term when deforming from Γ to Γ is negligible. Hence, we have b ( x, ξ ) (cid:39) πh ) n (cid:90) ( u,η ) ∈ Γ , | Re u |≤ δ e i (cid:104) u,ξ − η (cid:105) h a ( x, η, x + u )d η d u. (1.73)When deforming the contour of integration from Γ to Γ , we increased theimaginary part Im ( u, η ) of the phase by a term (cid:15) (cid:104) ξ (cid:105) − | ξ − η | . Since the imaginary part of the phase was already non-negative on Γ (thanks to the assumption (cid:15) (cid:28) (cid:15) ),we see that the part of the integral in (1.73) where | ξ − η | ≥ δ (cid:104) ξ (cid:105) is negligible andthus b ( x, ξ ) (cid:39) πh ) n (cid:90) ( u,η ) ∈ Γ | Re u |≤ δ, | ξ − η |≤ δ (cid:104) ξ (cid:105) e i (cid:104) u,ξ − η (cid:105) h a ( x, η, x + u )d η d u. The right hand side is now an integral over a bounded set, it is convenient to rescalethis integral, writing b ( x, ξ ) (cid:39) (cid:104) ξ (cid:105) n (2 πh ) n (cid:90) Γ e i (cid:104) ξ (cid:105) h (cid:104) u,v (cid:105) a ( x, ξ − (cid:104) ξ (cid:105) v, x + u )d u d v, (1.74)where the contour Γ is defined byΓ := { ( u, v ) : ( u, ξ − (cid:104) ξ (cid:105) v ) ∈ Γ , | Re u | ≤ δ and | v | ≤ δ } . We want now to approximate the right hand side of (1.74) by a G symbol. Itfollows from the Holomorphic Stationary Phase Method — Proposition 1.6 or [ Sj¨o82 ,Th´eor`eme 2.8] — that there is a formal analytic symbol (cid:80) k ≥ h k c k of order m suchthat the difference between (cid:80) ≤ k ≤ C − (cid:104) ξ (cid:105) /h h k c k ( x, ξ ), for C > b a realization of the formalanalytic symbol (cid:80) k ≥ h k c k (given by Lemma 1.7). Gevrey case.
We turn now to the case s >
1. In that case, we may directlywrite b ( x, ξ ) = 1(2 πh ) n (cid:90) R n × R n e i (cid:104) u,ξ − η (cid:105) h a ( x, η, x + u )d η d u. Indeed, a ( x, η, x + u ) vanishes when | u | > δ and the order of a is less than − n − b ( x, ξ ) = (cid:104) ξ (cid:105) n (2 πh ) n (cid:90) R n × R n e − i (cid:104) ξ (cid:105) h (cid:104) u,v (cid:105) a ( x, ξ + (cid:104) ξ (cid:105) v, x + u )d u d v. Using the cutoff functions χ again, we can write b ( x, ξ ) = b ( x, ξ ) + (cid:104) ξ (cid:105) n (2 πh ) n (cid:90) R n × R n e − i (cid:104) ξ (cid:105) h (cid:104) u,v (cid:105) (1 − χ ( | v | )) a ( x, ξ + (cid:104) ξ (cid:105) v, x + u )d u d v, (1.75)where b ( x, ξ ) = (cid:104) ξ (cid:105) n (2 πh ) n (cid:90) R n × R n e − i (cid:104) ξ (cid:105) h (cid:104) u,v (cid:105) χ ( | v | ) a ( x, ξ + (cid:104) ξ (cid:105) v, x + u )d u d v (1.76)is a symbol in S s,m (cid:0) T ∗ Ω (cid:1) according to Lemma 1.17. Here, it is very important that δ > G s in v . Indeed,without the cutoff functions the estimates on the derivatives would deteriorate for v (cid:39) − ξ/ (cid:104) ξ (cid:105) .Now, we need to prove that the remainder term in (1.75) is as small as announcedin (1.65). To do so, we will apply the non-stationary phase method to the integral (cid:90) R n e − i (cid:104) ξ (cid:105)(cid:104) v (cid:105) h (cid:104) u, v (cid:104) v (cid:105) (cid:105) a ( x, ξ + (cid:104) ξ (cid:105) v, x + u )d u. (1.77)Here, v and ξ are dealt with merely as parameters, so that the lack of regularity inthese variable will not matter. However, the integrand is uniformly G s in x . When | v | > δ/
2, the phase in (1.77) is non-stationary, so that we can apply Proposition1.1 (with a rescaling arguments as in § O G sx (cid:32) exp (cid:32) − (cid:18) (cid:104) ξ (cid:105) (cid:104) v (cid:105) Ch (cid:19) s (cid:33)(cid:33) . .3. GEVREY PSEUDO-DIFFERENTIAL OPERATORS 63 Thus, (1.65) follows from (1.75) after integration over v . (cid:3) Remark . Since the proof of Lemma 1.21 is based on an application of theStationary Phase Method (Proposition 1.6 or Lemma 1.17), we see that the symbol b has the usual asymptotic expansion (recall Remark 1.23 and Remark 1.24) b ( x, ξ ) ∼ (cid:88) α ∈ N n h | α | ∂ αξ ∂ αy a ( x, ξ, x ) i | α | α ! . When s = 1, this expansion holds in the sense of realization of formal analyticsymbol (see Definition 1.3). When s >
1, the estimate is not as good, but we stillknow that, for every N ∈ N , we have b ( x, ξ ) = (cid:88) | α | Apply Lemma 1.21 to the kernel K a ( y, x ) = 1(2 πh ) n (cid:90) R n e i (cid:104) x − y,ξ (cid:105) h ¯ a ( y, ξ )d ξ to find b ∈ S s,m (cid:0) T ∗ Ω (cid:1) such that K a ( y, x ) = K b ( x, y ) + O G s (cid:18) exp (cid:18) − Ch s (cid:19)(cid:19) . The result follows then by switching x and y back and taking complex conjugate. (cid:3) Using the Kuranishi trick, we deduce from Lemma 1.21 the following result thatjustifies Definition 1.6. This is a very classical procedure, see for instance [ Zwo12 ,Theorem 9.3]. In particular, the proof of Lemma 1.23 do not rely on the analyticversion of the Kuranishi trick Lemma 1.24 that we will discuss soon (there is nocomplex phase or contour shift here). Lemma . Let V be another open subset of R n and let κ : U → V be a G s diffeomorphism. Let a ∈ S s,m (cid:0) T ∗ U (cid:1) and Ω be a relatively compact subset of U .Then there is a symbol b ∈ S s,m (cid:0) T ∗ κ (Ω) (cid:1) such that K a (cid:16) κ − ( x ) , κ − ( y ) (cid:17) Jκ − ( y ) = K b ( x, y ) + O G s (cid:18) exp (cid:18) − Ch s (cid:19)(cid:19) on κ (Ω) . For several reasons, we may encounter operators whose kernel is an oscillatoryintegral, with a phase that is not exactly the phase (cid:104) x − y, ξ (cid:105) , but close to it. It isthen crucial to be able to recognize the kernel of a pseudor, and this is what theKuranishi trick is for. Let us be more precise an define the class of phases that wewill use. Definition . Let M be either a compact real-analytic manifold or an opensubset of R n . A phase on M is a holomorphic symbol Φ of order 1 defined for( α, y ) ∈ ( T ∗ M ) (cid:15) × ( M ) (cid:15) with d ( α x , y ) < δ (for some (cid:15), δ > 0) such that(i) if ( α, y ) ∈ T ∗ M × M then the imaginary part of Φ( α, y ) is non-negative;(ii) Φ( α, α x ) = 0 for α = ( α x , α ξ ) ∈ T ∗ M ;(iii) for α ∈ T ∗ M , we have d y Φ( α, α x ) = − α ξ .We say that Φ is an admissible phase if it satisfies in addition the coercivity condition:(iv) there is a constant C > α, y are real and Φ( α, y ) is defined,then Im Φ( α, y ) ≥ C − (cid:104) α (cid:105) d ( α x , y ) .The point (ii) and (iii) in Definition 1.7 may be stated loosely asΦ( x, ξ, y ) = (cid:104) x − y, ξ (cid:105) + O (cid:16) (cid:104)| ξ |(cid:105) | x − y | (cid:17) . Notice that, with this definition, the standard phase Φ( x, ξ, y ) = (cid:104) x − y, ξ (cid:105) is notan admissible phase: it does not satisfy the condition (iv). In some sense, (iv) issatisfied by the standard phase after a contour shift (this is basically the argumentin the proof of Lemma 1.18).If Φ is a phase in the sense of Definition 1.7 on some open subset U of R n and a ∈ S ,m (cid:0) T ∗ U × U (cid:1) is an analytic symbol, we may see using standard integrationby parts that the oscillating integral K Φ ,a ( x, y ) = 1(2 πh ) n (cid:90) R n e i Φ( x,ξ,y ) h a ( x, ξ, y )d ξ (1.78)defines a distribution near the diagonal of U × U . Adapting the proof of Lemma1.18, one can see that if x, y ∈ U are close enough to the diagonal – so that K Φ ,a isdefined near ( x, y ) – but not on the diagonal, then K Φ ,a is an O (exp( − / ( Ch ))) in G near ( x, y ).We will say that an operator whose kernel (modulo a small G error) takes theform K Φ ,a , is a pseudor with non-standard phase . There are two reasons for whichwe want to study such operators. The first one is that an analytic pseudor withnon-standard phase appears in the proof of Theorem 6. The other one is that wewill use pseudors with non-standard phases to prove Proposition 1.11. The mainresult about those operators is the following: Lemma . Let U be a bounded open subset of R n . Let Φ and Φ be phases on U in the sense of Definition 1.7. Let m ∈ R and a ∈ S ,m (cid:0) T ∗ U × U (cid:1) . Let Ω be a relatively compact open subset of U . Then there is a ∈ S ,m (cid:0) T ∗ Ω × Ω (cid:1) such that K Φ ,a ( x, y ) = K Φ ,a ( x, y ) near the diagonal of Ω × Ω . Proof of Lemma 1.24. Instead of considering the general case, it suffices toconsider the case when one of the phases Φ and Φ is the standard phase (cid:104) x − y, ξ (cid:105) .Thus, we pick some phase Φ as in Definition 1.7. According to Taylor’s formula(with integral remainder) we haveΦ( x, ξ, y ) = (cid:10) x − y, θ x,y ( ξ ) (cid:11) , (1.79) .3. GEVREY PSEUDO-DIFFERENTIAL OPERATORS 65 where θ x,y ( ξ ) = ξ − (cid:90) (1 − t ) D y,y Φ( x, ξ, x + t ( y − x ))( y − x )d t = ξ + O ( (cid:104)| ξ |(cid:105) | x − y | ) . From this, one may show that there are (cid:15) , (cid:15) > θ x,y is injective on ( R n ) (cid:15) and has a right inverse Υ x,y : ( R n ) (cid:15) → ( R n ) (cid:15) (here, we use the Kohn–Nirenbergmetric to define the Grauert tubes of R n ). We may then use Υ x,y as a change ofvariable to write(1.80) K Φ ,a ( x, y ) = 1(2 πh ) n (cid:90) θ x,y ( R n ) e i (cid:104) x − y,ξ (cid:105) h a ( x, Υ x,y ( ξ ) , y ) J Υ x,y ( ξ )d ξ. This is to be understood as an oscillatory integral, which can be put in convergentform using integration by parts. Then, if x and y are close enough, θ x,y ( R n ) remainsuniformly transverse to i R n , and may thus be written as a graph over R n . We maythus shift the contour in (1.80) to replace it by R n , using for instance the homotopy[0 , × θ x,y ( R n ) (cid:51) ( t, ξ ) (cid:55)→ Re ξ + it Im ξ . We obtain K ,a ( x, y ) = 1(2 πh ) n (cid:90) R n e i (cid:104) x − y,ξ (cid:105) h a ( x, Υ x,y ( ξ ) , y ) J Υ x,y ( ξ )d ξ = K ,b ( x, y ) , (1.81)where Φ ( x, ξ, y ) = (cid:104) x − y, ξ (cid:105) denotes the standard phase and the symbol b isdefined by b ( x, ξ, y ) = a ( x, Υ x,y ( ξ ) , y ) J Υ x,y ( ξ ). We end the proof by noticingthat we can reverse the argument since a is retrieved from b using the formula a ( x, ξ, y ) = b ( x, θ x,y ( ξ ) , y ) Jθ x,y ( ξ ). (cid:3) In order to prove the existence of an analytic principal symbol for a G pseudor(Proposition 1.10), it will be useful to understand the action of a pseudor on ananalytic family of coherent states. This is closely related to the FBI transform thatwill be the subject of the next chapter. Definition . A G family of coherent states of order (cid:96) ∈ R on M is a holomorphic function ( α, x ) (cid:55)→ u α ( x ) on (cid:0) T ∗ M (cid:1) (cid:15) × ( M ) (cid:15) (for some (cid:15) > η, C > α = ( α x , α ξ ) ∈ ( T ∗ M ) (cid:15) and x ∈ ( M ) (cid:15) we have:(i) if d ( α x , x ) ≥ η then | u α ( x ) | ≤ C exp (cid:18) − (cid:104)| α |(cid:105) Ch (cid:19) ;(ii) if d ( α x , x ) ≤ η then u α ( x ) = e i Φ( α,x ) h a ( α, x ) + O (cid:18) exp (cid:18) − (cid:104)| α |(cid:105) Ch (cid:19)(cid:19) , where a and Φ are analytic symbols respectively in S ,(cid:96) (cid:0) T ∗ M × M (cid:1) andin S , (cid:0) T ∗ M × M (cid:1) (maybe defined only for ( α x , x ) near the diagonal)that satisfy:(iii) Φ ( α, α x ) = 0;(iv) the derivative d x Φ( α, α x ) is elliptic of order 1 in the sense that | d x Φ( α, α x ) | ≥ (cid:104)| α |(cid:105) C when (cid:104)| α |(cid:105) ≥ C ; (v) if α and x are real then we have,Im Φ( α, x ) ≥ d ( α x , x ) C . We say that Φ and a are respectively the phase and the amplitude of ( u α ).We can now describe the action of G pseudors on coherent states. Notice thatwhen P is a differential operator, Proposition 1.12 is just a consequence of theLeibniz formula. Proposition . Let P be a G h -pseudor of order m ∈ R and ( u α ) be a G coherent state of order (cid:96) ∈ R on M . Then ( P u ) α ( x ) := ( P ( u α ))( x ) defines a G family of coherent states of order m + (cid:96) on M . We denote by b theamplitude of P u α . If the kernel of P is described in a coordinate patch U by (1.52) ,with p ∈ S ,m (cid:0) T ∗ U (cid:1) , then b is given at first order on U by (1.82) b ( α, x ) = p ( x, d x Φ ( α, x )) a ( α, x ) mod hS ,m + (cid:96) − . Proof. We take (cid:15) > P u α ( x ) for ( α, x ) ∈ (cid:0) T ∗ M (cid:1) (cid:15) × ( M ) (cid:15) . Choose η > (cid:15) (cid:28) η ). We start by investigating P u α ( x ) when the distance between x and α x is larger than 10 − η . We start byassuming that x ∈ M . In that case, we may choose balls D and D of radius 10 − η that does not intersect, with x ∈ D and Re α x ∈ D . We may also assume that x and Re α x remain at distance at least 10 − η from the boundaries of D and D (weonly need a finite number of such disks to cover all cases). Then write P u α ( x ) = (cid:32)(cid:90) D + (cid:90) D + (cid:90) M \ ( D ∪ D ) (cid:33) K P ( x, y ) u α ( y )d y, (1.83)where K P ( x, y ) denotes the kernel of P . Notice that all the terms in (1.83) makesense since the singular support of K P is contained in the diagonal of M × M . Theintegral over M \ ( D ∪ D ) is easily dealt with: by assumption, the kernel of P is an O (exp( − C/h )) and u α is an O (exp( − C (cid:104)| α |(cid:105) /h )) in G there. To tackle the integralover D , just notice that u α is an O (exp( − C (cid:104)| α |(cid:105) /h )) in G there. Hence, we mayapply Lemma 1.19 and find that the integral over D is an O (exp( − C (cid:104)| α |(cid:105) /h )) in G . It remains to deal with the integral over D . Notice that, up to smooth andsmall terms that we may neglect, the integral over D writes (cid:90) D e i Φ( α,y ) h a ( α, y ) K P ( x, y )d y. (1.84)Here, since x and y are uniformly away from each other the kernel of K P is an O (exp( − C/h )) in G . For small α , if α is sufficiently close to the reals, Im Φ cannotbe too negative, so this is sufficient to find a O (exp( − C/h )) bound in G for theintegral. Thanks to the assumption of ellipticity on the derivative of Φ, when α is large enough the phase is non-stationary in (1.84). Hence, the integral (1.84)is an O (exp( − C (cid:104)| α |(cid:105) /h )) as an application of Proposition 1.5. Hence, we provedthat P u α ( x ) is negligible when x and α x are away from each other and x is real.However, we proved an estimate in G so that this estimate remains true when x ∈ ( M ) (cid:15) with (cid:15) (cid:28) η .We want now to understand the kernel P u α ( x ) when d ( x, Re α x ) < η – as above,we may assume x real and then prove a bound in G . To do so, we may assume that x and Re α x are contained in a ball D of radius 10 η and then denote by D (cid:48) the ball .3. GEVREY PSEUDO-DIFFERENTIAL OPERATORS 67 of radius 100 η with the same center than D (we only need a finite number of such D to cover the whole manifold). Then, write P u α ( x ) = (cid:32)(cid:90) D (cid:48) + (cid:90) M \ D (cid:48) (cid:33) K P ( x, y ) u α ( y )d y We obtain a O (exp( − C (cid:104)| α |(cid:105) /h )) bound in G for the integral over M \ D (cid:48) as inthe previous case (both kernels are smooth and small there). By taking η smallenough, we may assume that D (cid:48) is contained in a coordinate patch and work inthese coordinates. When studying the integral over D (cid:48) , we will ignore the errorterms in (1.52) and in the definition of u α . Hence, we want to study the kernelgiven by the oscillating integral P u α ( x ) = 1(2 πh ) n (cid:90) R n × D (cid:48) e i (cid:104) x − y,ξ (cid:105) +Φ( y,α ) h p ( x, ξ ) a ( α, y ) d ξ d y. The oscillating integral here is well-defined because the singular support of a pseudo-differential operator is contained in the diagonal of M × M . In order to manipulatethis integral, we will approximate it by converging integrals. Indeed, we have(1.85) P u α ( x ) = lim δ → πh ) n (cid:90) R n × D (cid:48) e i (cid:104) x − y,ξ (cid:105) +Φ( y,α ) h e − δ | ξ | p ( x, ξ ) a ( α, y ) d ξ d y (cid:124) (cid:123)(cid:122) (cid:125) := k δ ( x,α ) . We already know by Proposition 1.9 that P u α is analytic. Hence, we only need toestablish L ∞ symbolic bounds on the amplitude b ( α, x ) := e − i Φ( α,x ) h P u α ( x ) . Since we are only searching for uniform symbolic estimates, we can work locallyin α ξ (provided we indeed get uniform estimates): we assume that α ξ remains atdistance less than η of a point ξ ∈ R n for the Kohn–Nirenberg metric. Changingvariable in (1.85), we find P u α ( x ) = (cid:18) (cid:104) ξ (cid:105) πh (cid:19) n (cid:90) R n × D (cid:48) e i (cid:104) ξ (cid:105) h Ψ x,α,ξ ( ξ,y ) e − δ (cid:104) ξ (cid:105) | ξ | p ( x, (cid:104) ξ (cid:105) ξ ) a ( α, y ) d ξ d y, (1.86)where the phase Ψ x,α,ξ is defined byΨ x,α,ξ ( ξ, y ) = (cid:104) x − y, ξ (cid:105) + Φ( y, α ) (cid:104) ξ (cid:105) . In order to regularize the integral in (1.86) and get rid of the approximation, weintroduce for ν > (cid:40)(cid:32) ξ + iν | ξ | ( x − y ) , y + iχ ( y ) ν ∇ y Ψ x,α,ξ ( ξ, y ) (cid:10)(cid:12)(cid:12) ∇ y Ψ x,α,ξ ( ξ, y ) (cid:12)(cid:12)(cid:11) (cid:33) : ξ ∈ R n , y ∈ D (cid:48) (cid:41) , where the bump function χ : R n → [0 , 1] takes value 1 on a neighbourhood of D and is supported in the interior of D (cid:48) . Along Γ, the imaginary part of the phasebecomes positive, as we explain now. If ξ, y ∈ R n and ξ is large enough then (cid:12)(cid:12) ∇ y Ψ x,α,ξ (Re ξ, Re y ) (cid:12)(cid:12) ≥ (cid:104)| ξ |(cid:105) C . Hence, we find by an application of Taylor’s formula that for ( y, ξ ) ∈ Γ with ξ largewe have (the constant C > x,α,ξ ( ξ, y ))= Im (cid:0) Ψ x,α,ξ (Re ξ, Re y ) (cid:1) + ν | Re ξ | | x − Re y | + νχ (Re y ) (cid:12)(cid:12) ∇ y Ψ x,α,ξ (Re ξ, Re y ) (cid:12)(cid:12) + O (cid:16) ν χ (Re y ) | Re ξ | (cid:17) ≥ νC (cid:16) | x − Re y | + χ (Re y ) (cid:17) (cid:104)| ξ |(cid:105) − Cν (cid:104)| ξ |(cid:105) − C(cid:15) (cid:104)| ξ |(cid:105)≥ (cid:18) ν (cid:18) C − Cν (cid:19) − C(cid:15) (cid:19) (cid:104)| ξ |(cid:105) ≥ ν (cid:104)| ξ |(cid:105) C . (1.87)Here, we used that if χ (Re y ) (cid:54) = 1 then | x − Re y | is uniformly bounded from below(because x is in D ) and we assumed that ν was small enough and that (cid:15) (cid:28) ν .Beware that (1.87) is only valid for large ξ . However, this bound is sufficient to see,using Stokes’ Formula and the dominated convergence theorem, that P u α ( x ) = (cid:18) (cid:104) ξ (cid:105) πh (cid:19) n (cid:90) Γ e i (cid:104) ξ (cid:105) h Ψ x,α,ξ ( ξ,y ) p ( x, (cid:104) ξ (cid:105) ξ ) a ( α, y ) d ξ d y. Here, there is no error term since we do not deform near the boundary of R n × D (cid:48) and the integrand is holomorphic. Notice that the deformation is possible herebecause p ( x, (cid:104) ξ (cid:105) ξ ) is holomorphic for | Im ξ | ≤ C − | Re ξ | for some C > | Im ξ | ≤ C − (cid:104) Re ξ (cid:105) , due to the rescaling).There is another consequence of (1.87): for some large A (independent of α , ξ ), we have (recall that (cid:104) ξ (cid:105) (cid:39) (cid:104)| α |(cid:105) ) P u α ( x ) = (cid:18) (cid:104) ξ (cid:105) πh (cid:19) n (cid:90) (cid:26) ξ ∈ Γ (cid:104)| ξ |(cid:105)≤ A (cid:27) e i (cid:104) ξ (cid:105) h Ψ x,α,ξ ( ξ,y ) p ( x, (cid:104) ξ (cid:105) ξ ) a ( α, y ) d ξ d y + O (cid:18) exp (cid:18) − (cid:104)| α |(cid:105) Ch (cid:19)(cid:19) . We would like to apply the Stationary Phase Method now, but there is an issue: dueto the rescaling, the function ξ (cid:55)→ p ( x, (cid:104) ξ (cid:105) ξ ) is not uniformly analytic. To make itworks, we need to get rid of small frequencies, and this is where the assumption ofellipticity on d y Φ is useful.We choose a point x ∈ D and a large constant B > 0. We denote by (cid:101) D theball of radius Bη centered at (cid:104) ξ (cid:105) − ∇ x Φ( x , ( x , ξ )). Then, if x ∈ D (cid:48) , y ∈ D and ξ ∈ R n \ D (cid:48) we have (cid:12)(cid:12) ∇ y Ψ x,α,ξ ( y ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) ∇ x Φ( α, y ) (cid:104) ξ (cid:105) − ξ (cid:12)(cid:12)(cid:12)(cid:12) ≥ (cid:12)(cid:12)(cid:12)(cid:12) ξ − ∇ x Φ( x , ( x , ξ )) (cid:104) ξ (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12) ∇ x Φ( α, y ) (cid:104) ξ (cid:105) − ∇ x Φ( x , ( x , ξ )) (cid:104) ξ (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) ≥ Bη − O ( η ) ≥ Bη , provided that B is large enough. Hence, by a an application of Taylor’s formula asabove, we see that if ( ξ, y ) ∈ Γ is such that Re ξ / ∈ (cid:101) D, (cid:104)| ξ |(cid:105) ≤ A and χ (Re y ) = 1 thenthe imaginary part of Ψ x,α,ξ ( ξ, y ) is uniformly positive (provided that (cid:15) (cid:28) ν and (cid:15) (cid:28) η ). But if χ (Re y ) (cid:54) = 1 then (cid:104) ξ (cid:105) − Im Φ ( α, y ) is uniformly positive (because y is away from α x ) and so is the imaginary part of Ψ x,α,ξ ( ξ, y ). Thus, if we introduce .3. GEVREY PSEUDO-DIFFERENTIAL OPERATORS 69 the new contour Γ (cid:48) = (cid:110) ( ξ, y ) ∈ Γ : (cid:104)| ξ |(cid:105) ≤ A and Re ξ ∈ (cid:101) D (cid:111) , we see that P u α ( x ) = (cid:18) (cid:104) ξ (cid:105) πh (cid:19) n (cid:90) Γ (cid:48) e i (cid:104) ξ (cid:105) h Ψ x,α,ξ ( ξ,y ) p ( x, (cid:104) ξ (cid:105) ξ ) a ( α, y ) d ξ d y + O (cid:18) exp (cid:18) − (cid:104)| α |(cid:105) Ch (cid:19)(cid:19) . The advantage here is that Γ (cid:48) is contained in a neighbourhood of size proportional to η of the critical point y = x and ξ = (cid:104) ξ (cid:105) − d x Φ ( α, x ) of the phase Ψ x,α,ξ . Moreover,due to the ellipticity assumption on d y Φ, when (cid:104) ξ (cid:105) is large, the projection of Γ (cid:48) onthe ξ variable is uniformly away from zero, and thus the integral has a holomorphicextension to a complex neighbourhood of Γ (cid:48) of fixed size. From this fact andthe positivity of the phase on the boundary of Γ (cid:48) , we see that we can apply theStationary Phase Method, Proposition 1.6 (see also Proposition 1.3). For this, itsuffices to observe that when x = α x and x and α are real, Im Ψ x,α,ξ ≥ (cid:48) ,Im Ψ > ∂ Γ (cid:48) and Ψ x,α,ξ has a unique critical point in a complex neighbourhoodof Γ (cid:48) , which is non-degenerate. Indeed, this critical point is given by y = α x = x , ξ = d x Φ( α, α x ). At the critical point, Ψ α x ,x,ξ vanishes and the Hessian is (cid:18) D x,x Φ( α, α x ) − − (cid:19) , which is invertible. Proposition 1.6 then applies. Consequently, we have that P u α ( x ) = e i Φ( α,x ) h b x ,ξ ( α, x ) + O (cid:18) exp (cid:18) − (cid:104)| α |(cid:105) Ch (cid:19)(cid:19) . Here b x ,ξ is a symbol of order m that satisfies uniform estimate in x and ξ (because the hypothesis of Proposition 1.6 are uniformly satisfied up to a factor (cid:104) ξ (cid:105) m ). Consequently, we have b ( α, x ) = b x ,ξ ( α, x ) + e − i Φ( α,x ) O (cid:18) exp (cid:18) − (cid:104)| α |(cid:105) Ch (cid:19)(cid:19) = b x ,ξ ( α, x ) + e − i Φ( α,x ) O (cid:18) exp (cid:18) − (cid:104)| α |(cid:105) Ch (cid:19)(cid:19) . On the second line here, we assumed that η was small enough in order to ensure thatthe imaginary part of Φ( α, x ) was smaller than (2 C ) − (cid:104)| α |(cid:105) (recall that Φ( α, α x ) = 0and that α x and x are at distance at most η ). Consequently, b inherits the estimateon b x ,ξ , so that b is indeed an analytic symbol of order m .Since the determinant of the Hessian at the critical point is ( − n , and the valueof the phase at the critical point is Φ ( α, x ), we get the first order approximation b ( α, x ) = b x ξ ( α, x ) mod hS ,m + (cid:96) − = p ( x, d x Φ( α, x )) a ( α, x ) mod hS ,m + (cid:96) − . (cid:3) Now that we have obtainedthe necessary basic results regarding the kernels of Gevrey pseudors, we can provetheir basic properties as operators that we stated in § Proof of Proposition 1.8. If K A ( x, y ) denotes the kernel of A , then the ker-nel of A ∗ is K A ( y, x ). Hence, it is clear that the kernel of A ∗ is O G s (exp( − / ( Ch ) /s ))away from the diagonal (because so is K A ). Then, it follows from Lemma 1.21 thatthe kernel of A ∗ is of the form (1.52) in local coordinates (that we may choosevolume preserving) as in the proof of Lemma 1.22. (cid:3) It is then straightforward to deduce the mapping properties of G s pseudors(Proposition 1.9) from Lemma 1.19 and Proposition 1.8. We turn now to the stabilityunder composition (Proposition 1.7). The proof relies on Lemmas 1.18,1.19 and1.21. Proof of Proposition 1.7. We start by proving that the kernel K P Q of P Q is small in G s away from the diagonal. We only need to consider the G s norm of K P Q on K = D × D when D and D are disjoint closed balls in M . Then, wemay choose disjoint closed balls D (cid:48) and D (cid:48) such that D and D are containedrespectively in the interior of D (cid:48) and D (cid:48) . If x ∈ D and y ∈ D , we may write K P Q ( x, y ) = (cid:90) M K P ( x, z ) K Q ( z, y )d z = (cid:32)(cid:90) D (cid:48) + (cid:90) D (cid:48) + (cid:90) M \ ( D (cid:48) ∪ D (cid:48) ) (cid:33) K P ( x, z ) K Q ( z, y )d z. This splitting of the integral makes sense, considering the singular supports of K P and K Q . The integral over M \ (cid:0) D (cid:48) ∩ D (cid:48) (cid:1) is easily dealt with since the integrand isan O (exp( − / ( Ch ) /s )) in G s . The integral over D (cid:48) is dealt with by Lemma 1.19.The integral over D (cid:48) is dealt with by Lemma 1.19 after using Lemma 1.21 to write K Q as the right quantization of a G s symbol.We need now to understand the kernel K P Q ( x, y ) near the diagonal, and thiscan be done in local coordinates. Here, the kernel K P may be written as a leftquantization modulo a small G s error, as in (1.52). We may decompose K Q similarlyand then, applying Lemma 1.21, we can express K Q as a right quantization, i.e. K Q ( x, y ) = 1(2 πh ) n (cid:90) R n e i (cid:104) x − y,ξ (cid:105) h q ( ξ, y )d ξ + O G s (cid:18) exp (cid:18) − Ch s (cid:19)(cid:19) , (1.88)where q is a symbol as in Definition 1.1. In order to fix notations, let us say that(1.52) and (1.88) hold in a ball D (cid:48) (in local coordinates) and that we want to computethe kernel K P Q ( x, y ) of P Q in a strictly smaller ball D . The remainder terms in(1.52) and (1.88) are dealt with as in the non-diagonal case, applying Lemma 1.19,and we will ignore them.Let then ψ be a C ∞ function from M to [0 , 1] that vanishes outside D (cid:48) and isidentically equal to 1 on a neighbourhood of D . We may write for x, y ∈ DK P Q ( x, y ) = (cid:90) D (cid:48) K P ( x, z ) K Q ( z, y ) ψ ( z )d z + (cid:90) M K P ( x, z ) K Q ( z, y ) (1 − ψ ( z )) d z. (1.89)The second term in the right hand side of (1.89) is a O (exp( − Ch /s )) in G s becauseit only involves off diagonal parts off K P and K Q . In order to deal with the firstterm in (1.89), we apply (1.52) and (1.88) (ignoring the remainder terms as we .3. GEVREY PSEUDO-DIFFERENTIAL OPERATORS 71 announced it) to write the oscillatory integral (cid:90) D (cid:48) K P ( x, z ) K Q ( z, y ) ψ ( z )d z = 1(2 πh ) n (cid:90) R n × R n × R n e i (cid:104) x − z,ξ (cid:105) + (cid:104) z − y,η (cid:105) h p ( x, ξ ) ψ ( z ) q ( η, y )d ξ d z d η = 1(2 πh ) n (cid:90) R n e i (cid:104) x − y,ξ (cid:105) h p ( x, ξ ) q ( ξ, y )d ξ + 1(2 πh ) n (cid:90) R n × R n × R n e i (cid:104) x − z,ξ (cid:105) + (cid:104) z − y,η (cid:105) h p ( x, ξ ) ( ψ ( z ) − q ( η, y )d ξ d z d η. (1.90)Here, we recognized a Fourier inversion formula. The first term in the right handside of (1.90) is of the form prescribed by (ii) in Definition 1.6 thanks to Lemma1.21. Hence, we only need to prove that the second term in the right hand side of(1.90) is an O (exp( − / ( Ch ) /s )) in G s . To do so, notice that the factor 1 − ψ ( z )in the integrand ensures that we only integrate over z ’s that are uniformly awayfrom x and y . Hence, the phase is non-stationary in η and ξ , and we may shift thecontour in η and ξ to prove that the integrable is negligible (as we did in the proofsof Lemmas 1.18 and 1.19, we do not repeat the details). Notice that we do not needto shift contour in z , so that it does not matter that ψ is only C ∞ . (cid:3) Now, we use Proposition 1.12 to construct a principal symbol for a G pseudorand prove Proposition 1.10. Proof of Proposition 1.10. When s > 1, the existence of Gevrey partitionsof unity make the proof an easy consequence of Lemmas 1.18 and 1.23 (see alsoRemark 1.27). This is a quite standard procedure and thus we do not detail it (seefor instance [ Zwo12 , Theorem 14.1] for the C ∞ version). Consequently, we turnnow to the case s = 1 which relies on an application of Proposition 1.12.We can find a phase function Φ, defined near the diagonal of ( T ∗ M ) (cid:15) × ( M ) (cid:15) so that on the reals Im Φ ≥ 0, andΦ( α, x ) = (cid:104) x − α x , α ξ (cid:105) + O ( (cid:104)| α |(cid:105)| α x − x | ) , and for real α, x , we have Im Φ( α, x ) ≥ (cid:104)| α |(cid:105)| α x − x | . Then, using the ∂ trick(Lemma 1.10), we can find a G family of coherent states of order 1 with phase Φand amplitude 1, denoted u α .Let P ∈ G Ψ m . According to the previous proposition, P u α is another G family of coherent states, and this implies that p ( α ) := P u α ( α x )defines an element of S ,m ( T ∗ M ). Since d x Φ( α, α x ) = α ξ , then (1.82) implies that p is a G principal symbol for P . (cid:3) Finally, we use the notion of pseudor with non-standard phase that we introducedin § Proof of Proposition 1.11. Thanks to Lemma 1.23, the case s > C ∞ case: there are G s partitions of unity. In order to give aconstruction for the case s = 1, choose an admissible phase Φ ∗ on M , and definethe phase Φ by Φ( x, α ) = − Φ ∗ (¯ α, ¯ x ) . We can find a kernel k ( x, α ) holomorphic on (cid:0) M × T ∗ M (cid:1) (cid:15) such that when α x and x are close k ( x, α ) = e i Φ( x,α ) h + O L ∞ (cid:18) exp (cid:18) − (cid:104)| α |(cid:105) Ch (cid:19)(cid:19) (1.91)and k ( x, α ) is exponentially small if α x and x are away from each other. This followsfrom an application of the ∂ trick (Lemma 1.10). Taking p a G symbol, one maycheck that K P ( x, α x ) := 1(2 πh ) n (cid:90) T ∗ αx M k ( x, α ) p ( α )d α ξ is a well defined as an oscillatory integral. It is a distribution on M × M . It is O (exp( − /Ch )) in G away from the diagonal. Moreover, near the diagonal wemay apply the Kuranishi trick (Lemma 1.24) in local coordinates to see that K P is the kernel of a G pseudor P with principal symbol p (notice that our definitionof Φ ensures that we retrieve an admissible phase in local coordinates, because wechanged the sign but we also switched variables). We may then set Op( p ) = P andthe proof is over. (cid:3) Remark . Notice that the principal symbol of the G pseudor with kernel K Φ ,a defined by (1.78) is a ( x, ξ, x ). However, the Op we have constructed does notsatisfy the usual relations Op(1) = 1. It was the price to pay to be able to build G pseudors. This could be worked around, but it is not necessary for our purposes. When restricting to G pseudors, we canobtain stronger statements, in particular when they are elliptic. In this section weprove Theorem 4 and 5. A parametrix construction. We study now the construction of a parametrix foran elliptic G pseudor A . We will start by focusing on the construction of the formalsymbol of the parametrix of A in the Euclidean case. We will then use this localconstruction to prove Theorem 4.Notice that given formal analytic symbols a = (cid:80) k ≥ h k a k and b = (cid:80) k ≥ h k b k ,we can always define the formal symbol of the composition a b by a b = (cid:88) k ≥ (cid:88) | α | + (cid:96) + m = k h k α ! ∂ αξ a (cid:96) ∂ αx b m . It is easy to show that it defines in fact a formal analytic symbol. Let a and b berealizations respectively of a and b (in the sense of Definition 1.3). Let A and B are G pseudors given in local coordinates by the the formula (1.52) with symbolsrespectively a and b . We know by Proposition 1.7 that C = AB is a G pseudor, sothat it is given in local coordinates by the formula (1.52) with a certain symbol c .Then, recall Remark 1.27, the symbol c is obtained in the proof of Proposition 1.7by an application of Lemma 1.21, and, consequently, c is a realization of the formalsymbol a b . The first step in a parametrix construction is hence the following: Lemma . Let U be an open subset of R n and let m ∈ R . Let the formalsymbol a = (cid:80) k ≥ h k a k ∈ F S ,m (cid:0) T ∗ U (cid:1) be elliptic in the sense that a satisfies (1.53) in any relatively compact subset of U (with a constant C that may depend onthe subset). If Ω is a relatively compact open subset of U , then there is a formalanalytic symbol b of order − m on Ω such that b a = 1 . The usual C ∞ parametrix construction gives a candidate for b , built by induction,and implies that there can be at most one solution to this problem. However, it isnot clear then that the formal symbol b that we obtain is indeed a formal analytic .3. GEVREY PSEUDO-DIFFERENTIAL OPERATORS 73 symbol. To do so, we could probably work as in [ BDMK67 ], but we rather adaptthe proof of [ Sj¨o82 , Theorem 1.5]. Since we only want to establish the regularity ofthe solution, we may assume that Ω is a ball.With a formal analytic symbol a = (cid:80) k ≥ h k a k ∈ F S ,m (cid:0) T ∗ U (cid:1) , we associatethe formal sum of differential operators F ( a ) = (cid:88) k ≥ h k A k , where A k is defined by the finite sum A k = (cid:88) (cid:96) + | α | = k i | α | α ! ∂ αξ a (cid:96) ( x, ξ ) ∂ αx . (1.92)If b = (cid:80) k ≥ h k b k belongs to F S ,m (cid:0) T ∗ U (cid:1) , we may define similarly a formal sum F ( b ) = (cid:80) k ≥ h k B k of differential operator. Then, it is natural to define thecomposition F ( a ) ◦ F ( b ) by F ( a ) ◦ F ( b ) = (cid:88) k ≥ h k (cid:88) k + k = k A k ◦ B k . (1.93)Notice that with this convention we have that F ( a ) ◦ F ( b ) = F ( a b ). Moreover,we may retrieve the formal symbol a from the operator F ( a ) by noticing that a k = A k (1). Hence, the associativity of ◦ implies the associativity of (cid:15) > t ∈ [0 , (cid:15) ] and λ ∈ N the open subset V t,λ of (cid:0) T ∗ U (cid:1) (cid:15) by V t, = (cid:110) ( x, ξ ) ∈ (cid:0) T ∗ U (cid:1) (cid:15) : d ( x, Ω) < t and | Re ξ | < (cid:111) and for λ ≥ V t,λ = (cid:110) ( x, ξ ) ∈ (cid:0) T ∗ U (cid:1) (cid:15) : d ( x, Ω) < t and 2 λ − < | Re ξ | < λ +1 (cid:111) . Then, we denote by B (cid:0) V t,λ (cid:1) the Banach space of bounded holomorphic functionson V t,λ . It follows then by Cauchy’s formula that there are constants C, R > s < t ∈ [0 , (cid:15) ] , λ ∈ N and α ∈ N d the norm of ∂ αx as an operator from B (cid:0) V t,λ (cid:1) to B (cid:0) V s,λ (cid:1) satisfies (cid:107) ∂ αx (cid:107) t,s,λ ≤ CR | α | α ! ( t − s ) −| α | . Now, using Cauchy’s formula again and recalling that a is a formal analytic symbol,we see that, provided that (cid:15) is small enough, there are constants C, R > (cid:96), λ ∈ N and α ∈ N d the function ∂ αξ a (cid:96) is bounded by CR | α | + (cid:96) α ! (cid:96) !2 λ ( m − (cid:96) −| α | ) .Hence, up to taking C, R > A k , defined by (1.92), as anoperator from B ( V t,λ ) to B (cid:0) V s,λ (cid:1) satisfies (cid:107) A k (cid:107) t,s,λ ≤ CR k k !2 λ ( m − k ) ( t − s ) − k , for all k, λ ∈ N and 0 ≤ s < t ≤ (cid:15) . Define then for every k ∈ N the quantity (theindex m reminds us that a has order m ) f k,m ( a ) = sup ≤ s Lemma . Let a and b be formal sums of symbols as above. Assume thatthere are m , m ∈ R and ρ > such that (cid:107) a (cid:107) m ,ρ and (cid:107) b (cid:107) m ,ρ are finite. Then (cid:107) a b (cid:107) m + m ,ρ is finite and (cid:107) a b (cid:107) m + m ,ρ ≤ (cid:107) a (cid:107) m ,ρ (cid:107) b (cid:107) m ,ρ . Proof. To do so, we only need to prove that for all k ∈ N we have f k,m + m ( a b ) ≤ (cid:88) k + k = k f k ,m ( a ) f k ,m ( b ) . (1.96)We recall that F ( a b ) = F ( a ) ◦ F ( b ) is given by the formula (1.93). But then, if0 ≤ s < t ≤ (cid:15) and λ ∈ N , we have (with k + k = k and any s < u < t ) (cid:13)(cid:13) A k ◦ B k (cid:13)(cid:13) t,s,λ ≤ (cid:13)(cid:13) A k (cid:13)(cid:13) t,u,λ (cid:13)(cid:13) B k (cid:13)(cid:13) u,s,λ ≤ k k λ ( m − k ) ( t − u ) − k f k ,m ( a ) k k λ ( m − k ) ( u − s ) − k f k ,m ( b ) ≤ k k k k ( t − u ) − k ( u − s ) − k λ ( m + m − k ) f k ,m ( a ) f k ,m ( b ) . Then, we may choose u such that t − u = ( t − s ) k k + k and u − s = ( t − s ) k k + k tofind that (cid:13)(cid:13) A k ◦ B k (cid:13)(cid:13) t,s,λ ≤ k k ( t − s ) − k λ ( m + m − k ) f k ,m ( a ) f k ,m ( b ) . From this, (1.96) follows, and the proof of the lemma is thus over. (cid:3) We can now prove Lemma 1.25. Proof of Lemma 1.25. Recall that a is supposed to be an elliptic symbol oforder m . Hence, if we define b = a then b is an analytic symbol of order − m .Thus, for some ρ > (cid:107) a (cid:107) m,ρ < + ∞ and (cid:107) b (cid:107) − m,ρ < + ∞ . Then, bysymbolic calculus we have b a = 1 − r, where r is a formal analytic symbol of order 0, with no leading coefficient (thatis r is h times a formal analytic symbol of order − < ρ ≤ ρ , (cid:107) r (cid:107) ,ρ ≤ ρρ (cid:107) − b a (cid:107) ,ρ ≤ ρρ (cid:0) (cid:107) a (cid:107) m,ρ (cid:107) b (cid:107) − m,ρ (cid:1) , so that (cid:107) r (cid:107) ,ρ < ρ is small enough. Thus, we want to use Von Neumann’sargument to say that b = (cid:80) k ≥ r k b is the symbol of a parametrix for a .Applying Lemma 1.26, we see that for every (cid:96) ∈ N (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:96) (cid:88) p =0 r p b (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − m,ρ ≤ (cid:107) b (cid:107) − m,ρ − (cid:107) r (cid:107) ,ρ . .3. GEVREY PSEUDO-DIFFERENTIAL OPERATORS 75 Since r has no leading coefficient, if (cid:96) is large enough (depending on k ), then f k, − m ( b ) = f k, − m (cid:16)(cid:80) (cid:96)p =0 r p b (cid:17) . In particular, for all N ≥ 0, for (cid:96) N largeenough, N (cid:88) k =0 ρ k f k, − m ( b ) = N (cid:88) k =0 ρ k f k, − m (cid:32) (cid:96) N (cid:88) p =0 r p b (cid:33) ≤ (cid:107) b (cid:107) − m,ρ − (cid:107) r (cid:107) ,ρ . Letting N → + ∞ , we deduce (cid:107) b (cid:107) − m,ρ ≤ (cid:107) b (cid:107) − m,ρ − (cid:107) r (cid:107) ,ρ < + ∞ , so that b is a formal analytic symbol for the reason described above. (cid:3) Remark . It will be useful to obtain a version with parameters of Lemma1.25. We pick a symbol a ∈ S ,m ( T ∗ U ), elliptic of order m > 0. We assume thatthere is V ⊆ C , an open set, conical at infinity, so that V ∩ a ( T ∗ R n ) = ∅ . For (cid:15) > W = { z ∈ C : d ( z, C \ V ) > (cid:15) | z |} , then, for some δ > x, ξ ) ∈ ( T ∗ R n ) (cid:15) , we have | a ( x, ξ ) − s | > (cid:15) | a ( x, ξ ) | for all s ∈ W . Then for s ∈ W , the family s (cid:55)→ a − s is a bounded family of G symbols of order − m uniformly elliptic. In particular,denoting by b ( s ) the formal symbol such that b ( s ) a − s ) = 1, we see that b ( s ) is abounded family of formal symbols when s varies in W . Moreover, from Remark 1.18and Lemma 1.7, we see that we may choose for s ∈ W a realization b ( s ) of b ( s ), sothat b ( s ) defines a bounded family of analytic symbol with measurable dependenceon s .We are now ready for the: Proof of Theorem 4. We only construct a left parametrix for P . The con-struction of a right parametrix is similar, and then a standard argument ensuresthat the left and right parametrix coincide.By compactness of M , we may cover M by a finite number of balls (cid:0) D j (cid:1) j ∈ J such that each D j is a relatively compact subset a coordinate patch U j . Since P isan elliptic G pseudor, for each j ∈ J there is a symbol a j , elliptic of order m on U j such that in the coordinates on U j the kernel of P is K j ( x, y ) = 1(2 πh ) n (cid:90) R n e i (cid:104) x − y,ξ (cid:105) h a j ( x, ξ )d ξ + O G (cid:18) exp (cid:18) − Ch (cid:19)(cid:19) . We apply Lemma 1.25 to a j to find a formal symbol b j such that b j a j = 1. Then,choose a realization b j for b j , and define for x, y in a neighbourhood of D j , Q j ( x, y ) := 1(2 πh ) n (cid:90) R n e i (cid:104) x − y,ξ (cid:105) h b j ( x, ξ )d ξ. (this in the coordinate charts of U j ). Let us assume that D j ∩ D (cid:96) (cid:54) = ∅ , and consider D j ∩ D (cid:96) ⊂ D ⊂ D (cid:48) ⊂ U j ∩ U (cid:96) open balls relatively compact one in another. Thanks to the definition of b j , and since away from the diagonal the kernel of G pseudorsare O (exp( − / ( Ch ))) in G , we have in D × D ( Q j − Q (cid:96) )( x, y )= (cid:90) D (cid:48) × D (cid:48) ( Q j − Q (cid:96) )( x, z ) P ( z, z (cid:48) ) Q j ( z (cid:48) , y )d z d z (cid:48) + O G (cid:18) exp (cid:18) − Ch (cid:19)(cid:19) , = O G (cid:18) exp (cid:18) − Ch (cid:19)(cid:19) . Indeed, we can compute the kernel of this composition as in the proof of Proposition1.7, recalling that the symbols that appear in this computation have the usualasymptotic expansions (see Remark 1.27), so that our choice of symbols for Q j and Q (cid:96) ensure that they are local parametrices for P (up to a negligible kernel). Hence,the distribution g j,(cid:96) ( x, y ) := ( Q j − Q (cid:96) )( x, y ), for j, (cid:96) ∈ J , is in fact an analyticfunction. Moreover, g j,(cid:96) has a holomorphic extension to a complex neighbourhoodof D j ∩ D (cid:96) which is an O (exp( − / ( Ch ))). It follows from the analytic continuationprinciple that, wherever it makes sense, these holomorphic extensions satisfy Q j ( x, y ) = Q (cid:96) ( x, y ) + g j,(cid:96) ( x, y ) and g j,k ( x, y ) = g j,(cid:96) ( x, y ) + g (cid:96),k ( x, y ) . (1.97)For j ∈ J , we choose a complex neighbourhood W j ⊆ (cid:102) M of D j small enough so that g j,(cid:96) is well-defined on (cid:0) W j ∩ W (cid:96) (cid:1) for (cid:96) ∈ J . Then, we pick U ⊆ (cid:102) M × (cid:102) M an openset such that U does not encounter the diagonal. Up to taking the complexification (cid:102) M of M smaller, we may assume that U ∪ { W j × W j | j ∈ J } is an open cover of (cid:102) M × (cid:102) M . Then, we pick a corresponding C ∞ partition of unity (( χ j ) j ∈ J , χ U ). As afirst approximation for the kernel of the parametrix Q , we define the distribution Q ( x, y ) = (cid:88) j χ j ( x, y ) Q j ( x, y )on M × M . Of course, due to the use of a C ∞ partition of unity, it is very unlikelyfor Q ( x, y ) to be the kernel of G pseudor. We will use a ¯ ∂ trick to correct that.For (cid:15) > f of type (0 , 1) on ( M × M ) (cid:15) by f ( x, y ) = − Q j ( x, y ) ¯ ∂χ U ( x, y ) − (cid:80) (cid:96) ∈ J g (cid:96),j ( x, y ) ¯ ∂χ (cid:96) ( x, y ) if ( x, y ) ∈ W j × W j x, y ) / ∈ (cid:83) j ∈ J W j × W j . Here, the holomorphic extension of Q j ( x, y ) away from the diagonal is given byLemma 1.18 and if one of the g (cid:96),j does not makes sense then the ¯ ∂ in factor vanishesso that the sum is well-defined. The fact that f is well-defined despite the possibleintersection between cases in its definition comes from (1.97) (use also the fact that¯ ∂ (cid:16) χ U + (cid:80) j ∈ J χ j (cid:17) = 0). From the estimate on the g (cid:96),j and Lemma 1.18, we seethat f is an O (exp( − /Ch )) on ( M × M ) (cid:15) .It follows then from Lemma 1.10 that there are (cid:15) > r on ( T ∗ M ) (cid:15) which is an O (exp( − /Ch )) and such that ¯ ∂r = f . We use r to definea new approximation of the kernel of our parametrix: Q ( x, y ) = Q ( x, y ) + r ( x, y ) . Notice that for x, y in a neighbourhood of D j , we have Q ( x, y ) − Q j ( x, y ) = Q j ( x, y ) χ U ( x, y ) + (cid:88) (cid:96) ∈ J χ (cid:96) ( x, y ) g (cid:96),j ( x, y ) + r ( x, y ) . (1.98) .3. GEVREY PSEUDO-DIFFERENTIAL OPERATORS 77 The right hand side of (1.98) has a smooth extension to W j × W j which is holomorphic(because ¯ ∂r = f ) and an O (exp( − /Ch )) (due to Lemma 1.18 and the estimates onthe g (cid:96),j ’s and on r ). We just proved that Q ( x, y ) = Q j ( x, y ) + O G (cid:18) exp (cid:18) − Ch (cid:19)(cid:19) (1.99)on a neighbourhood of D j × D j . Working similarly, we see that Q ( x, y ) is an O (exp( − /Ch )) in G away from the diagonal. Thus, Q ( x, y ) is the kernel of a G pseudor that we also denote by Q . From (1.99) and the definition of the Q j ’s, wesee that Q P = 1 + O G (cid:18) exp (cid:18) − Ch (cid:19)(cid:19) . The right hand side here takes the form 1 + R , with (cid:107) R (cid:107) < E ,R ) (cid:48) to E ,R (for some large R and provided h is small enough). In thatspace, we obtain the 1 + R is invertible, with inverse 1 + R (cid:48) , so that R (cid:48) = − R − R (1 + R (cid:48) ) R. From this formula, we deduce that R (cid:48) has kernel O G (exp( − /Ch )), so that Q =(1 + R (cid:48) ) Q is a G pseudor (due to Proposition 1.7), inverse of P . (cid:3) Functional calculus for analytic pseudors. This last section is dedicated tothe proof of Theorem 5. Consequently, we denote by A a self-adjoint G pseudor,classically elliptic of order m > s, + ∞ [.Let us start a few reductions. First of all, up to adding a constant to A , wemay assume that s = 0. Then, for any integer (cid:96) ∈ N , we may write f ( A ) = (1 + A ) (cid:96) (1 + A ) − (cid:96) f ( A ) , hence, we may replace the function f by the function z (cid:55)→ f ( z )(1+ z ) (cid:96) , and assumethat N < − − nm . Let us state two lemmas before proving Theorem 5. Recallthat A admits a G principal symbol a in the sense of Proposition 1.10. Since A isself-adjoint, we may assume that a is real-valued (up to replace it by ( a + ¯ a ) / Lemma . Let δ > . For h small enough, a is valued in [ − δ, + ∞ [ . Proof. The proof is by contradiction. During this proof, we will write a = a h and A = A h to make the dependence on h clearer. Assume that there is a sequence( h k ) k ∈ N converging to 0 and a sequence of points α k ∈ T ∗ M such that for every k ∈ N we have a h k ( α k ) < − δ . Since we assumed that A h is classically elliptic, thereis a constant C > α ∈ T ∗ M and h > | a h ( α ) | ≥ (cid:104) α (cid:105) m C − C. Hence, there is a constant B > | a h ( α ) | ≥ (cid:104) α (cid:105) ≥ B . Consequently,the sign of a h is constant on the connected components of (cid:8) α ∈ T ∗ M : (cid:104) α (cid:105) ≥ B (cid:9) .This sign has to be positive, otherwise it would follow from the Weyl’s law that A has negative eigenvalues.It follows that the α k ’s remain in a compact subset of T ∗ M . Hence, upto extracting a subsequence, we may assume that ( α k ) k ∈ N converges to some α ∞ = (cid:0) α ∞ ,x , α ∞ ,ξ (cid:1) ∈ T ∗ M . By uniform continuity of the a h k ’s we see that there isa neighbourhood V of α ∞ in T ∗ M such that for k large enough and α ∈ V we have a h k ( α ) < − δ . Now let ϕ be a C ∞ function from M to R such that d ϕ ( α ∞ ,x ) = α ∞ . Then, we choose a C ∞ function χ from M to R such that χ ( α ∞ ,x ) = 1 and d ϕ ( x ) ∈ V for every x in the support of χ . By the ( C ∞ ) stationary phase method, we find that A h k (cid:16) χe i ϕhk (cid:17) = k → + ∞ a h k (d ϕ ) χ + O C ∞ ( h k ) . However, for k large enough, the spectrum of A h k is by assumption contained in R + and thus0 ≤ (cid:68) A h k (cid:16) χe i ϕhk (cid:17) , χe i ϕhk (cid:69) = k → + ∞ (cid:90) M a h k (d ϕ ( x )) | χ ( x ) | d x + O ( h k ) ≤ − δ (cid:90) M | χ ( x ) | d x, where the last line holds for k large enough and is absurd. (cid:3) Lemma . Let (cid:15) ∈ ]0 , (cid:15) [ . For h small enough, we have f ( A ) = 12 iπ (cid:90) ∂U ,(cid:15) f ( w ) ( w − A ) − d w, (1.100) where the integral converges in L operator norm for instance (recall that we reducedto the case N < − − nm ). Proof. Let ( λ k ) k ∈ N denotes the sequence of eigenvalues of A (ordered increas-ingly and counted with multiplicity) and ( ψ k ) k ∈ N denotes an associated orthonormalsystems of eigenvectors. By a ( C ∞ ) parametrix construction, we see that the re-solvent of A is bounded from L to the Sobolev space H m , hence we have (it alsofollows from Weyl’s law) (cid:88) k ∈ N | λ k | nm < + ∞ . The operator f ( A ) is defined to be f ( A ) = (cid:88) k ≥ f ( λ k ) (cid:104)· , ψ k (cid:105) ψ k , and, since N ≤ − − nm , we see that this sum converges in the L operator norm.Now, notice that (cid:88) k ≥ Leb (cid:18)(cid:20) λ k − λ − ( nm ) k , λ k + λ − ( nm ) k (cid:21)(cid:19) < + ∞ , hence, since Leb ( R + ) = + ∞ but the Lebesgue measure of any compact subset of R + is finite, we may find a sequence ( r (cid:96) ) (cid:96) ∈ N of positive real numbers that tends to+ ∞ , such that for every natural integers k, (cid:96) we have | r (cid:96) − λ k | ≥ λ − ( nm ) k . It follows that for (cid:96) large enough and all k ∈ N we have | r (cid:96) − λ k | ≥ r − ( nm ) (cid:96) . (1.101)Now, set for (cid:96) ∈ N V (cid:96) = U ,(cid:15) ∩ { z ∈ C : Re z < r (cid:96) } , I (cid:96) = U ,(cid:15) ∩ { z ∈ C : Re z = r (cid:96) } , and J (cid:96) = { z ∈ C : Re z ≤ r l } ∩ ∂U ,(cid:15) . .3. GEVREY PSEUDO-DIFFERENTIAL OPERATORS 79 Then, for (cid:96) large, we write Cauchy’s formula (cid:88) k ≥ λ k ≤ r (cid:96) f ( λ k ) (cid:104)· , ψ k (cid:105) ψ k = 12 iπ (cid:90) ∂V (cid:96) f ( w ) ( w − A ) − d w = 12 iπ (cid:90) J (cid:96) f ( w ) ( w − A ) − d w + 12 iπ (cid:90) I (cid:96) f ( w ) ( w − A ) − d w. (1.102)Using the bound (cid:107) ( w − A ) − (cid:107) L → L ≤ | Im w | − , we see that ( w − A ) − is uniformlybounded in the L operator norm on ∂U ,(cid:15) . Hence, since N < − 1, the integral inthe right hand side of (1.100) converges. Moreover, we see that the integral over J (cid:96) in (1.102) tends to the right hand side of (1.100) when (cid:96) tends to + ∞ . Consequently,we only need to prove that the integral over I (cid:96) in (1.102) tends to 0 when (cid:96) tendsto + ∞ to end the proof of the lemma.To do so, notice that, for (cid:96) large enough, when w ∈ I (cid:96) then, thanks to (1.101),the distance between w and the spectrum of A is greater than r − (2+ n/m ) (cid:96) , so thatwe have (cid:107) ( w − A ) − (cid:107) L → L ≤ r n/m(cid:96) . Then, using the decay assumption on f andthe fact that the length of I (cid:96) is a O ( r (cid:96) ), we find that the integral over I (cid:96) in (1.102)is a O ( r N +3+ n/m(cid:96) ). Hence, this integral tends to 0 since we reduced to the case N < − − n/m . (cid:3) We are now ready to prove Theorem 5. Proof of Theorem 5. We want to apply Lemma 1.28 with (cid:15) = (cid:15) . To doso, we need to check that for w ∈ ∂U , (cid:15)/ , the operator ( w − A ) − is a semi-classical G pseudor with principal symbol ( w − a ) − . Recall Remark 1.29 and apply Lemma1.27 with δ = (cid:15) to see that w − a for w ∈ ∂U , (cid:15)/ is uniformly elliptic. We maythen apply Theorem 4 to see that ( w − A ) − is a G pseudor.We explain now why Definition 1.6 remains valid after averaging over ∂U , (cid:15)/ .We prove that f ( A ) satisfies (ii) in Definition 1.6 (the point (i) is easier to check).Since ( w − A ) − for w ∈ ∂U , (cid:15)/ is a G pseudor, its kernel may be written in localcoordinates: ( w − A ) − ( x, y ) = 1(2 πh ) n (cid:90) R n e i (cid:104) x − y,ξ (cid:105) h p w ( x, ξ )d ξ + R w ( x, y ) , (1.103)where p w is an analytic symbol and R w is an O (exp( − /Ch )) in G . Moreover, itfollows from Remark 1.29 (see also Remark 1.18) and the proof of Theorem 4 that p w and R w satisfies uniform bound in w and have a measurable dependence on w .Consequently, we may average (1.103) over ∂U , (cid:15)/ and find with Lemma 1.28 andFubini’s Theorem that (the decay assumption on f ensures the integrability) f ( A )( x, y ) = 1(2 πh ) n (cid:90) R n e i (cid:104) x − y,ξ (cid:105) h (cid:32)(cid:90) ∂U , (cid:15)/ f ( w ) p w ( x, ξ )d w (cid:33)(cid:124) (cid:123)(cid:122) (cid:125) := q ( x,ξ ) d ξ + (cid:90) ∂U , (cid:15)/ f ( w ) R w ( x, y )d w (cid:124) (cid:123)(cid:122) (cid:125) := R ( x,y ) . (1.104)Due to the uniform bounds on p w and R w , we see that q is an analytic symboland that R is an O (exp( − /Ch )) in G . Here, we applied Fubini’s Theorem in an oscillating integral, this is made rigorous by integrating against test functions anddoing a finite number of integration by parts. Finally, notice that q is given at firstorder by f ( a ) since p w is given at first order by ( w − a ) − . (cid:3) HAPTER 2 FBI transform on compact manifolds The purpose of this chapter is to present some features of FBI transforms in theGevrey classes. We consider fixed ( M, g ) a compact analytic Riemannian manifoldwithout boundary. In § T ∗ M , introduce the so-called I-Lagrangians, and the correspondingI-Lagrangian (functional) spaces. We also study the relation between the regularityof distributions and the decay of their FBI transforms.In the next section § h -pseudors on I-Lagrangianspaces. The most technical and crucial arguments of the paper are contained in § § § M .For the convenience of the reader, we start with a summary of the technicalresults. Results: I-Lagrangian deformations and Gevrey pseudors. We recallhere some definitions and notations that are exposed in greater detail in § M by d vol g , or just d x as we will take chartswith Jacobian identically equal to 1. The cotangent space T ∗ M will be endowedwith the corresponding (analytic) Kohn–Nirenberg metric g KN . In local charts, itis equivalent to g flat KN = d x + d ξ (cid:104) ξ (cid:105) . Given an analytic manifold X , we will denote by ( X ) (cid:15) its (cid:15) -Grauert tube (see § X . In thecase of T ∗ M , the tube ( T ∗ M ) (cid:15) is an asymptotically conical complex neighbourhood.It will be convenient to introduce for α ∈ ( T ∗ M ) (cid:15) the Japanese brackets (cid:104) α (cid:105) and (cid:104)| α |(cid:105) defined by (1.6) in § |(cid:104) α (cid:105)| , Re (cid:104) α (cid:105) and (cid:104)| α |(cid:105) are comparable on ( T ∗ M ) (cid:15) for (cid:15) > h > 0. Unless specified, all the estimates are supposed to be uniform in h .We recall that an admissible phase in the sense of Definition 1.7, is a holomorphicsymbol Φ( α, x ) of order 1 on T ∗ M × M , defined for x close to α x , so thatΦ( α, x ) = (cid:104) α x − x, α ξ (cid:105) + O ( (cid:104)| α |(cid:105)| α x − x | ) , Im Φ ≥ C (cid:104)| α |(cid:105)| α x − x | . Here, we recall that we write α = ( α x , α ξ ) for α ∈ T ∗ M .In general, an FBI transform T (on M ) is a linear map between D (cid:48) ( M ) and C ∞ (( T ∗ M )). Away from the “diagonal” { x = α x } , its kernel is assumed to decayat a sufficient rate (which depends on the context), and near the diagonal, (possiblymodulo a remainder) it takes the form e i Φ T ( α,x ) h a ( α, x ) , where a is elliptic in some symbol class, and Φ T is an admissible phase. 812 2. FBI TRANSFORM ON COMPACT MANIFOLDS Definition . An analytic FBI transform is an FBI transform T such thatfor some C, (cid:15) , (cid:15) , η > 0, the kernel K T of T is holomorphic in ( M × T ∗ M ) (cid:15) andfor ( x, α ) therein satisfies(i) for d ( x, α x ) > (cid:15) , we have | K T ( α, x ) | ≤ Ce − η (cid:104)| α |(cid:105) h ;(ii) for d ( x, α x ) ≤ (cid:15) ,(2.1) (cid:12)(cid:12)(cid:12)(cid:12) K T ( α, x ) − e i Φ T ( α,x ) h a ( α, x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ Ce − η (cid:104)| α |(cid:105) h ;Φ T being an admissible phase (as defined in Definition 1.7), and a being a semi-classical analytic symbol, elliptic in the symbol class h − n/ S ,n/ (with the subtletythat a ( α, x ) is maybe only defined for α x and x close to each other).An adjoint analytic FBI transform is an operator S : C ∞ c ( T ∗ M ) → C ∞ ( M )whose kernel K S ( x, β ) satisfies that ( α, x ) (cid:55)→ K S ( x, α ) is the kernel of an analyticFBI transform. Remark . While it does not appear in the notation, the symbol a is allowedto depend on the small implicit parameter h > h ). We say that Φ T is the phase of T and that a is its symbol. Theclass of analytic symbol S ,n/ is defined in Definition 1.4.If S is an adjoint analytic FBI transform, we say that Φ S ( x, α ) and b ( x, α ) arerespectively the phase and the symbol of S if ( α, x ) (cid:55)→ − Φ T ( x, α ) and ( α, x ) (cid:55)→ b ( x, α ) are the phase and the symbol of the analytic FBI transform with kernel( α, x ) (cid:55)→ K S ( x, α ).Recall that the fact that K T is the kernel of T means that if u is a smoothfunction on M then T u is defined by the formula T u ( α ) = (cid:90) M K T ( α, x ) u ( x )d x. (2.2)Notice that if T is an analytic FBI transform, its adjoint T ∗ with respect to the L spaces on M and T ∗ M is an adjoint analytic FBI transform, since the kernel of T ∗ is given by ( x, α ) (cid:55)→ K T ( x, α ) (the cotangent bundle T ∗ M is endowed with itscanonical volume form). Remark . If S is an adjoint analytic FBI transform and P is a G order 0elliptic h -pseudor, it follows from Proposition 1.12 in § P S is still an adjointanalytic FBI transform. In fact, if P is a general G h -pseudor (not necessarilyelliptic of order 0), then P S still satisfies all the points in the definition of an adjointanalytic FBI transform but one: the symbol may not be elliptic in the good symbolclass.Similarly, if T is an analytic FBI transform and P a G h -pseudor, then we maywrite T P = (cid:0) P ∗ T ∗ (cid:1) ∗ . We know that T ∗ is an adjoint analytic FBI transform andthat P ∗ is a G h -pseudor (according to Proposition 1.8). Hence, the discussionabove implies that T P satisfies the conditions of Definition 2.1, except that thesymbol a may not be in the good symbol class, and is not necessarily elliptic. Remark . Notice that if T is an analytic FBI transform, then it satisfiesDefinition 2.2 with arbitrarily small (cid:15) , up to taking smaller η and (cid:15) (with (cid:15) (cid:28) (cid:15) )and larger C . This elementary fact will be useful later. It implies that, while T is aglobal object, its most relevant properties are local.Our first result, which is an extension of claims (1.10)-(1.12) in [ Sj¨o96a ], isthe following (its proof may be found in § a priori that thecomposition of an analytic FBI transform and an adjoint analytic FBI transformmakes sense, we will see in Proposition 2.3 how to define this composition. . FBI TRANSFORM ON COMPACT MANIFOLDS 83 Theorem . There exists an analytic FBI transform T , and h , R > suchthat for < h < h , the composition T ∗ T makes sense and is the identity operatoron ( E ,R ) (cid:48) . The space ( E ,R ) (cid:48) that appears in Theorem 6 is defined precisely in § R ”. Thedifference with the transform in [ Sj¨o96a ] is that our transform is globally analyticinstead of only near the diagonal, so that it acts on ( E ,R ) (cid:48) instead of just usualdistributions.We will from now on take T to be an analytic FBI transform given by Theorem6, and denote S = T ∗ . Since its kernel has a holomorphic extension to a complexneighbourhood of M × T ∗ M , for some hyperfunction u , it makes sense to considerholomorphic extensions of T u to (cid:0) T ∗ M (cid:1) (cid:15) . We will prove (Proposition 2.1) that if u is G s for some s ≥ 1, when α is at distance (cid:46) C − h − /s (cid:104)| α |(cid:105) /s from T ∗ M , with C > u , | T u ( α ) | ≤ C exp (cid:32) − (cid:18) (cid:104)| α |(cid:105) Ch (cid:19) s (cid:33) when (cid:104)| α |(cid:105) is large enough.The relation between the FBI transform and the symplectic geometry on T ∗ M is quite deep. The canonical symplectic form ω = d ξ ∧ d x extends to ( T ∗ M ) (cid:15) as acomplex symplectic form. Its imaginary part ω I = Im ω is a real symplectic form.Following Sj¨ostrand, we say that a submanifold Λ ⊂ ( T ∗ M ) (cid:15) is I-Lagrangian if itis Lagrangian for ω I . Given a real-valued function G , we will denote by H ω I G itsHamiltonian vector field with respect to ω I , in the sense that d G ( · ) = ω I ( · , H ω I G ).To be able to use almost analytic extensions of Gevrey symbols defined on T ∗ M , we will have to restrict our attention to families of Lagrangian submanifolds(Λ h ) h> such that as h → • the Λ h ’s are uniformly close to T ∗ M in C ∞ (for the Kohn–Nirenbergmetric), • in C topology, Λ h is at distance ≤ τ h − /s (cid:104)| α |(cid:105) /s from T ∗ M (with τ small).A particular case of such families are given by Λ := e H ωIG ( T ∗ M ), where G is a symbolof order ≤ /s , so that H ω I G is O ( τ h − /s (cid:104)| α |(cid:105) /s ) (for the Kohn–Nirenberg metric).For such examples, we can find a global function H on Λ such that if θ = ξ · d x isthe holomorphic Liouville one form, d H = − Im θ | Λ . We will call such examples( τ , s ) -Gevrey adapted Lagrangians (see Definition 2.2 for a precise statement).In § G s h -pseudors, in loose terms those h -pseudors whose symbol satisfies estimates of theform | ∂ αx ∂ βξ p | ≤ CR | α | + | β | ( α ! β !) s (cid:104) ξ (cid:105) m −| β | . The main result of the present chapter is the following (see Proposition 2.11 for aslightly more general statement): Theorem . Let P be a G s pseudor of order m . Denote by p its principalsymbol, and also one of its Gevrey almost analytic extensions. Then, there exists R, τ > such that, if Λ is a ( τ , s ) -Gevrey adapted Lagrangians, then there is h > such that for < h < h and for u ∈ E ,R ( M ) (cid:90) Λ T ( P u )( α ) T u ( α ) e − H ( α ) /h d α = (cid:90) Λ ( p ( α ) + O ( h (cid:104)| α |(cid:105) m − )) | T u ( α ) | e − H ( α ) /h d α. Additionally, E ,R ( M ) is dense for all h > sufficiently small in H := (cid:26) u ∈ ( E ,R ) (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) Λ | T u | e − H/h d α < ∞ (cid:27) . The space E ,R ( M ) is a space of analytic functions with radius of convergence ∼ /R – see § H in Theorem 7is the following. Certain projectors on the image of T have a structure of FIO withcomplex phase. However, to apply the techniques of [ MS75 ], we need the imaginarypart of some phase function to be positive. It is only after the conjugation by e H/h that this property is satisfied – see Lemma 2.13 and more explanations in § Letus start by recalling a few facts on the symplectic geometry of ( T ∗ M ) (cid:15) for (cid:15) > CdS01 ]. The usual symplectic form ω = d ξ ∧ d x of T ∗ M can be extended to a complex-linear symplectic form on ( T ∗ M ) (cid:15) , still denotedby ω . The Liouville 1-form θ = ξ · d x can also be extended, so that ω = d θ still isan exact form.We let ω R = Re ω and ω I = Im ω . Notice that ω R and ω I are real symplecticforms on ( T ∗ M ) (cid:15) . In local charts with ˜ x = x + iy , ˜ ξ = ξ + iη , the expression for ω is given by ω = d ξ ∧ d x − d η ∧ d y (cid:124) (cid:123)(cid:122) (cid:125) = ω R + i (d η ∧ d x + d ξ ∧ d y ) (cid:124) (cid:123)(cid:122) (cid:125) = ω I . We can also express the Liouville 1-form: θ = ξ · d x − η · d y + i ( ξ · d y + η · d x ) . Following Sj¨ostrand, we will denote with an I objects of symplectic geometric definedthrough the use of ω I . For example, the I-Hamiltonian (i.e. w.r.t. ω I ) vector fieldof a C function f is given in the coordinates above by H ω I f = ∇ η f · ∂∂x − ∇ x f · ∂∂η + ∇ ξ f · ∂∂y − ∇ y f · ∂∂ξ = n (cid:88) j =1 ∂f∂η j ∂∂x j − ∂f∂x j ∂∂η j + ∂f∂ξ j ∂∂y j − ∂f∂y j ∂∂ξ j , (2.3)so that d f = ω I ( · , H ω I f ).One finds directly that T ∗ M is a I-Lagrangian submanifold of ( T ∗ M ) (cid:15) . Theidea of [ HS86 ] is to replace it by another I-Lagrangian submanifold. However, wewill not work with any I-Lagrangian subspace of (cid:0) T ∗ M (cid:1) (cid:15) , we will only consideradapted I-Lagrangians as we define now (we explain in Remark 2.7 how one coulddeal with slightly more general I-Lagrangians). Definition . Let s ≥ τ ≥ 0. Let Λ be an I-Lagrangian in (cid:0) T ∗ M (cid:1) (cid:15) .We say that Λ is a ( τ , s )- adapted Lagrangian if it takes the formΛ = e H ωIG (cid:0) T ∗ M (cid:1) . (2.4)Here, we assume that G is a real-valued function so that G := h − /s G is asymbol (in the usual Kohn–Nirenberg class of symbol) on (cid:0) T ∗ M (cid:1) (cid:15) of order 1 /s ,supported in some ( T ∗ M ) (cid:15) with (cid:15) < (cid:15) . Additionally, we require that (we use .1. BASIC PROPERTIES OF THE FBI TRANSFORM 85 the covariant derivatives associated to the Kohn–Nirenberg metric to measure thederivatives of G ) sup α ∈ ( T ∗ M ) (cid:15) k ≤ (cid:13)(cid:13)(cid:13) ∇ k G ( α ) (cid:13)(cid:13)(cid:13) KN (cid:104)| α |(cid:105) s ≤ τ . (2.5) Remark . If G is as in Definition 2.2, then one easily sees that the vectorfield H ω I G is complete, so that we can define a ( τ , s )-adapted Lagrangian Λ bythe formula (2.4) (the manifold Λ is then I-Lagrangian since exp( H ω I G ) is an I -symplectomorphism). Notice also that the assumptions on the symbol G imposethat it depends on h , but in a uniform fashion as h → h tends to 0).In the applications, the dependence of G on h and τ will be fairly explicit since G will be of the form τ h − /s G with G of order 1 /s satisfying the assumptionsof Definition 2.2 with τ = h = 1.One may notice that if τ ≥ τ and s ≥ ˜ s then any ( τ , s )-adapted Lagrangianis also ( τ , ˜ s )-adapted. Remark . Let m ∈ R and Ω be a manifold on which there is a notionof Kohn–Nirenberg metric and of Japanese bracket (the main examples are themanifolds T ∗ M, ( T ∗ (cid:102) M ) (cid:15) and an adapted Lagrangian Λ), then we define as usual theKohn–Nirenberg class of symbol S mKN (Ω) as the space of C ∞ functions a : Ω → C such that for every k ∈ N we have (using the covariant derivative associated to theKohn–Nirenberg metric): sup α ∈ Ω (cid:13)(cid:13)(cid:13) ∇ k a ( α ) (cid:13)(cid:13)(cid:13) KN (cid:104)| α |(cid:105) m < + ∞ . For instance, in Definition 2.2 we ask for G ∈ S /sKN (cid:16)(cid:0) T ∗ M (cid:1) (cid:15) (cid:17) .Since the adapted Lagrangians are uniformly smooth submanifolds with respectto the Kohn–Nirenberg, and image of T ∗ M under a uniformly smooth flow, thesymbol class S mKN (Λ) is well defined, and to check that a ∈ S mKN (Λ), we cancompute the derivatives either directly on Λ, or through the pullback by exp( H ω I G ),with covariant derivatives or with partial derivatives in coordinates.The notion of ( τ , s )-adapted Lagrangian is tailored so that it makes sense torestrict the almost analytic extension of a G s symbol (as defined in Remark 1.10)to a ( τ , s )-adapted Lagrangians when τ is small. More precisely, if G is as inDefinition 2.2, it follows from the local expression 2.3 for H ω I G that the norm of H ω I G for the Kohn–Nirenberg metric is O ( τ h − /s (cid:104)| α |(cid:105) /s − ). This essentially proves: Lemma . Let s ≥ and T ≥ . There is a constant C > such that, forevery τ ∈ [0 , T ] , if Λ is a ( τ , s ) -adapted Lagrangian and α ∈ Λ then | Im α | ≤ Cτ h − s (cid:104)| α |(cid:105) s − (On a Grauert tube, we can define | Im α | in a coordinate-free way, see Section1.1.1.2). In particular, for every (cid:15) > , there is a τ > such that, for every τ ∈ [0 , τ ] , any ( τ , s ) -adapted Lagrangian is contained in (cid:0) T ∗ M (cid:1) (cid:15) . In view of this lemma, it will be convenient to consider for (cid:15) > δ ∈ [0 , T ∗ M ) (cid:15),δ := (cid:110) α ∈ ( T ∗ M ) (cid:15) | | Im α | ≤ (cid:15)h − δ (cid:104)| α |(cid:105) δ − (cid:111) . Observe that ( T ∗ M ) (cid:15), = ( T ∗ M ) (cid:15) for (cid:15) small enough. In order to define weighted L spaces on adapted Lagrangians, we need todefine a volume form on such Lagrangians. Since we have the Kohn–Nirenbergmetric on Lagrangians, we could take the associated volume form; however it is notintrinsically related to the geometry of the problem, so it is more relevant to useanother one. To do so, we use the following lemma. Lemma . Let s ≥ . There exists T > and a constant C > such that, forevery τ ∈ [0 , T ] , if Λ is a ( τ , s ) -adapted Lagrangian then the restriction of ω R to Λ (that we will also denotes by ω R ) is symplectic.Moreover, if G is as in Definition 2.2, and J G denotes the Jacobian of exp( H ω I G ) from T ∗ M to Λ (both Lagrangians are endowed with the volume form ω nR /n ! ), thenfor every α ∈ T ∗ M we have | J G ( α ) − | ≤ Cτ h − s (cid:104)| α |(cid:105) s − . Proof. Let us use the notations from Definition 2.2. The first idea here wouldbe to try and prove that exp( H ω I G ) ∗ ω R is close to ω R . For this, one would computein local coordinates L H ωIG ω R = d( ı H ωIG ω R ) , = d (cid:2) ∇ ξ G · d η − ∇ η G · d ξ + ∇ x G · d y − ∇ y G · d x (cid:3) , = i d( ∂ − ∂ ) G = i ( − ∂∂ + ∂∂ ) G, = − i∂∂G. If G | T ∗ M is real-analytic, and G = Re ˜ G , where ˜ G is a holomorphic extension of therestriction of G to T ∗ M , then this vanishes. However, in general, the coefficients of ∂∂G are symbols of order 1 /s , and thus not bounded: it seems we cannot deduceanything directly from this identity.However, this problems stems from the fact that we are not measuring the sizeof L H ωIG ω R in a sensible way. Indeed, the Kohn–Nirenberg metric gives us a metricon the bundle of forms, so thatd x j , d y j , d ξ j (cid:104)| α |(cid:105) , and d η j (cid:104)| α |(cid:105) , j = 1 . . . n, form a basis in which the matrix (and its inverse) of g KN is bounded. We can thusreplace it by the metric that makes them an orthogonal basis. Computing normswith this new metric, we find (cid:107) ∂∂G (cid:107) = O (cid:16) τ h − s (cid:104)| α |(cid:105) s (cid:17) , and (cid:107) ω R (cid:107) = O ( (cid:104)| α |(cid:105) ) , (cid:107) ω − R (cid:107) = O ( (cid:104)| α |(cid:105) − ) . According to Taylor’s formula, ( ω R ) | Λ is thus symplectic, and the Jacobian J G closeto 1. (cid:3) From now on, if Λ is an adapted Lagrangian, we will just denote by d α the2 n -form ω nR /n !, which induces a volume form on Λ. We will denote the correspondingduality pairing(2.7) (cid:104) f, g (cid:105) Λ = (cid:90) Λ f g d α. The natural space in our setting will not be L (Λ , d α ) but rather L (Λ , e − H/h d α ),where H is an action associated with Λ, solving(2.8) d H = − Im θ | Λ . .1. BASIC PROPERTIES OF THE FBI TRANSFORM 87 Since Λ is I-Lagrangian, we deduce that there are local solutions to this equation.However, since Λ is assumed to be of the form (2.4), we can find an explicit global solution, given by(2.9) H := (cid:90) (cid:16) e ( τ − H ωIG (cid:17) ∗ ( G − Im θ ( H ω I G ))d τ. For a proof, we follow the arguments after equation 1.17 in [ Sj¨o96a ]. Observe thatd(Im θ ( H ω I G )) = Im d( ı H ωIG θ ) = Im( L H ωIG θ − ı H ωIG d θ ) = L H ωIG Im θ + d G. In particular, we getd H = (cid:90) τ (cid:16) e ( τ − H ωIG (cid:17) ∗ L H ωIG ( − Im θ )d τ = − Im θ + (cid:16) e − H ωIG (cid:17) ∗ Im θ. The second term vanishes when restricting to Λ because Im θ vanishes on T ∗ M .Computing in local coordinates, we getIm θ ( H ω I G ) = Im[( ξ + iη )( ∇ η G + i ∇ ξ G )] = η · ∇ η G + ξ · ∇ ξ G. Hence, Im θ ( H ω I G ) is a symbol of order 1 /s and, from (2.3), we get that the derivative H ω I G (Im θ ( H ω I G )) is a symbol of order − /s . In particular, we find that(2.10) H = G − Im θ ( H ω I G ) + O C (cid:18)(cid:16) τ h − s (cid:17) (cid:104)| α |(cid:105) s − (cid:19) . The explicit formula (2.9) for H has another consequence. For every s ≥ T ≥ 0, there is a constant C > τ , s )-adapted Lagrangian with τ ≤ T then, for every α ∈ Λ, we have | H ( α ) | ≤ Cτ h − s (cid:104)| α |(cid:105) s . (2.11)Let us give a word on the simplest way the symbol G in Definition 2.2 can bechosen. Let us consider a real valued symbol G of order 1 /s on T ∗ M . Denoting by˜ G one of its almost analytic extensions, we put G = τ h − /s Re ˜ G . In that case,if p ∈ S m is another real valued symbol with almost analytic extension (cid:101) p , we findthat for α ∈ T ∗ M and τ near 0 we have(2.12) (cid:101) p ( e τH ωIG ( α )) = p ( α ) + iτ h − /s { G , p } ( α ) + O ( τ h − s (cid:104)| α |(cid:105) m + s − ) , where { G , p } = ∇ ξ G · ∇ x p − ∇ x G · ∇ ξ p. More generally, to obtain Formula (2.12), instead of taking G = Re ˜ G , one cantake, as Sj¨ostrand did in [ Sj¨o96a ], any symbol G that extends G , and such that ∂ y,η G = 0 on T ∗ M .When m = 1, in the case that s ≥ 2, the remainder in (2.12) is the symbol ofa bounded operator. However, we will be interested in the case that s can be anynumber in [1 , + ∞ [. Then, the remainder is not bounded as α → ∞ . To circumventthis, in Section 3.1.1, we will build G directly on ( T ∗ M ) (cid:15) . This construction willnot yield a G such that ∇ y,η G = 0 on T ∗ M . Since Sj¨ostrand only consideredcompactly supported deformations, he could ignore this subtlety. In [ Sj¨o96a ], thefact that ∇ y,η G vanishes on T ∗ M served two purposes. The first is Formula (2.12),and the second is a quick proof that the Jacobian of exp( H ω I G ) is close to 1. For therest, that assumption is not necessary, and we can remove it.Equipped with this viaticum of symplectic geometry, we can define the functionalspaces associated to adapted Lagrangians. Beforehand, we have to check that thereactually exists an analytic FBI transform in the sense of Definition 2.1. First, weobserve that −(cid:104) α ξ , exp − α x ( x ) (cid:105) + i (cid:104) α (cid:105) d ( x, α x ) 28 2. FBI TRANSFORM ON COMPACT MANIFOLDS defines an admissible phase in the sense of Definition 1.7. From now on, we fix anadmissible phase Φ T (not necessarily equal to the phase above). Lemma . Let a be an elliptic symbol in h − n S , n . There exist an analyticFBI transform with symbol a and phase Φ T . Proof. We consider e i Φ T ( α,x ) /h a ( α, x ). It is holomorphic and well-definedwhen ( α, x ) ∈ (cid:0) T ∗ M (cid:1) (cid:15) × ( M ) (cid:15) satisfies d ( x, α x ) < δ , for some (cid:15), δ > 0. Moreover,for some C, (cid:15) > 0, it is smaller than C exp( − C − (cid:104)| α |(cid:105) /h ) when d (Re x, Re α x ) > (cid:15) .Hence, if we choose a bump function χ : M × M → [0 , 1] such that χ ( x, y ) = 1 if d ( x, y ) ≤ (cid:15) and χ ( x, y ) = 0 if d ( x, y ) ≥ δ/ (cid:15) is arbitrarilysmall), then the Cauchy–Riemann operator applied to( α, x ) (cid:55)→ χ (Re α x , Re x ) e i Φ T ( α,x ) h a ( α, x )(2.13)gives an O (exp( − C − (cid:104)| α |(cid:105) /h )) for ( α, x ) ∈ (cid:0) T ∗ M (cid:1) (cid:15) × ( M ) (cid:15) . In particular, wecan apply Lemma 1.10 to (2.13), and we find a globally analytic kernel K T ( α, x )satisfying the desired properties. (cid:3) From now on, in this section, we will assume that T is some fixed analytic FBItransform, and S a fixed adjoint analytic FBI transform (see Definition 2.1), withsymbols and phases respectively a and Φ T and b and Φ S . Since K T is analytic,there is a R > T u ( α ) is well-defined by Formula (2.2) for u ∈ ( E ,R ) (cid:48) ,and α ∈ ( T ∗ M ) (cid:15) (where (cid:15) is from Definition 2.1).We will also, for τ > τ , G ∈ S KN (cid:16)(cid:0) T ∗ M (cid:1) (cid:15) (cid:17) , and action H ). Notice that, sinceΛ ⊂ (cid:0) T ∗ M (cid:1) (cid:15) , for u ∈ (cid:16) E ,R (cid:17) (cid:48) the FBI transform T u is well-defined on Λ (providedthat τ is small enough).We define the FBI transform T Λ associated with Λ by the formula T Λ u = ( T u ) | Λ .We also define S Λ on C ∞ c (Λ) by S Λ v ( x ) = (cid:90) Λ K S ( x, α ) v ( α )d α. We can now define the scale of spaces that we are going to use in the following.First of all, let us denote for k ∈ R the weighted L space on Λ L k (Λ) = L (cid:16) Λ , (cid:104)| α |(cid:105) k e − Hh d α (cid:17) , (2.14)where we recall that d α denotes the volume form associated to the symplectic form ω R on Λ and that H is the action defined by (2.9) and that satisfies (2.8). For k ∈ R we define then the space (for R > H k Λ = (cid:110) u ∈ ( E ,R ) (cid:48) : T Λ u ∈ L k (Λ) (cid:111) endowed with the norm (we will see later that this is indeed a norm)(2.16) (cid:107) u (cid:107) H k Λ = (cid:107) T Λ u (cid:107) L k (Λ) and its analogue on the FBI side(2.17) H k Λ , FBI = (cid:110) T Λ u : u ∈ H k Λ (cid:111) ⊂ L k (Λ) . We will also need the spaces H ∞ Λ = (cid:92) k ∈ R H k Λ and H ∞ Λ , FBI = (cid:92) k ∈ R H k Λ , FBI . (2.18) .1. BASIC PROPERTIES OF THE FBI TRANSFORM 89 We cannot say much about these spaces yet, but their basic properties will followfrom the study of the FBI transform T Λ in § § Remark . Definition 2.2 of adapted Lagrangians may seem restrictive. Thereare two fundamental items that we will need in order to work with a manifoldΛ. First, we need to control the distance between Λ and T ∗ M (this is done usingLemma 2.1) and ensure that some transversality conditions satisfied by T ∗ M remaintrue for Λ. Hence, it is natural to ask for Λ to be close to T ∗ M in some C k sense.The second required item is a global solution to the equation (2.8) on Λ (wediscuss in Remark 2.7 how one could relax this condition). This is ensured by thefact that the Lagrangian Λ is of the form (2.4) for a real-valued function G . Let usnow explain briefly why it is reasonable to ask for Λ to be of the form (2.4) if wewant (2.8) to have a global solution H on Λ.By the Weinstein Tubular Neighbourhood Theorem, T ∗ M has a neighbourhoodin (cid:0) T ∗ M (cid:1) (cid:15) which is symplectomorphic (when endowed with ω I ) to a neighbourhoodof the zero section in T ∗ (cid:0) T ∗ M (cid:1) (endowed with its canonical symplectic form). Then,if Λ is a I-Lagrangian C ∞ close to T ∗ M , it corresponds through this symplectomor-phism to a Lagrangian submanifold (cid:101) Λ of T ∗ (cid:0) T ∗ M (cid:1) close to the zero section. Hence (cid:101) Λ is the graph of a 1-form γ on T ∗ M (we ignore non-compactness issues in thisinformal discussion). The fact that (cid:101) Λ is Lagrangian implies that γ is closed. Inthese new coordinates, the existence of a global solution to (2.8) is equivalent to theexactness of γ . Hence, there is a function f defined on T ∗ M such that d f = − γ .Now, if we define G on T ∗ (cid:0) T ∗ M (cid:1) by G ( x, ξ ) = f ( x ), we see that (cid:101) Λ is the image of T ∗ M by the time 1 map of the Hamiltonian flow defined by G .Consequently, having Λ of the form (2.4) is the exact geometric condition thatwe need if we want (2.8) to have a global solution H on Λ. Remark . As we explained in Remark 2.6, we make the assumption that Λis of the form (2.4) in order to ensure that (2.8) as a global solution H on Λ. Letus describe a possible way to work without the existence of such a H . Since − Im θ is closed, it defines an element of the homology group H (Λ). Thus, we can use itto define a line bundle over Λ in the following way: if (cid:98) Λ denotes the universal coverof Λ then we define an action of the fundamental group π (Λ) of Λ on (cid:98) Λ × C by c · ( x, u ) = (cid:18) c · x, e (cid:82) c ρh u (cid:19) , where c · x denotes the action of π (Λ) on (cid:98) Λ and ρ is the lift of − Im θ to (cid:98) Λ. Then,the quotient of (cid:98) Λ × C by this action defines a complex line bundle L over Λ (thatmay depend on h ). Then, let H : (cid:98) Λ → R be such that d H = ρ and defines the map A from (cid:98) Λ × C to itself by A ( x, u ) = (cid:16) x, e − H ( x ) h u (cid:17) , and notice that for ( x, u ) ∈ (cid:98) Λ × C we have A ( c · ( x, u )) = (cid:18) c · x, e − H ( c · x )+ (cid:82) c ρh u (cid:19) = (cid:16) c · x, e − H ( x ) h u (cid:17) = c ∗ A (( x, u )) , where ∗ denotes the action of π ( M ) on the first coordinates only (that is c ∗ ( x, u ) =( c · x, u )). Consequently, A defines in the quotient a vector bundle morphism A : L → Λ × C that can be used to replace the multiplication by e − Hh . For instance,instead of working with the space L (Λ , e − H/h d α ) (as we will do below), we wouldhave to work in this case with the space of sections u of L such that A u belongs to L (Λ , d α ).Of course, to deal with such a Lagrangian, we should use a FBI transform thatsends functions on M to sections of L . It is likely that most of the analysis may beadapted to this more general case. Notice however that some additional difficultiesarise (for instance, one has to say something about the dependence of L on h whenapplying H¨ormander’s solution to the ¯ ∂ equation in order to construct the kernel ofthe FBI transform). However, since we were not aware of any problem that wouldrequire to work with Lagrangians that are not of the form (2.4), we did not pursuethis line of work. Gevrey regularity on the FBI side. To simplify manipulations later on,let us here study the continuity of analytic FBI transforms on some functionalspaces. We will develop the idea that the regularity of an ultradistribution u on M translates into decay for its FBI transform T u .The notations are the same as in the previous paragraph § T is some fixedanalytic FBI transform, and S a fixed adjoint analytic FBI transform (see Definition2.1), with symbols and phases respectively a and Φ T and b and Φ S . For somesmall τ > 0, we denote by Λ a ( τ , G ∈ S KN (cid:16)(cid:0) T ∗ M (cid:1) (cid:15) (cid:17) , and action H ).In order to highlight the idea that the regularity of an ultradistribution on M may be understood through the decay of its FBI transform, let us introduce thefollowing norms. Let s ≥ r ∈ R . If f is a function from Ω ⊂ ( T ∗ M ) (cid:15) to C , let (cid:107) f (cid:107) Ω ,s,r := sup α ∈ Ω | f ( α ) | exp (cid:16) − r (cid:104)| α |(cid:105) s (cid:17) ∈ R + ∪ { + ∞} . Then we define F s,r (Ω) to be the Banach space of continuous function f : Ω → C such that (cid:107) f (cid:107) Ω ,s,r < + ∞ .We define then the following spaces of functions on the Lagrangian Λ. For s ≥ G s (Λ) := (cid:91) r< F s,r (Λ) , and the space of functions diverging slower than any exponential U s (Λ) := (cid:92) r> F s,r (Λ) . These spaces are endowed respectively with the inductive and projective limitstructure (in the category of locally convex topological vector spaces). Notice that G s (Λ) is dense in U s (Λ) (just multiply by a bump function) and that the L pairing(2.7) in Λ gives a natural duality bracket between U s (Λ) and G s (Λ). The spaces G s (Λ) and U s (Λ) are natural analogues of G s ( M ) and U s ( M ) on the FBI side.Indeed, we have the following results. Proposition . Let ˜ s > s ≥ and assume that Λ is a ( τ , ˜ s ) -Gevrey adaptedLagrangian with τ small enough. Then, the transform T Λ is continuous from G s ( M ) to G s (Λ) and from U s ( M ) to U s (Λ) . Proposition . Let ˜ s > s ≥ and assume again that Λ is a ( τ , ˜ s ) -Gevreyadapted Lagrangian with τ small enough. Then, the transform S Λ is continuousfrom G s (Λ) from G s ( M ) and admits a continuous extension from U s (Λ) to U s ( M ) . .1. BASIC PROPERTIES OF THE FBI TRANSFORM 91 If Propositions 2.1 and 2.2 give a good idea of the link between regularity on M and decay on the FBI side, they are not precise enough for what we intend to do. Inparticular, when dealing with G s Anosov flows, we want to consider ( τ , s )-adaptedLagrangians, in order to get the best results possible. Thus, we need more preciseestimates that explains how ( τ , s )-adapted Lagrangians relate with G s functionsand associated ultradistributions. We will then deduce Propositions 2.1 and 2.2from these estimates.2.1.2.2. Explicit estimates. The main goal of this section set, we go into details.We want to understand first the growth of T Λ u when u is either a Gevrey functionor an ultradistribution. Since T Λ u is the restriction of T u to Λ, and in view ofLemma 2.1, we will study the growth of T u inside a sub-conical neighbourhood of T ∗ M , as defined in (2.6).We start by studying T u when u is smooth. To do so, we use the non-stationaryphase method – Proposition 1.5. Since the phase of T is non-stationary only forlarge α , we need another bound for small α ’s. We use the following elementarybound: there is C > α ∈ ( T ∗ M ) (cid:15) and every bounded function u on M , we have | T u ( α ) | ≤ C (cid:107) u (cid:107) L ∞ ( M ) h − n (cid:104)| α |(cid:105) n exp (cid:18) C (cid:104)| α |(cid:105) | Im α | h (cid:19) . (2.19)The bound (2.19) follows from an immediate majoration of the kernel of T , usingthe fact that the imaginary part of Φ T is positive on T ∗ M × M . Notice also that,since T u is holomorphic, the bound (2.19) is in fact a symbolic estimate.To understand T u ( α ) for large α when u is Gevrey, we will rely on the followingestimate. Lemma . Let s ≥ . For every R > , there are constants C, τ > suchthat, for every α ∈ (cid:0) T ∗ M (cid:1) τ , /s such that (cid:104)| α |(cid:105) ≥ C , and every u ∈ E s,R ( M ) , wehave | T u ( α ) | ≤ C (cid:107) u (cid:107) E s,R exp (cid:32) − C (cid:18) (cid:104)| α |(cid:105) h (cid:19) s (cid:33) . (2.20) In particular, if r ≥ − /Ch /s and if Λ is a ( τ , s ) -adapted Lagrangian with τ smallenough, then T Λ is bounded from E s,R ( M ) to F s,r (Λ) . Then, we want to understand the growth of T u ( α ) when u is an ultradistribution.To do so, we only need to understand the size of the kernel of T in Gevrey norms.We will prove the following estimate, using the Bochner–Martinelli Trick, Lemma1.4. Lemma . Let s ≥ . For every (cid:15) > , there are constants C, τ , R > suchthat for every α ∈ ( T ∗ M ) τ , /s we have (cid:107) y (cid:55)→ K T ( α, y ) (cid:107) E s,R ≤ C exp (cid:32) (cid:15) (cid:18) (cid:104)| α |(cid:105) h (cid:19) s (cid:33) . (2.21) In particular, for u ∈ (cid:16) E s,R ( M ) (cid:17) (cid:48) and α ∈ ( T ∗ M ) τ , /s , we have | T u ( α ) | ≤ C (cid:107) u (cid:107) ( E s,R ) (cid:48) exp (cid:32) (cid:15) (cid:18) (cid:104)| α |(cid:105) h (cid:19) s (cid:33) . Consequently, if r ≥ (cid:15)h − /s and Λ is a ( τ , s ) -adapted Lagrangian for τ smallenough, then T Λ is bounded from (cid:16) E s,R ( M ) (cid:17) (cid:48) to F s,r (Λ) . The same argument gives an estimate on the kernel of S . Lemma . Let s ≥ . For every (cid:15) > , there are constants C, τ , R > suchthat for every α ∈ ( T ∗ M ) τ , /s we have (cid:107) y (cid:55)→ K S ( α, y ) (cid:107) E s,R ≤ C exp (cid:32) (cid:15) (cid:18) (cid:104)| α |(cid:105) h (cid:19) s (cid:33) . (2.22) In particular, if r < − (cid:15)h − /s and Λ is a ( τ , s ) -adapted Lagrangian with τ smallenough then S Λ is bounded from F s,r (Λ) to G s ( M ) . It is understood that, since T u is holomorphic, the estimates from Lemmas 2.4and 2.5 are in fact C ∞ estimates, due to Cauchy’s formula. Proof of Lemma 2.4. Choose τ (cid:28) α ∈ (cid:0) T ∗ M (cid:1) τ , /s , write T u ( α ) = (cid:90) M K T ( α, y ) u ( y )d y. (2.23)Then, we choose a small ball D in M and assume that Re α x remains in D , uniformlyaway from the boundary of D (at a distance much larger than τ ). The results forall α follows then immediately from a compactness argument. We split then theintegral (2.23) into the integral over D and the integral over M \ D , that we denoterespectively by T D u ( α ) and T M \ D u ( α ). The quantity T M \ D u ( α ) is defined byintegration over y ’s that remain uniformly away from Re α x and hence the definitionof K T implies that (recall Remark 2.3, we bound the L ∞ norm by the E s,R norm) (cid:12)(cid:12)(cid:12) T M \ D u ( α ) (cid:12)(cid:12)(cid:12) ≤ C (cid:107) u (cid:107) E s,R exp (cid:18) − (cid:104)| α |(cid:105) Ch (cid:19) . (2.24)Consequently, we may focus on T D u ( α ). Up to an error term which satisfies thesame kind of bound as T M \ D u ( α ), and that we will consequently ignore, we have T D u ( α ) = (cid:90) D e i Φ T ( α,y ) h a ( α, y ) u ( y )d y. (2.25)By taking D small enough, we may work in local coordinates. We want now to applyProposition 1.5. To do so, we need to check the hypotheses (i)-(iii). Hypothesis (i)is an immediate consequence of the point (i) in Definition 1.7 of an admissible phasethat is satisfied by Φ T . Hypothesis (ii) follows from the point (iv) in Definition1.7 and the fact that α x remains uniformly away from the boundary of D . Finally,hypothesis (iii) is satisfied because d y Φ T ( α, α x ) = − α ξ .We can consequently apply Proposition 1.5, ending the proof of the lemma. (cid:3) Proof of Lemma 2.5. We will apply the Bochner–Martinelli trick Lemma1.4 to the function f α,h : y (cid:55)→ K T ( α, y ) exp (cid:32) − (cid:15) (cid:18) (cid:104)| α |(cid:105) h (cid:19) s (cid:33) with “ λ = (cid:104)| α |(cid:105) /h ”. Since the imaginary part of Φ T ( α, y ) is non-negative when α and y are real, we see that if α ∈ ( T ∗ M ) τ , /s and y is at distance at most τ ( (cid:104)| α |(cid:105) /h ) /s − of M we have (cid:12)(cid:12) f α,h ( y ) (cid:12)(cid:12) ≤ C exp (cid:32) − (cid:15) (cid:18) (cid:104)| α |(cid:105) h (cid:19) s (cid:33) , for some C > 0. Point (ii) in Lemma 1.4 is trivially satisfied since f α,h is holomorphic.We can consequently apply the Bochner–Martinelli Trick with “ λ = (cid:104)| α |(cid:105) /h ” to .1. BASIC PROPERTIES OF THE FBI TRANSFORM 93 find that f α,h is uniformly bounded in E s,R ( M ) for some R > 0. The bound (2.21)follows. (cid:3) The proof of Lemma 2.6 is similar and we omit it. We are now in position toprove Propositions 2.1 and 2.2. Proof of Proposition 2.1. We assume that ˜ s > s ≥ τ , ˜ s )-adapted Lagrangian with τ > R > α ∈ Λ is large enough then α ∈ ( T ∗ M ) τ , /s (this is a consequence of Lemma 2.1 because ˜ s > s ). Consequently,if u ∈ E s,R ( M ), we can use the estimate (2.20) to bound T Λ u ( α ) for large α . Forsmall α , we can always use the bound (2.19). Hence, we see that T Λ is boundedfrom E s,R ( M ) to F s,r (Λ) for any r > − C − h − /s . The operator norm of T Λ maydepend on h , but we do not care about it here. We just proved that T Λ is boundedfrom G s ( M ) to G s (Λ).We turn to the continuity from U s ( M ) to U s (Λ). To do so choose r > (cid:15) = rh /s / h of the bound, and can consequently work with h fixed). Using the notations fromLemma 2.5, we see as above that if α ∈ Λ is large enough then α ∈ ( T ∗ M ) τ , /s .Consequently, if u ∈ U s ( M ) ⊆ ( E s,R ( M )) (cid:48) (where R > T Λ u ( α ) for large α in term of the normof u in ( E s,R ( M )) (cid:48) . For small α , we just apply an elementary bound, using thatthe kernel of T is analytic and a compactness argument. Hence, we see that T Λ isbounded from U s ( M ) to F s,r (Λ). Since r > T Λ is bounded from U s ( M ) to U s (Λ). (cid:3) Proof of Proposition 2.2. The continuity of S Λ from G s (Λ) to G s ( M )follows from Lemma 2.6, using Lemma 2.1 as in the proof of Proposition 2.1.To extend S Λ from U s (Λ) to U s ( M ), just notice that the formal adjoint of S is(by definition) an analytic FBI transform. Consequently, the formal adjoint of S Λ isbounded from G s ( M ) to G s (Λ) by Proposition 2.1. Since the duality pairing (2.7)embeds G s (Λ) into the strong dual of U s (Λ), it follows that S Λ has a continuousextension from U s (Λ) to U s ( M ). (cid:3) The next section is dedicated to the study of the composition S Λ T Λ . However,due to the sometimes intricate boundedness properties of T Λ and S Λ , it is notalways clear that this composition makes sense. We end this section by a result thatexplains why this composition is well-defined. Proposition . Let R > be large enough. Then for R > large enoughand τ small enough, there is a r > such that, if Λ is a ( τ , -adapted Lagrangianwith τ small enough, then T Λ is continuous from ( E ,R ) (cid:48) to F ,r (Λ) , and S Λ hasa continuous extension from F ,r (Λ) to ( E ,R ) (cid:48) , so that S Λ T Λ has a continuousextension from ( E ,R ) (cid:48) to ( E ,R ) (cid:48) (here r may depend on h but the quantificationon R and τ does not).In the same fashion, for every R > large enough, there is a r > , such that,provided that τ is small enough, for every k ∈ R , the operator S Λ is bounded from L k (Λ) to ( E ,R ) (cid:48) and T Λ is bounded from ( E ,R ) (cid:48) to F ,r (Λ) , so that T Λ S Λ isbounded from L k (Λ) to F ,r (Λ) . Proof. Choose R > 0. Let Λ be a ( τ , τ > C > τ is small enough, S Λ isbounded from ( E ,R ( M )) (cid:48) to F s,r (Λ) with r = ( Ch ) − . Now, applying Lemma 2.5 with (cid:15) = 1 /C , we see that, provided that τ is small enough, there is R > T Λ is bounded from ( E ,R ( M )) (cid:48) to F ,r (Λ). Consequently, the composition S Λ T Λ is well-defined and continuous from ( E ,R ) (cid:48) to ( E ,R ) (cid:48) . Of course, we canalways take R larger, since it only makes ( E ,R ) (cid:48) smaller.The proof of the second statement is similar. One just need in addition to noticethat it follows from (2.11) that, for any c > 0, if Λ is a ( τ , τ small enough then we may use (2.7) to pair elements of L k (Λ) with elementsof F ,r (Λ) where r = − ch . Thus S Λ has a continuous extension from L k (Λ) to (cid:16) E ,R (cid:17) (cid:48) if τ is small enough. (cid:3) Remark . Integrating by parts, one can show that T maps continuously C ∞ ( M ) to the space of maps on T ∗ M that decay faster than any power of (cid:104)| α |(cid:105) − at infinity ( h being fixed). Similarly, basic estimates on the kernel of S shows thatit maps the space of maps on T ∗ M that decay faster than any power of (cid:104)| α |(cid:105) − atinfinity continuously into C ∞ ( M ). Consequences of Theorem 6 and reduction of its proof. This section isdevoted to proving the existence of an FBI transform T such that T ∗ T = 1 (Theorem6), some of its consequences and its proof. The first consequence of Theorem 6 is Corollary . There is an R > such that E ,R ( M ) is dense in C ∞ ( M ) ,and, for every R > large enough, there is an R > such that, E ,R is dense in ( E ,R ) (cid:48) for the topology induced by ( E ,R ) (cid:48) . Proof. If u belongs to C ∞ ( M ) or ( E ,R ) (cid:48) , we want to approximate u by u n := T ∗ (cid:16) χ (cid:0) n (cid:104)| α |(cid:105) (cid:1) T u (cid:17) , where χ ∈ C ∞ c ( R ) is equal to 1 around 0. It follows from Lemma 2.5 that the u n ’sbelongs to E ,R for R large enough (that does not depend on u ). If u ∈ C ∞ ( M ),then it follows from Remark 2.8 that u n tends, when n tends to + ∞ , to T ∗ T u = u in C ∞ ( M ). Now, if u ∈ ( E ,R ) (cid:48) , we see as in the proof of Proposition 2.3 that u n tends to T ∗ T u = u in ( E ,R ) (cid:48) provided that R (cid:29) R . (cid:3) Now that we have this density statement, we can consider the composition S Λ T Λ for Λ an adapted Lagrangian. Lemma . Assume that T is given by Theorem 6, and that S = T ∗ is thecorresponding adjoint transform. Then, for every R > large enough, if Λ is a ( τ , -Gevrey adapted Lagrangian with τ small enough, then, for h small enough, S Λ T Λ = I when acting on ( E ,R ) (cid:48) . Proof. First of all, recall that, thanks to Proposition 2.3, the operator S Λ T Λ is well-defined on ( E ,R ) (cid:48) when R is large enough (taking value in a space which is a priori larger).From Proposition 2.3, we see that, provided that τ is small enough, there are R , R > S Λ T Λ is continuous from ( E ,R ) (cid:48) to ( E ,R ) (cid:48) . Moreover, wemay assume that R is arbitrarily large (since it makes ( E ,R ) (cid:48) smaller) and henceapply Corollary 2.1 to see that, provided that R > E ,R is densein ( E ,R ) (cid:48) for the topology of ( E ,R ) (cid:48) . Consequently, we only need to prove that S Λ T Λ u = u for u ∈ E ,R .However, for u ∈ E ,R , it follows from Lemmas 2.1 and 2.4 that, provided that τ is small enough (depending on R ), the integral in the following right-hand side .1. BASIC PROPERTIES OF THE FBI TRANSFORM 95 is convergent(2.26) S Λ T Λ u ( x ) = (cid:90) Λ K S ( x, α ) T u ( α )d α. Moreover, the integrand in (2.26) is holomorphic and rapidly decreasing (due toLemmas 2.4 and 2.6). Hence, we may apply Stokes’ Formula to shift contoursin (2.26) and replace the integral over Λ by an integral over T ∗ M , proving that S Λ T Λ u = T ∗ T u = u , and hence the lemma. (cid:3) Remark . Mind that S Λ is not a right inverse for T Λ . Indeed, T Λ is notsurjective, since its image only contains smooth function. In fact, we will form theprojector Π Λ = T Λ S Λ on the image of T Λ . The study of the operator Π Λ , and moregenerally of operators of the form T Λ P S Λ , will be carried out in § § Corollary . Let s ≥ and Λ be a ( τ , s ) -adapted Lagrangian with τ smallenough. Under the assumptions of the previous lemma, for every k ∈ R , the space H k Λ (defined by (2.15) ) is a Hilbert space and the space H k Λ , FBI is a closed subspaceof L k (Λ) .Moreover, for R large enough, τ small enough (depending on R ), and all k ∈ R , (2.27) E s,R ⊆ H k Λ ⊆ (cid:16) E s,R ( M ) (cid:17) (cid:48) . If ≤ ˜ s < s then, provided that τ is small enough, for every k ∈ R , we have G ˜ s ( M ) ⊆ H k Λ ⊆ U ˜ s ( M ) . All these inclusions are continuous. Proof. Since T Λ has a left inverse, the equation (2.16) indeed defines a norm(provided that R is chosen large enough in (2.15) so that Lemma 2.7 applies). Noticethen that the spaces H k Λ and H k Λ , FBI are isometric, so that the completeness of H k Λ is equivalent to the closedness of H k Λ , FBI .In order to prove the closedness of H k Λ , FBI in L k (Λ), we will explain how R ischosen in (2.17). First of all, choose R large enough so that Lemma 2.7 applies.We want also that R is large enough so that, by Proposition 2.3 and for τ smallenough, S Λ is bounded from L k (Λ) to ( E ,R ) (cid:48) , and Π Λ is bounded from L k (Λ) tosome F ,r .With these conditions in mind, let ( u n ) n ∈ N be a sequence in H k Λ , FBI thatconverges to u in the space L k (Λ). Then (Π Λ u n ) n ∈ N converges to Π Λ u in F ,r ,hence pointwise. But, since Π Λ u n = u n for n ∈ N (because our choice of R ensuresthat Lemma 2.7 applies), we see that u = Π Λ u = T Λ S Λ u ∈ H , FBI (here we use thefact the S Λ u ∈ ( E ,R ) (cid:48) thanks to our choice of R ). It follows that H k Λ , FBI is closedin L k (Λ).The last statement follows from Propositions 2.1, 2.2. To get the more preciseestimates (2.27), one just need to apply the more precise estimate Lemma 2.4,recalling (2.11) to control the size of H . This gives immediately the first inclusion,the second one is obtained by a duality argument. (cid:3) The proof of Theorem 6 relies on the following lemma: Lemma . Let T and S be respectively an analytic and an adjoint analyticFBI transform, whose phases satisfy Φ S ( x, α ) = − Φ T ( α, x ) . Then the operator ST is a G pseudor of order on M . Moreover, its principal symbol is α (cid:55)→ n ( hπ ) n a ( α, α x ) b ( α x , α ) (cid:113) det d x Im Φ T ( α, α x ) , where a and b are the symbols of T and S respectively. In particular, the analyticpseudor ST is elliptic of order . Proof of Theorem 6. We start by taking an analytic transform T withsymbol a ( α, x ) = 2 − n ( hπ ) − n (det d x Im Φ T ( α, α x )) / , and applying Lemma 2.8,we deduce that P = ( T ) ∗ T is an elliptic G pseudor with principal symbol 1.Additionally, it is self-adjoint on L . Since the principal symbol of P is 1, it followsfrom G˚arding’s inequality that there is C > h small enough, thespectrum of P is contained in [ C − , C ]. In particular, the inverse square root of P is well-defined (by the functional calculus for self-adjoint operators) and given bythe formula P − = 12 iπ (cid:90) γ ( w − P ) − d w √ w , (2.28)where γ is a curve in { z ∈ C : Re z > } that turns around the segment [ C − , C ].Since the principal symbol of P is 1, we see that all the w − P for w ∈ γ aresemi-classically elliptic and hence it follows from Theorem 4 that ( w − P ) − is asemi-classical G pseudor. Moreover, this is true with uniform estimates when w ∈ γ ,and thus it follows from (2.28) that P − is a semi-classical G pseudor (we applyDefinition 1.6 of a G pseudor and Fubini Theorem as in the proof of Theorem 5). Itfollows from Proposition 1.12 (see also Remark 2.2) that T = T P − is an analyticFBI transform, and moreover it satisfies that T ∗ T = P − P P − = I. (cid:3) Product of a transform and an adjoint transform. This subsection isdevoted to the proof of Lemma 2.8.In order to manipulate converging integrals, we start by introducing a regular-ization procedure. From (2.26) with Λ = T ∗ M , we notice that if u ∈ G ( M ) then(by dominated convergence and Fubini’s Theorem) ST u ( x ) = lim (cid:15) → (cid:90) T ∗ M e − (cid:15) (cid:104) α (cid:105) K S ( x, α ) T u ( α )d α = lim (cid:15) → (cid:90) M K (cid:15) ( x, y ) u ( y )d y, where the kernel K (cid:15) is defined by K (cid:15) ( x, y ) = (cid:90) T ∗ M e − (cid:15) (cid:104) α (cid:105) K S ( x, α ) K T ( α, y )d α. (2.29)We start by showing that, outside of a small neighbourhood of the diagonal,the kernel of ST is a rapidly decaying (when h tends to 0) analytic function. To doso, choose some small η > C > x, y ∈ M are such that d ( x, y ) > η , then | K S ( x, α ) K T ( α, y ) | ≤ C exp (cid:18) − (cid:104)| α |(cid:105) Ch (cid:19) . (2.30) .1. BASIC PROPERTIES OF THE FBI TRANSFORM 97 This follows from the definition of K T and K S since either x or y is at distance atleast η/ α x . Estimate (2.30) implies that for such x and y the kernel K (cid:15) ( x, y )converges when (cid:15) tends to 0 to a kernel K ( x, y ) which is a O (exp( − /Ch )). SinceEstimate (2.30) remains true when x and y are in a small tube ( M ) ˜ (cid:15) , this is in factan estimate in G .Hence, we only need to understand the kernel of ST in an arbitrarily smallneighbourhood of the diagonal in M × M . By a standard compactness argument,we only need to understand the kernel of ST for x and y in a fixed ball D of radius10 η . To do so, let D (cid:48) and D (cid:48)(cid:48) be respectively a ball of radius 100 η and 1000 η withthe same center as D . When x and y are in D , we can split the integral (2.29)into the integral over T ∗ D (cid:48) and the integral over T ∗ (cid:0) M \ D (cid:48) (cid:1) . The integral over T ∗ (cid:0) M \ D (cid:48) (cid:1) is dealt with as in the case d ( x, y ) > η . By taking η small enough, wemay assume that when x, y and α x belongs to D (cid:48) , the kernels K T and K S bothsatisfy an estimate as equation (2.1), denoting by a the symbol of T , and by b thatof S . The O (exp( − C − (cid:104) α (cid:105) /h )) remainders are tackled as above, so that we focusnow on the kernel (cid:101) K (cid:15) ( x, y ) = (cid:90) T ∗ D (cid:48) e − (cid:15) (cid:104) α (cid:105) e i Φ S ( x,α )+Φ T ( α,y ) h a ( α, y ) b ( x, α )d α, for x, y ∈ D . We want to recognize, when (cid:15) tends to 0, the kernel of an analyticpseudor (with non-standard phase) and to compute its symbol. By taking η smallenough, we may assume that we work in local coordinates α = ( z, ξ ). We willassume (as we may) that these coordinates satisfy det g z = 1. We can also replacethe regularization factor exp( − (cid:15) (cid:104) α (cid:105) ) by the simpler exp( − (cid:104) ξ (cid:105) ): testing againstan analytic function, we see that the limiting distribution is independent on theregularization. Hence, we have(2 πh ) − n (cid:101) K (cid:15) ( x, y )= (cid:90) R n e − (cid:15) (cid:104) ξ (cid:105) (cid:32)(cid:18) (cid:104) ξ (cid:105) πh (cid:19) n (cid:90) D (cid:48) e i (cid:104) ξ (cid:105) h Ψ x,ξ,y ( z ) (2 πh ) n a ( z, ξ, y ) b ( x, z, ξ ) (cid:104) ξ (cid:105) n d z (cid:33) d ξ, (2.31)where the phase Ψ x,ξ,y is defined byΨ x,ξ,y ( z ) = (cid:104) ξ (cid:105) − (Φ S ( x, ( z, ξ )) + Φ T (( z, ξ ) , y ))= (cid:104) ξ (cid:105) − (cid:16) Φ T (( z, ξ ) , y ) − Φ T ((¯ z, ¯ ξ ) , ¯ x ) (cid:17) . To put the kernel into pseudo-differential form, it suffices formally to eliminate thevariable z , and this is done by Holomorphic Stationary Phase, Proposition 1.6 (seealso Remark 1.9). Let us check that the hypotheses of Proposition 1.6 are satisfied.Hypothesis (i) follows from Condition (i) in Definition 1.7 of an admissible phase.Hypothesis (ii) follows from Condition (iv) in the Definition 1.7 and the fact that x and y remains at uniform distance from the boundary of D (cid:48) . Finally, from (iii) inDefinition 1.7, we see that Ψ x,ξ,x has a critical point at z = x , which is uniformlynon-degenerate thanks to (iv) in Definition 1.7 (the imaginary part of the Hessianis uniformly definite positive).Moreover, the associated critical value is 0. Thus,provided that η is small enough, we find that the inner integral in (2.31) is givenfor ( x, ξ, y ) ∈ (cid:0) T ∗ D (cid:1) (cid:15) × ( D ) (cid:15) (for some (cid:15) > 0) and arbitrarily large C by e i Φ ST ( x,ξ,y ) h (cid:88) ≤ k ≤(cid:104)| ξ |(cid:105) / ( C h ) h k c k ( x, ξ, y ) + O (cid:18) exp (cid:18) − (cid:104)| ξ |(cid:105) Ch (cid:19)(cid:19) . (2.32) Here, (cid:80) k ≥ h k c k is a formal analytic symbol of order 1 and Φ ST ( x, ξ, y ) denotesthe critical value of z (cid:55)→ (cid:104) ξ (cid:105) Ψ x,ξ,y ( z ).Let us check that Φ ST is a phase in the sense of Definition 1.7. The holomorphy ofΦ ST follows from the Implicit Function Theorem, and the symbolic estimates from thefact that Ψ x,ξ,y is uniformly bounded. Point (ii) in Definition 1.7 is satisfied becausewhen x = y the critical point of Ψ x,ξ,x is x so that Φ ST ( x, ξ, x ) = (cid:104) ξ (cid:105) Ψ x,ξ,x ( x ) = 0.From the Implicit Function Theorem, we also see thatd y Φ ST ( x, ξ, x ) = d y Φ T ( x, ξ, x ) = − ξ, since Φ T is itself a phase. This proves that Φ ST satisfies point (iii) of Definition 1.7.It remains to check (i), i.e. that for α, x real, Im Φ ST ≥ 0. This follows from the“Fundamental” Lemma 1.16Now, let c be a realization of the formal symbol (cid:80) k ≥ h k c k . Since the imaginarypart of Φ ST is non-negative for real ( x, ξ, y ), we see that for ( x, ξ, y ) ∈ (cid:0) T ∗ D (cid:1) (cid:15) × (cid:0) D (cid:15) (cid:1) , the factor e i Φ ST ( x,ξ,y ) /h in (2.32) is an O (exp( C(cid:15) (cid:104)| ξ |(cid:105) /h )). Consequently, bytaking (cid:15) small enough, we may move the error term in (2.32) out of the parenthesis.Thus, the inner integral in (2.31) is given by e i Φ ST ( x,ξ,y ) h c ( x, ξ, y )up to an O (cid:16) exp (cid:16) − C − (cid:104)| ξ |(cid:105) /h (cid:17)(cid:17) error. Plugging this expression into (2.31) andusing Lemma 1.5, we see that (cid:101) K (cid:15) ( x, y ) converges when (cid:15) tends to 0 to the kernel K Φ TS ,c defined by (1.78), up to an O G (exp( − / ( Ch ))). By Lemma 1.24, we knowthat K Φ TS ,c is in fact the kernel of a G pseudor. Thus, ST is itself a G pseudor.According to Remark 1.24 and the proof of Lemma 1.24, the principal symbol of ST is given in our coordinates by c ( x, ξ, x ) = 2 n ( hπ ) n a ( x, ξ, x ) b ( x, ξ, x ) (cid:113) det d x Im Φ T ( x, ξ, x ) , for x, ξ real. (cid:3) In this section, the purpose of defining the H k Λ ’s will be explained. We fix ananalytic FBI transform T given by Theorem 6 and denote by S = T ∗ its adjoint.For an adapted Lagrangian Λ, we will study the operator Π Λ = T Λ S Λ defined before.We will find that it acts naturally on the space L (Λ). More generally, given P acontinuous operator on some suitable spaces of ultradistributions, it make sense toconsider the operator T Λ P S Λ acting on L (Λ), or equivalently the action of P on H . In this section, we will consider the case that P is a G s pseudor. In § T Λ P S Λ . In § § To study the continuityproperties of pseudors on the spaces H k Λ associated to I-Lagrangians, we will needto understand precisely the asymptotics of the kernel of T Λ P S Λ . In this section weobtain such technical estimates.As a precaution, we start by observing that, according to the results of § T Λ P S Λ maps C ∞ c (Λ) to elements of G s (Λ) ⊂ D (cid:48) (Λ), so it has awell-defined Schwartz kernel K T P S with respect to d α , which we can study. Theanalyticity of the FBI transform implies that K T P S , a priori defined only on Λ × Λ, .2. LIFTING PSEUDO-DIFFERENTIAL OPERATORS 99 is actually the restriction of a holomorphic function on ( T ∗ M ) (cid:15) × ( T ∗ M ) (cid:15) thatdoes not depend on Λ, as long as Λ ⊂ ( T ∗ M ) (cid:15) . We still denote it by K T P S .The kernel K T P S ( α, β ) behaves as an oscillatory integral, with large parameter λ := (cid:104)| α |(cid:105) + (cid:104)| β |(cid:105) h . For convenience, we recall that γ ∈ ( T ∗ M ) τ , /s means that γ ∈ T ∗ (cid:102) M and | Im γ | ≤ τ ( h/ (cid:104)| γ |(cid:105) ) − /s . When estimating K T P S , we have to distinguish between two regimes: far and closefrom the diagonal. For the first regime, we obtain Lemma . Let s ≥ . Let P be a G s semi-classical pseudo-differential operatorof order m on M . Let η > be small enough. Then there are constants C , τ > such that for α, β in ( T ∗ M ) τ , /s , and d KN ( α, β ) > η/ , the following holds: K T P S ( α, β ) = O C ∞ (cid:32) exp (cid:32) − (cid:18) (cid:104)| α |(cid:105) + (cid:104)| β |(cid:105) C h (cid:19) s (cid:33)(cid:33) The proof of this Lemma relies mostly on non-stationary phase estimates. Forthe second regime, we have to use a stationary phase method. For this reason, theproof is more subtle. It is the most technical part of this article. Lemma . Let s ≥ . Let P be a G s semi-classical pseudo-differentialoperator of order m on M . Let η > be small enough. Let Λ be a ( τ , s ) -adaptedLagrangian with τ small enough. Then there exists a constant C > such that for α, β in Λ with d KN ( α, β ) ≤ η , with λ = (cid:104)| α |(cid:105) /h , K T P S ( α, β ) = e ih Φ TS ( α,β ) e ( α, β ) + O C ∞ (cid:32) exp (cid:32) − λ s − C − λ s d KN ( α, β ) C (cid:33)(cid:33) , (2.33) where Φ T S ( α, β ) is the critical value of y (cid:55)→ Φ T ( α, y ) + Φ S ( β, y ) and e is a symbolof order m in the Kohn–Nirenberg class h − n S mKN (Λ × Λ) , given at first order onthe diagonal by (2.34) e ( α, α ) = 1(2 πh ) n p ( α ) mod h − n +1 S m − KN (Λ) , where p is an almost analytic extension of a representative of the principal symbolof P (see Remark 1.10). For the proof of Lemma 2.10, it will not be useful that Λ is Lagrangian, onlythat it is C close to the reals. Actually, in the proof of Lemma 2.10, the mainpoint will be that ( α, β ) is in the region described in Figure 1, which contains aneighbourhood of the diagonal of Λ × Λ, if Λ is ( τ , s )-adapted. Remark . There is a similar statement in [ LL97 ]. However, we make threeimprovements of this result. First, we get rid of the condition s = 3, which wasmade in [ LL97 ] in view of the application they had in mind. Then, we do notassume that G is compactly supported and hence deal with both the limit h → (cid:104)| α |(cid:105) → + ∞ . Finally, we deal here with a pseudo-differential operator P while[ LL97 ] only considered differential operators. On a technical level, there is anothernotable difference: we implement the non-stationary phase method in Gevrey classesusing almost analytic extensions and shifting of contours rather than integrations byparts and formal norms. We hope that this makes the proof easier to understand. 00 2. FBI TRANSFORM ON COMPACT MANIFOLDS (cid:104) α (cid:105) Re( α − β ) | Im( α − β ) || Im( α + β ) | δh − /s δ ˜ h − /s τ ˜ h − / (2 s − τ ˜ h − /s τ ˜ h − / s | R e ( α − β ) | ∼ Figure 1. Region near the diagonal in the complex where K T P S can be controlled. Here, ˜ h = h/ (cid:104)| α |(cid:105) . Remark . Since it is not necessary for our purposes, we did not investigateultradifferentiability of the kernel. However, the proof we give should imply a G s − regularity without much effort if Λ were itself a G s − manifold. We suspect thatthere is a way to obtain a G s estimate in the case that Λ is a G s manifold, but forwant of application we did not investigate further.The proof of Lemmas 2.9 and 2.10 are slightly different in the cases s > s = 1. To apply a combination of stationary and non-stationary phase method,we split the domain of integration in several regions. Several times, the imaginarypart of the phase possibly vanishes near the boundary of those domains. Since it isnon-stationary when this happens, one solution is to use contour deformations nearthose boundaries. Another solution, only available when s > 1, is to introduce G s cutoff functions. Since this lightens the proof, we will cover in full detail the case s > s = 1 at the end of this section. Proof of Lemma 2.9 in the case s > . In the proof, η > τ > independently of h > α and β ; it will be the casethat τ (cid:28) η . The cutoffs are assumed to be G s , throughout the proof. We willdenote K T P S just as K .Throughout this proof, we assume that that α, β are in ( T ∗ M ) τ , /s , and that d ( α x , β x ) > η/ 2. Since τ (cid:28) η , we have that d ( α x , β x ) ∼ d (Re α x , Re β x ) in thisregion. We may find bump functions χ α and χ β that take value 1 on a neighbourhoodrespectively of Re α x and Re β x and whose support do not intersect (we only need afinite number of such pairs of bump functions to deal with the whole manifold bycompactness). Then we split the integral K ( α, β ) = (cid:90) M × M K T ( α, x ) K P ( x, y ) K S ( y, β ) d x d y (2.35)into K ( α, β ) = (cid:90) M × M K T ( α, x ) K P ( x, y ) (cid:0) − χ β ( y ) (cid:1) K S ( y, β ) d x d y + (cid:90) M × M K T ( α, x ) (1 − χ α ( x )) K P ( x, y ) χ β ( y ) K S ( y, β ) d x d y + (cid:90) M × M K T ( α, x ) χ α ( x ) K P ( x, y ) χ β ( y ) K S ( y, β ) d x d y. (2.36)We notice that the first two integrals in the right hand side have a symmetricbehaviour, so that it suffices to deal with the first one. For this, we also notice that .2. LIFTING PSEUDO-DIFFERENTIAL OPERATORS 101 it is nothing else than T (cid:32) P (cid:16) y (cid:55)→ (cid:0) − χ β ( y ) (cid:1) K S ( y, β ) (cid:17)(cid:33) ( α ) . Since χ β is equal to 1 near Re β x , it follows from the assumptions on K S (see Defi-nition 2.1) that the function y (cid:55)→ (cid:0) − χ β ( y ) (cid:1) K S ( y, β ) is an O (exp( − (cid:104)| β |(cid:105) /Ch ))in some E s,R ( M ). Then it follows from Lemmas 1.18 and 1.19 that P (cid:16) y (cid:55)→ (cid:0) − χ β ( y ) (cid:1) K S ( y, β ) (cid:17) . is an O (exp( − C − ( (cid:104)| β |(cid:105) /h ) /s )) in some E s,R ( M ). Finally, it follows from Lemma2.4 and the bound (2.19) that the first term (and hence also the second one) in(2.36) is O C ∞ (cid:32) exp (cid:32) − (cid:18) (cid:104)| α |(cid:105) + (cid:104)| β |(cid:105) Ch (cid:19) s (cid:33)(cid:33) , provided that τ is small enough. To deal with the last integral in (2.36), we start bynoticing that, since P is G s pseudo-local (according to Definition 1.6), the function( x, y ) (cid:55)→ χ α ( x ) K P ( x, y ) χ β ( y )is an O (exp( − C − h − /s )) in some space E s,R ( M × M ). Then we may apply thesame argument as in the proof of Lemma 2.4 to see that this last integral is also ofthe expected size. More precisely, we perform a non-stationary phase (Proposition1.5) separately in both variables x and y .Now, we move to the study of the kernel K T P S ( α, β ) when the distance betweenRe α x and Re β x is less than 10 η . To do so, the compactness of M allows us toassume that both Re α x and Re β x belongs to a same ball D of radius 100 η , andthat they remain uniformly away from the boundary of D (recall that we assume τ (cid:28) η ). We introduce then a G s bump function χ supported in D that takes value1 on a neighbourhood of Re α x and Re β x . Reasoning as in the previous case, wesee that the kernel K T P S ( α, β ) is given, up to a negligible term, by the integral (cid:90) D × D K T ( α, x ) χ ( x ) K P ( x, y ) χ ( y ) K S ( y, β )d x d y. We may rewrite this, up to negligible terms, as the oscillating integral1(2 πh ) n (cid:90) D × R n × D e i Φ T ( α,x )+ (cid:104) x − y,ξ (cid:105) +Φ S ( y,β ) h a ( α, x ) χ ( x ) p ( x, ξ ) χ ( y ) b ( y, β ) (cid:124) (cid:123)(cid:122) (cid:125) := f α,β ( x,ξ,y ) d x d ξ d y, (2.37)where p is the symbol of P as in (1.52) from Definition 1.6, Φ T and Φ S are thephases of T and S respectively and a and b their symbols. The phaseΨ α,β ( x, ξ, y ) = Φ T ( α, x ) + (cid:104) x − y, ξ (cid:105) + Φ S ( y, β )will play a crucial role in the following. From now on, we may and will assume thatwe are working in coordinates (by choosing η very small).Let A > (cid:12)(cid:12) α ξ − β ξ (cid:12)(cid:12) ≥ Aη max ( (cid:104)| α |(cid:105) , (cid:104)| β |(cid:105) ) (ofcourse, we need that A does not depend on η , provided that η is small enough). Todo so, we compute the gradient of the phase Ψ α,β , which is given by ∇ Ψ α,β ( x, ξ, y ) = ∇ x Φ T ( α, x ) + ξx − y ∇ y Φ S ( y, β ) − ξ . 02 2. FBI TRANSFORM ON COMPACT MANIFOLDS Notice that if x and y belong to the support of χ then we have ∇ x Φ T ( α, x ) + ξ = ξ − α ξ + O ( η (cid:104)| α |(cid:105) ) ∇ y Φ S ( y, β ) − ξ = β ξ − ξ + O ( η (cid:104)| β |(cid:105) ) . (2.38)Hence, when (cid:12)(cid:12) α ξ − β ξ (cid:12)(cid:12) ≥ Aη max ( (cid:104)| α |(cid:105) , (cid:104)| β |(cid:105) ), we have, for some constant C > |∇ x Φ T ( α, x ) + ξ | + (cid:12)(cid:12) ∇ y Φ T ( y, β ) − ξ (cid:12)(cid:12) ≥ (cid:12)(cid:12) α ξ − ξ (cid:12)(cid:12) + (cid:12)(cid:12) β ξ − ξ (cid:12)(cid:12) − Cη max ( (cid:104)| α |(cid:105) , (cid:104)| β |(cid:105) ) ≥ (cid:12)(cid:12) α ξ − β ξ (cid:12)(cid:12) − Cη max ( (cid:104)| α |(cid:105) , (cid:104)| β |(cid:105) ) ≥ ( A − C ) η max ( (cid:104)| α |(cid:105) , (cid:104)| β |(cid:105) ) . Hence, if A ≥ C , then when (cid:12)(cid:12) α ξ − β ξ (cid:12)(cid:12) ≥ Aη max ( (cid:104)| α |(cid:105) , (cid:104)| β |(cid:105) ), the phase Ψ α,β isnon-stationary. We can then apply Proposition 1.1 to find that (2.37) is negligible(using the large parameter λ = max ( (cid:104)| α |(cid:105) , (cid:104)| β |(cid:105) ) /h and a suitable rescaling as in § x and y , so that we can apply Proposition 1.1 at fixed ξ and then integrate over ξ .Finally, the imaginary part of the phase is not necessarily positive on the boundaryof the domain of integration. However, the amplitude vanishes near the boundary,so this is not a problem either.This closes the proof of Lemma 2.9 (cid:3) We now turn to the proof of Lemma 2.10. We will isolate the technical difficultiesin two lemmas – 2.11 and 2.12. Proof of Lemma 2.10 in the case s > . For this proof, we may work lo-cally in the cotangent space (provided that we get uniform estimates) and assumethat we are considering α = ( α x , α ξ ) , β = ( β x , β ξ ) ∈ Λ which lie in a neighbour-hood of size η > α = ( α ,x , α ,ξ ) ∈ T ∗ M for the Kohn–Nirenbergmetric (as in the proof of Lemma 2.9, we assume that τ (cid:28) η ). Then we have (cid:104)| α |(cid:105) ∼ (cid:104)| β |(cid:105) ∼ (cid:104) α (cid:105) , and we will be looking for estimates suitably uniform as (cid:104) α (cid:105) → ∞ . The large parameter in the oscillating integrals will be λ = (cid:104) α (cid:105) /h . Asin the proof of Lemma 2.9, we will denote K = K T P S .Since α x and β x are close, we can work in local coordinates and rewrite thekernel K T P S as in (2.37). In order to get rid of the oscillating integral, we introducea cutoff function θ ( ξ ) that takes value 1 when ξ is at distance less than 100 η of α ,ξ and supported in a ball of radius proportional to η (all these distances have to beunderstood using the Kohn–Nirenberg metric). The cutoff θ depend on α , but itbelongs uniformly to the symbol class S s, (cid:0) T ∗ R n (cid:1) (once η has been fixed), so thatthis dependence will not matter. Then, instead of considering the integral (2.37),we may study (cid:98) K ( α, β ) := 1(2 πh ) n (cid:90) D × R n × D e i Ψ α,β ( x,ξ,y ) h θ ( ξ ) f α,β ( x, ξ, y )d x d ξ d y. (2.39)Indeed, it follows from (2.38) and the same non-stationary argument as in the proofof Lemma 2.9 that the difference between (2.37) and (2.39) is negligible (controlledby exp( − λ /s )). It is then useful to change variable and write (cid:98) K ( α, β ) = (cid:18) λ π (cid:19) n/ (cid:90) D × R n × D e iλ (cid:101) Ψ α,β ( x,ξ,y ) g α,β ( x, ξ, y )d x d ξ d y (2.40)where (cid:101) Ψ α,β ( x, ξ, y ) = Φ T ( α, x ) + Φ S ( y, β ) (cid:104) α (cid:105) + (cid:104) x − y, ξ (cid:105) (2.41) .2. LIFTING PSEUDO-DIFFERENTIAL OPERATORS 103 and(2.42) g α,β ( x, ξ, y ) = (cid:18) λ π (cid:19) − n/ θ ( (cid:104) α (cid:105) ξ ) f α,β ( x, (cid:104) α (cid:105) ξ, y ) = O (cid:18) (cid:104) α (cid:105) m h n (cid:19) . This estimate on the amplitude follows from the fact that the symbol of T is a G symbol of order h − n/ (cid:104)| α |(cid:105) n/ – recall m is the order of P . Here, we assume that η is small enough so that the size of any point in the support of θ is comparableto (cid:104) α (cid:105) when (cid:104) α (cid:105) holds to + ∞ . The estimate (2.42) holds as a G s estimate in allvariables. We will denote by (cid:101) D a disc of radius proportional to η (for the Euclideanmetric) such that θ ( (cid:104) α (cid:105) · ) is supported in (cid:101) D . Notice that we may assume that D and (cid:101) D only depend on α .In order to apply the steepest descent method to study (cid:98) K ( α, β ), we need tounderstand the critical points of the phase (cid:101) Ψ α,β . Notice first that when α = β , thephase (cid:101) Ψ α,α has a critical point at x = y = α x and ξ = α ξ . Moreover, this criticalpoint is non-degenerate since the Hessian of (cid:101) Ψ α,α is given there by(2.43) (cid:104) α (cid:105) − D x,x Φ T ( α, α x ) I I − I − I (cid:104) α (cid:105) − D x,x Φ S ( α x , α ) and the determinant of this matrix is( − n det (cid:16) (cid:104) α (cid:105) − (cid:16) D x,x Φ T ( α, α x ) + D x,x Φ S ( α x , α ) (cid:17)(cid:17) (cid:54) = 0 . This determinant is in fact bounded away from zero uniformly in α and α since theimaginary part of the matrix (cid:104) α (cid:105) − (cid:16) D x,x Φ T ( α, α x ) + D x,x Φ S ( α x , α ) (cid:17) is uniformlydefinite positive (this is part of Definition 1.7 of an admissible phase). Hence, theHessian (2.43) of the phase (cid:101) Ψ α,α is uniformly invertible.We return now to our situation where d KN ( α, α ) , d KN ( β, α ) ≤ η . It followsfrom the Implicit Function Theorem that, provided η is small enough, (cid:101) Ψ α,β has aunique critical point z ( α, β ) = ( x ( α, β ) , ξ ( α, β ) , y ( α, β )) near ( α ,x , α ,ξ , α ,x ), andthis critical point is non-degenerate. Notice that if y c ( α, β ) denotes the critical pointof the phase y (cid:55)→ Φ T ( α, y ) + Φ S ( y, β ) (which is defined by similar considerations)then we have x ( α, β ) = y ( α, β ) = y c ( α, β )and ξ ( α, β ) = − ∇ x Φ T ( α, y c ( α, β )) (cid:104) α (cid:105) = ∇ y Φ S ( y c ( α, β ) , β ) (cid:104) α (cid:105) . The critical value of (cid:101) Ψ α,β is given by (cid:101) Ψ α,β ( x ( α, β ) , ξ ( α, β ) , y ( α, β )) = (cid:104) α (cid:105) − Φ T S ( α, β ) , where the phase Φ T S is the critical value of y (cid:55)→ Φ T ( α, y ) + Φ S ( y, β ). We can applythe Holomorphic Morse Lemma 1.15 to (cid:101) Ψ. Provided η is small enough, there existholomorphic Morse coordinates ρ α,β defined on a fixed complex neighbourhood W of D × (cid:101) D × D and whose image contains a ball of fixed radius in C n such that forevery z = ( x, ξ, y ) ∈ W we have (cid:101) Ψ α,β ( z ) = Φ T S ( α, β ) (cid:104) α (cid:105) + i ρ α,β ( z ) . (2.44)Since the proof of Lemma 1.15 is constructive – see [ Sj¨o82 , Lemma 2.7] – we obtainbounds on the derivatives of ρ α,β and ρ − α,β in function of the derivatives of (cid:101) Ψ α,β .Via symbolic estimates, we deduce that these estimates are uniform as (cid:104) α (cid:105) → ∞ . 04 2. FBI TRANSFORM ON COMPACT MANIFOLDS Differentiating (2.44), we find that for z = z ( α, β ),(2.45) i t (cid:0) Dρ α,β ( z ( α, β )) (cid:1) Dρ α,β ( z ( α, β )) = D z,z (cid:101) Ψ α,β ( z ( α, β )) . We denote Dρ α,α = A + iB the decomposition into real and imaginary part at z = z ( α, α ), and deduce t AA − t BB = Im D z,z (cid:101) Ψ α,β ( z ( α, β )) . When α = β , we know that Im D z,z (cid:101) Ψ α,α ( z ( α, α )) ≥ 0. Hence t AA ≥ t BB . Itfollows that the kernel of A is contained in that of B . However, since Dρ α,α ( z ( α, α ))is invertible, these two kernels cannot intersect non-trivially. Hence A is invertible,and this comes with uniform estimates, so it remains true when α (cid:54) = β , provided η is small enough.Then, it follows by the Implicit Function Theorem that V α,β := ρ α,β ( D × (cid:101) D × D )is a graph over the reals, i.e. V α,β = (cid:8) w + iF α,β ( w ) : w ∈ Re V α,β (cid:9) , where the function F α,β from a neighbourhood of 0 in R n to R n is real-analytic(with symbolic estimates in α and β ). Then, we use the coordinates ρ α,β to changevariables and write (cid:98) K ( α, β ) as the contour integral (cid:98) K ( α, β ) = e ih Φ TS ( α,β ) (cid:18) λ π (cid:19) n (cid:90) V α,β e − λ w g α,β ◦ ρ − α,β ( w ) Jρ − α,β ( w )d z, (2.46)where Jρ − α,β = det (cid:16) Dρ − α,β (cid:17) denotes the Jacobian of ρ − α,β . Then, we use thehomotopy ( t, w ) (cid:55)→ w + itF α,β ( w ) to shift contour in (2.46) and replace the integralover V α,β by an integral over Re V α,β . We denote by σ α,β ( w ) the amplitude in theprevious integral, to find, using Stokes’ Formula that(2.47) (cid:98) K ( α, β ) = e ih Φ TS ( α,β ) (cid:18) λ π (cid:19) n (cid:90) Re V α,β e − λ w ˜ σ α,β ( w )d w + D ( α, β ) , where ˜ σ α,β denotes a G s almost analytic extension for σ α,β (obtained by replacing χ, θ and p by their almost analytic extensions given either by Lemma 1.1 or Remark1.10) and D ( α, β ) is the main part in Stokes’ Formula. The main technical point inthe proof of Lemma 2.10 will be to get the following bound on D ( α, β ). Lemma . Assume that η and τ are small enough. Then there exists C > such that D ( α, β ) = O C ∞ (cid:32) exp (cid:32) − λ s − C − λ s d KN ( α, β ) C (cid:33)(cid:33) . Let us postpone the proof of Lemma 2.11 to the end of this section and explainhow it allows us to end the proof of Lemma 2.10. From Lemma 2.11, the term D ( α, β ) may be considered as part of the error term in (2.33). Then, we knowthat Re V α,β contains a fixed ball D . From coercivity of the phase w (cid:55)→ w , wemay split the integral in (2.47) between the integral over D and the integral overRe V α,β \ D and consider the latter as part of the error term in (2.47). Indeed, itfollows from an L estimate that the integral over Re V α,β \ D is a O C ∞ (cid:18) exp (cid:18) − λC (cid:19)(cid:19) . Notice indeed that the dependence on α and β of the domain of integration issuperficial since the integrand is supported away from the boundary of the domain .2. LIFTING PSEUDO-DIFFERENTIAL OPERATORS 105 of integration Re V α,β . The integral over D e ( α, β ) := (cid:18) λ π (cid:19) n (cid:90) D e − λ w ˜ σ α,β ( w )d w is a Gaussian integral with a G s amplitude, so that it is G s symbol. It will besufficient to study it via the C ∞ stationary phase method. In particular, it is givenat first order by e ( α, β ) = 1(2 πh ) n e ( α, β ) mod h − n +1 S m − KN , where the symbol e ( α, β ) is given on the diagonal by e ( α, α ) = (2 πh ) n ˜ σ α,α (0) = (2 πh ) n ˜ g α,α ◦ ρ − α,α (0) Jρ − α,β (0))= ˜ p ( α ) (2 πh ) n (cid:104) α (cid:105) − n Jρ − α,α (0) a ( α, α x ) b ( α x , α ) . Here, the bump function χ and θ do not play any role since they take the value 1respectively at α x and α ξ . Recalling (2.45) and the expression (2.43) for the Hessianof (cid:101) Ψ α,α we find that (cid:104) α (cid:105) − n Jρ − α,α (0) = (cid:16) det (cid:16) − i (cid:16) D x,x Φ T ( α, α x ) + D x,x Φ S ( α, α x ) (cid:17)(cid:17)(cid:17) − , for a certain determination of the square root. Then, we have e ( α, α ) = ˜ p ( α ) (2 πh ) n a ( α, α x ) b ( α x , α ) (cid:16) det (cid:16) − i (cid:16) D x,x Φ T ( α, α x ) + D x,x Φ S ( α, α x ) (cid:17)(cid:17)(cid:17) . (2.48)In this expression, we may replace a and b by their principal part. It follows thenfrom the fact that ST = I and from the expression given in Lemma 2.8 for theprincipal symbol of ST that(2 πh ) n a ( α, α x ) b ( α x , α ) (cid:16) det (cid:16) − i (cid:16) D x,x Φ T ( α, α x ) + D x,x Φ S ( α, α x ) (cid:17)(cid:17)(cid:17) = 1(2.49)when α is real. Then, by analytic continuation principle, this formula remains truefor complex α . We see that the good determination of the square root in (2.48)gives a right hand side equals to 1 in (2.49) by a homotopy argument (we have thegood sign on the reals).In fact, if we have the wrong determination of the square root in (2.48), itmeans that we misoriented V α,β in (2.46), and this two signs error cancel. Finally,with (2.48) and (2.49), we find that e ( α, α ) = ˜ p ( α ), and Lemma 2.10 is proved inthe case s > (cid:3) In order to complete the proof of Lemma 2.10 in the case s > 1, we need nowto establish Lemma 2.11. Proof of Lemma 2.11. We recall that we are studying points α = ( α x , α ξ )and β = ( β x , β ξ ) ∈ Λ that lie in a neighbourhood of α = ( α ,x , α ,ξ ) ∈ T ∗ M . Firstof all, notice that we have D ( α, β ) = e ih Φ TS ( α,β ) (cid:18) λ π (cid:19) n (cid:90) B ∗ (cid:18) e − λ ˜ w ( ρ α,β ) ∗ ( ¯ ∂ ˜ g α,β ∧ d z )( ˜ w ) (cid:19) , where B denotes the homotopy ( t, w ) (cid:55)→ w + itF α,β ( w ) and the integral is over[0 , × Re V α,β . We need to understand how the imaginary part of the phase and thedecay of ¯ ∂ ˜ g α,β collaborate to make the integrand small. The idea will be to reduce 06 2. FBI TRANSFORM ON COMPACT MANIFOLDS this to some positivity estimate on the phase. To do so write z tα,β ( w ) = ρ − α,β ( B ( t, w )).In view of the estimate given in Remark 1.7, we observe that( ρ α,β ) ∗ ( ¯ ∂ ˜ g α,β ∧ d z )( B ( t, w )) = O C ∞ (cid:32) exp (cid:32) − C | Im z tα,β | s − (cid:33)(cid:33) . Next, we observe thatIm (cid:18) Φ T S ( α, β ) (cid:104) α (cid:105) + i B ( t, w ) (cid:19) = Im (cid:16) (cid:101) Ψ α,β ( z α,β ( w ) (cid:17) + 1 − t | F α,β ( w ) | . From this, we deduce that we have pointwise bounds e ih Φ TS ( α,β ) (cid:18) λ π (cid:19) n B ∗ (cid:18) e − λ w ( ρ α,β ) ∗ ( ¯ ∂ ˜ g α,β ∧ d z )( w ) (cid:19) = O L ∞ (cid:32) λ n exp (cid:32) − λ Im (cid:16) (cid:101) Ψ( z ( w ) (cid:17) − λ − t | F ( w ) | − C | Im z t | s − (cid:33)(cid:33) . (we omitted the α, β dependence in the second line). Each time we differentiatewith respect to α, β , we lose at most a power of λ in this estimate, so that it willsuffice for our purpose to prove that, for w ∈ Re V α,β and t ∈ [0 , λ Im (cid:16) (cid:101) Ψ α,β ( z α,β ( w ) (cid:17) + λ − t | F α,β ( w ) | + 1 C | Im z tα,β ( w ) | s − is larger than C − λ s − + C − λ s d KN ( α, β )for some C > 0. To obtain such an estimate, we start by observing that z α,β ( w ) isalways a real point. Since ρ α,β is uniformly bi-Lipschitz, we deduce that, for some C > | Im z tα,β ( w ) | ≤ | Im z α,β ( w ) | + C ( | t − || F α,β ( w ) | ) ≤ C (1 − t ) | F α,β ( w ) | . This suggests to consider for u = (cid:12)(cid:12) F α,β ( w ) (cid:12)(cid:12) the infimum (we perform the change ofvariable “ r = 1 − t ”)(2.51) inf r ∈ [0 , λru + 1( ru ) s − . The function of r above has a global minimum over R ∗ + , attained at r ∗ = (cid:18) s − λu s − s − (cid:19) − s . Thus, the infimum we look for is always greater than the global minimum whichhappens to be(2.52) s ( s − s − ( λu ) /s . However, if the argument r ∗ of the minimum is greater than 1, then the infimum(2.51) is attained at r = 1 (we compute the infimum of a decreasing function of r over [0 , r ∗ ≥ 1, the infimum (2.51) is greater than λu + u − s − ≥ u − s − = (cid:16) r ss − ∗ ( s − λ (cid:17) s − ≥ ( s − s − λ s − . .2. LIFTING PSEUDO-DIFFERENTIAL OPERATORS 107 On the other hand, if r ∗ ≤ 1, then we may bound from below the global infimum(2.52) by noticing that( λu ) s = r − s − ∗ ( s − s − s (2 s − λ s − ≥ s − s − s (2 s − λ s − . Finally, we find that there is a constant C that only depends on s such that theinfimum (2.51) is bounded from below by(2.53) 1 C (cid:16) ( λu ) s + λ s − (cid:17) . This gives a lower bound for the quantity in (2.50) in the form(2.54) λ Im (cid:16) (cid:101) Ψ α,β ( z α,β ( w ) (cid:17) + λ s − C + (cid:0) λ | F α,β ( w ) | (cid:1) s C , for some C > 0. Consequently, the key estimate Lemma 2.11 will be a consequenceof the following positivity estimate on the phase (cid:101) Ψ α,β : Lemma . Let C > . Then, if η and τ are small enough, there is a constant C (cid:48) > such that for w ∈ Re V α,β we have λ Im (cid:16) (cid:101) Ψ α,β ( z α,β ( w )) (cid:17) + λ s − C + (cid:0) λ | F α,β ( w ) | (cid:1) s C ≥ λ s − C (cid:48) + λ s d KN ( α, β ) C (cid:48) . (2.55) (cid:3) In the regions where the imaginary part of the phase alone is not sufficient tocontrol the right hand side, we will see that | F | is of the same order as |∇ z Φ | , sothat this estimates expresses the fact that Im Ψ and ∇ z Ψ cannot be simultaneouslytoo small. Proof of Lemma 2.12. Recall that the phase (cid:101) Ψ α,β is defined by (2.41). Weset z α,η ( w ) = ( x α,β ( w ) , ξ α,β ( w ) , y α,β ( w )). When there is no ambiguity, we will writerespectively z, x, ξ and y instead of z α,β ( w ), x α,β ( w ), ξ α,β ( w ) and y α,β ( w ). Noticethat, since x, y and ξ are realIm (cid:101) Ψ α,β ( z ) = Im (cid:18) Φ T ( α, x ) + Φ S ( y, β ) (cid:104) α (cid:105) (cid:19) . Let us now obtain some useful preliminary estimates. First, notice that Λ is C close to T ∗ M . The arguments that led to Lemma 2.1 shows that since α, β ∈ Λ wehave, for some C > λ = (cid:104) α (cid:105) /h ), d KN (Im α, Im β ) ≤ Cτ λ s − d KN ( α, β ) , (2.56)in addition to Lemma 2.1 of course. In particular, provided τ is small enough, forsome C > 0, we have1 C d KN (Re α, Re β ) ≤ d KN ( α, β ) ≤ Cd KN (Re α, Re β ) . The first estimate we prove on the phase is that (for some C > (cid:18) Φ T ( α, x ) + Φ S ( y, β ) (cid:104) α (cid:105) (cid:19) ≥ C (cid:16) | x − α x | + | y − β x | (cid:17) − Cτ λ s − − − Cτ λ s − ( | α x − x | + | y − β x | + d KN ( α, β )) . (2.57) 08 2. FBI TRANSFORM ON COMPACT MANIFOLDS For this, denote respectively by Q T,α and Q S,β the second derivatives of r (cid:55)→ T ( α, r ) at r = α x and r (cid:55)→ S,β ( r, β ) at r = β x . Taylor’s formula at order 2gives thenΦ T ( α, x ) + Φ S ( y, β ) = (cid:10) α ξ , α x − x (cid:11) + (cid:10) β ξ , y − β x (cid:11) + Q T,α ( y − α x , y − α x ) + Q S,β ( y − β x , y − β x )+ (cid:104) α (cid:105)O (cid:16) | x − α x | + | y − β x | (cid:17) . (2.58)We start by estimating the imaginary part of the first line in the right hand side of(2.58). To do so, we writeIm (cid:0)(cid:10) α ξ , α x − x (cid:11) + (cid:10) β ξ , y − β x (cid:11)(cid:1) = (cid:10) Im α ξ , Re α x − x (cid:11) + (cid:10) Im β ξ , y − Re β x (cid:11) + (cid:10) Re α ξ − Re β ξ , Im α x (cid:11) + (cid:10) Re β ξ , Im α x − Im β x (cid:11) . Since | Im α | , | Im β | (cid:46) τ λ /s − , and according to (2.56), we estimate each term tofind 1 (cid:104) α (cid:105) (cid:12)(cid:12) Im (cid:0)(cid:10) α ξ , α x − x (cid:11) + (cid:10) β ξ , y − β x (cid:11)(cid:1)(cid:12)(cid:12) ≤ Cτ λ s − ( | Re α x − x | + | y − Re β x | + d KN ( α, β )) . We also computeIm( Q T,α ( x − α x , x − α x )) = Im Q T,α ( x − Re α x , x − Re α x ) − Im Q T,α (Im α x , Im α x ) + 2 Re Q T,α (Im α x , x − Re α x ) . By definition of an admissible phase, the matrix Im Q T,α / (cid:104) α (cid:105) is uniformly definitepositive (provided that τ is small enough). Using | Im α | (cid:46) τ λ /s − again, we findthat Im (cid:0) Q T,α ( x − α x , x − α x ) (cid:1) (cid:104) α (cid:105)≥ C | x − Re α x | − Cτ λ s − | x − Re α x | − Cτ λ s − ≥ C | x − α x | − Cτ λ s − | x − α x | − Cτ λ s − . We have a similar estimate for the term involving Q S,β and (2.57) follows (the O in (2.58) is negligible thanks to the positive quadratic term). The λ /s − term iscontrolled by λ / (2 s − − because for s ≥ 1, we have 2 /s − ≤ / (2 s − C > 0, for τ small enough we have: λ (cid:16) | x − α x | + | y − β x | (cid:17) + λ s − + (cid:0) λ | F α,β ( w ) | (cid:1) s ≥ Cτ λ s ( | α x − x | + | y − β x | + d KN ( α, β ))(2.59)We get rid first of the term Cτ λ /s ( | α x − x | + | y − β x | ) on the right hand side. Todo so, there are two possibilities, the first one is that | α x − x | + | β x − y | ≥ λ s − , (2.60)in which case Cτ λ s ( | α x − x | + | y − β x | ) is controlled by the term C − λ ( | α x − x | + | y − β x | ) on the left hand side (provided that τ is small enough). If (2.60) doesnot hold then Cτ λ s ( | α x − x | + | β x − y | ) ≤ Cτ λ s − ≤ Cτ λ s − .2. LIFTING PSEUDO-DIFFERENTIAL OPERATORS 109 and this term is controlled by C − λ s − on the left hand side of (2.59).Dealing with this first term, we have further reduced our problem, and it sufficesnow to prove that, given C > 0, for τ > λ s − + (cid:0) λ | F α,β ( w ) | (cid:1) s + λ ( | x − α x | + | y − β x | ) ≥ Cτ λ s d KN ( α, β ) . Without loss of generality, we may assume that d KN ( α, β ) ≥ λ s − , (2.62)since otherwise the right hand side of (2.61) is controlled by λ s − in the left handside. We may also assume that | x − α x | + | y − β x | ≤ (cid:15)d KN ( α, β ) , (2.63)where (cid:15) > λ ( | x − α x | + | y − β x | ) would control λd KN ( α, β ) , itself larger than λ /s d KN ( α, β ), thanks to (2.62).It follows from (2.63) that | α x − β x | ≤ | x − α x | + | y − β x | + | x − y | ≤ | x − y | + (cid:15)d KN ( α, β ) , and thus | α x − β x | ≤ C | x − y | + C(cid:15) | α ξ − β ξ | / (cid:104) α (cid:105) . We obtain, using (2.56) andassuming that τ is small enough, that(2.64) d KN ( α, β ) ≤ C | x − y | + C | Re α ξ − Re β ξ |(cid:104) α (cid:105) . We want now to give a lower bound on (cid:12)(cid:12) F α,β ( w ) (cid:12)(cid:12) . To do so, recall the gradientof the phase (cid:101) Ψ α,β : ∇ z (cid:101) Ψ α,β ( z ) = ∇ x Φ T ( α,x ) (cid:104) α (cid:105) + ξx − y ∇ y Φ S ( y,β ) (cid:104) α (cid:105) − ξ Hence, it follows from Definition 1.7 of an admissible phase and Taylor’s formulathat (the value of C > (cid:12)(cid:12)(cid:12) Re ∇ z (cid:101) Ψ α,β ( z ) (cid:12)(cid:12)(cid:12) ≥ C (cid:12)(cid:12)(cid:12)(cid:12) Re α ξ (cid:104) α (cid:105) − ξ (cid:12)(cid:12)(cid:12)(cid:12) + 1 C (cid:12)(cid:12)(cid:12)(cid:12) ξ − Re β ξ (cid:104) α (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) + 1 C | x − y |− C ( | x − α x | + | y − β x | ) ≥ C (cid:18) | Re α ξ − Re β ξ |(cid:104) α (cid:105) + | x − y | (cid:19) − C(cid:15)d KN ( α, β ) ≥ (cid:18) C − C(cid:15) (cid:19) d KN ( α, β ) ≥ C d KN ( α, β )(2.65)where we applied (2.64) and (2.63) (assuming that (cid:15) > | Im α | , | Im β | (cid:46) τ λ /s − , (2.63) and (2.64),that (cid:12)(cid:12)(cid:12) Im ∇ z (cid:101) Ψ α,β ( z ) (cid:12)(cid:12)(cid:12) ≤ C ( τ + (cid:15) ) d KN ( α, β ) . (2.66)Now we compute the gradient of the phase (cid:101) Ψ α,β in Morse coordinates and find i t Dρ α,β ( z ) − ∇ z (cid:101) Ψ α,β ( z ) = w + iF α,β ( w ) . 10 2. FBI TRANSFORM ON COMPACT MANIFOLDS Hence, we have F α,β ( w ) = Re( t Dρ α,β ( z ) − ) Re ∇ z (cid:101) Ψ α,β ( z ) − Im( t Dρ α,β ( z ) − ) Im ∇ z (cid:101) Ψ α,β ( z ) . Following the argument after (2.45), applied to Dρ − α,β , we see that its real partis uniformly invertible. Indeed, it suffices to notice that the imaginary part of − ( D z,z (cid:101) Ψ α,α ) − is positive at the critical point.Hence, it follows from (2.65) and (2.66) that (for some C > τ and (cid:15) are small enough) (cid:12)(cid:12) F α,β ( w ) (cid:12)(cid:12) ≥ C d KN ( α, β ) − C ( τ + (cid:15) ) d KN ( α, β ) ≥ C d KN ( α, β ) . Notice also that, since the quantity | F α,β ( w ) | remains bounded, we have the lowerbound | F α,β ( w ) | /s ≥ | F α,β ( w ) | /C , and (2.61) holds for τ small enough, so thatthe proof of Lemma 2.12 is complete. (cid:3) It remains to explain why Lemmas 2.9 and 2.10 remains true in the case s = 1.The main difficulty (that makes us distinguish that case) is the lack of partitionof unity in the real-analytic category. This issue is solved by an application ofProposition 1.12. In fact, most of the proof of Lemma 2.10 in the case s > P . In the case s = 1, these integrals are null, hence a muchless complicated proof. Sketch of proof of Lemmas 2.9 and 2.10 in the case s = 1 . We startwith an application of Proposition 1.12 to compute the kernel of the operator P S .We see, as in Remark 2.2, that the kernel of P S has the same properties as the kernelof an adjoint analytic FBI transform, except maybe the order and the ellipticity ofthe symbol. Hence, we will assume for simplicity that P = I and compute only thekernel of T S . The only difference in the proof in the general case is when computingthe principal part of the symbol, but this computation has already been tackled inthe case s > s = 1.The kernel of T S is given by the formula K T S ( α, β ) = (cid:90) M K T ( α, y ) K S ( y, β ) d y. (2.67)This is much simpler than the formula (2.35) that we used in the case s > K T S ( α, β ) is negligible when α x and β x are away from each other, we apply the same non-stationary phase argumentas in the proof of Lemma 2.4 (or as in the case s > y is near α x , the phasein K T ( α, y ) is non-stationary and K T ( y, β ) is small in G . The converse happenswhen y is near β x , and when y is away from α x and β x both K T ( α, y ) and K S ( y, β )are small. Here, notice that we do not need partitions of unity to split the integral(2.67) due to the coercivity of the imaginary parts of the phases Φ T and Φ S .When α x and β x are closed, we may just neglect the y ’s in (2.67) that are awayfrom α x and β x , and work in local coordinates, studying the integral on some ball D (the points α x and β x remain uniformly away from the boundary of D ) (cid:98) K ( α, β ) := (cid:90) D e i Φ T ( α,y )+Φ S ( y,β ) h a ( α, y ) b ( β, y ) d y. (2.68)If α ξ and β ξ are away from each other (for the Kohn–Nirenberg metric), it followsfrom Definition 1.7 of an admissible phase that the phase in (2.68) is non-stationary .2. LIFTING PSEUDO-DIFFERENTIAL OPERATORS 111 (provided that α x and β x are close enough), and we may apply the non-stationaryphase method Proposition 1.2 after a proper rescaling as in § s = 1.We turn now to the proof of Lemma 2.10: we want consequently to understand(2.68) when α and β are near the diagonal. As in the case s > 1, we will rely on anapplication of the Stationary Phase Method. However, the situation is much simplerhere, since we can apply Proposition 1.6. In particular, we do not need to restrictto α, β ∈ Λ: the estimate (2.33) hold for α, β ∈ ( T ∗ M ) (cid:15) for some small (cid:15) > P ) in the case s = 1. Moreover, the error term in (2.33) takes thesimpler form O (exp( − λ/C )) in the case s = 1 (we recall that λ denotes the largeparameter (cid:104)| α |(cid:105) /h ). Finally, notice that, applying Proposition 1.6, we only need tostudy α = β real, since this implies an estimate for (cid:98) K ( α, β ) when α, β ∈ ( T ∗ M ) (cid:15) are close to each other.Choose a reference point z in the interior of D . We apply Proposition 1.6with “Φ( x, ξ, y ) = Φ( α x , β x , α ξ , β ξ , y ) = (Φ T ( α, y ) + Φ S ( y, β )) / (cid:104) α (cid:105) ”, “ x = ( z, z )”and “ F = (cid:8) ( α ξ , α ξ ) : α ξ ∈ R n (cid:9) ”. This will allow us to understand (cid:98) K ( α, β ) for α, β ∈ ( T ∗ M ) (cid:15) close to the diagonal and such that α x and β x are close to z .Lemma 2.10 follows then by compactness of M and a partition of unity argument(we do not claim holomorphicity for the symbol e ). Let us check the hypotheses ofProposition 1.6.When α = β is real the phase y (cid:55)→ (Φ T ( α, y ) + Φ S ( y, β )) / (cid:104) α (cid:105) has non-negativeimaginary part for real y , because so do Φ T ( α, y ) and Φ S ( y, α ) by assumption.When y is in ∂D , the imaginary part of the phase is uniformly positive since z / ∈ ∂D and Φ T is admissible. Finally, the phase has a critical point at y = z which isnon-degenerate (the Hessian of the phase has uniformly definite positive imaginarypart). Hence, this critical point is isolated by the Implicit Function Theorem, andthus the only critical point of the phase in D if η is small enough.From Proposition 1.6, we see that for α, β ∈ ( T ∗ M ) (cid:15) near the diagonal we have (cid:98) K ( α, β ) = e i Φ TS ( α,β ) h (cid:88) ≤ k ≤ λ/C h k e k ( α, β ) + O (cid:18) exp (cid:18) − λC (cid:19)(cid:19) , (2.69)for some formal analytic symbol (cid:80) k ≥ h k e k and arbitrarily large C . We mayreplace the sum in (2.69) by any realization e of the formal symbol (cid:80) k ≥ h k e k .It follows from Lemma 1.16 that Im Φ T S ( α, β ) is non-negative when α and β arereal. Consequently, if α, β ∈ ( T ∗ M ) (cid:15) , the factor e i Φ TS ( α,β ) h is an O (exp( C(cid:15) λ )).Hence, taking (cid:15) > K T S has the announced local structure nearthe diagonal. In fact, we get a much better estimate since we have an asymptoticexpansion for e that holds in the sense of realization of formal analytic symbols(since Φ T S has non-negative imaginary part on the real). (cid:3) Now that we have establisheda precise description of the kernel K T P S , we will be able to understand the actionof pseudors P on the spaces H k Λ , or equivalently the action of T Λ P S Λ on L k (Λ).Recall that this is the exponentially weighted space L (Λ , (cid:104)| α |(cid:105) k e − H ( α ) /h d α ). Themain result of this section is: Proposition . Let P be a G s pseudor of order m on M . If τ is smallenough and Λ is a ( τ , s ) -Gevrey adapted Lagrangian, then, for h small enough andevery k ∈ R , the operator T Λ P S Λ is bounded from the space L k (Λ) to the space L k − m (Λ) , with norm uniformly bounded when h tends to . 12 2. FBI TRANSFORM ON COMPACT MANIFOLDS In particular, P is bounded from H k Λ to H k − m Λ and Π Λ = T Λ S Λ is bounded from L k (Λ) to itself. In fact, we can say a bit more about the operator T Λ P S Λ . We are indeed alreadyable to prove a weak version of the multiplication formula Proposition 2.11 (whichis itself a precise version of Theorem 7). We recall that the notion of G s principalsymbol has been defined in Proposition 1.10, and that the associated almost analyticextensions are discussed in Remark 1.10. Proposition . Under the assumption of Proposition 2.4, if we denote by p Λ the restriction of an almost analytic extension of a G s principal symbol p to Λ , thenwe have T Λ P S Λ = p Λ Π Λ + O L k (Λ) → L k − m + 12 (Λ) ( h )= Π Λ p Λ + O L k (Λ) → L k − m + 12 (Λ) ( h ) . We will also need the following corollary of Proposition 2.4, which implies thatthe spaces H k Λ are legitimate functional spaces. Corollary . Assume that τ is small enough and that Λ is a ( τ , -adaptedLagrangian. Then, for h small enough, there is R > such that, for all k ∈ R ,the space E ,R ( M ) is dense in H k Λ . Moreover, if u ∈ L ( M ) ∩ H k Λ , then there isa sequence ( u n ) n ∈ N in E ,R ( M ) such that ( u n ) n ∈ N converges to u both in L ( M ) and in H k Λ . Finally, we will prove that in the absence of deformation, the H k Λ ’s are just theusual Sobolev spaces. Corollary . Let h > be small enough. Then, for every k ∈ R , the space H kT ∗ M is the usual semi-classical Sobolev space of order k on M , with uniformlyequivalent norms as h tends to . For the proof of these results, it is convenient to introduce the “reduced” kernel(2.70) K Λ T P S ( α, β ) := e H ( β ) − H ( α ) h K T P S ( α, β ) . This is now only defined on Λ × Λ, contrary to K T P S . Now we have all the piecesto explain the reason why we have to work with I-Lagrangians, and why the weight H was introduced in § K T P S is exponentially small far from thediagonal of Λ × Λ, sufficiently so that it suffices to study its behaviour near thediagonal. There, we have an expansion in the form(2.71) K T P S ( α, β ) = e ih Φ TS ( α,β ) a ( α, β ) + O (cid:32) exp (cid:32) − λ s − + λ s d KN ( α, β ) C (cid:33)(cid:33) . This expansion suggests to deal with T P S as an FIO with complex phase on Λ, inthe spirit of [ MS75 ]. For this machinery to work, we need to have a lower boundon Im Φ T S . When α, β are real, Im Φ T S ≥ 0, and only vanishes when α = β . Thisimplies that T P S is microsupported near the diagonal. However, the restriction ofIm Φ T S to Λ × Λ is not necessarily non-negative (a necessary assumption to applythe machinery from [ MS75 ]). Thankfully, we can get around this using the action H introduced in § Lemma . There is η > such that, if τ is small enough and Λ is a ( τ , -adapted Lagrangian, then there is C > such that, for every α, β ∈ Λ with d KN ( α, β ) ≤ η , we have C (cid:104)| α |(cid:105) d KN ( α, β ) ≤ Im Φ T S ( α, β ) + H ( α ) − H ( β ) ≤ C (cid:104)| α |(cid:105) d KN ( α, β ) . (2.72) .2. LIFTING PSEUDO-DIFFERENTIAL OPERATORS 113 The idea behind this statement is that for α, β close enough,Φ T S = (cid:104) α x − β x , β ξ (cid:105) + O ( (cid:104)| α |(cid:105) d KN ( α, β ) ) , and Im(Φ T S − (cid:104) α x − β x , β ξ (cid:105) ) ≥ (cid:104)| α |(cid:105) C d KN ( α, β ) . In particular, the methods of [ MS75 ] would prove that Π Λ is bounded on anexponentially weighted space L (Λ , e − H ( α ) /h dα ), if we had for (cid:15) > | H ( β ) − H ( α ) − Im( (cid:104) α x − β x , β ξ (cid:105) ) | ≤ (cid:15) (cid:104)| α |(cid:105) d KN ( α, β ) . This implies exactly that d H = − Im θ | Λ , where θ is the canonical Liouville 1-form.The existence of a solution implies that Λ is an exact I-Lagrangian, and the desiredbound holds if Λ is sufficiently C close to the reals. This is the reason why we haveto work with adapted Lagrangians, and introduce the weight H . Proof of Lemma 2.13. Define the function A ( α, β ) = Im Φ T S ( α, β )+ H ( α ) − H ( β ) near the diagonal on Λ × Λ. By definition, A satisfies symbolic estimates (oforder 1). First, since Φ T S ( α, α ) = 0, we have that A ( α, α ) = 0. Then using (2.8)and the fact that d α Φ T S ( α, α ) = d α Φ T ( α, α x ) = θ | Λ ( α )(2.73)and d β Φ T S ( α, α ) = d α Φ S ( α, α x ) = − θ | Λ ( α ) , (2.74)we find that the differential of A vanishes on the diagonal. The expression (2.73) and(2.74) follows from the definition of the phase Φ T S . Since A vanishes at order 2 onthe diagonal of Λ × Λ and is a symbol of order 1, the inequality of the right in (2.72)holds. Hence the left inequality is equivalent to the existence of a constant C > α ∈ Λ the Hessian ∇ β,β A | β = α is larger than C − (cid:104)| α |(cid:105) g KN .Hence, the general case follows from the case Λ = T ∗ M by a perturbation argument.Consequently, we only need to make the proof in the case Λ = T ∗ M .Notice that in the case Λ = T ∗ M , we have H = 0, and hence A = Im Φ T S .Recall that Φ T S ( α, β ) is the critical value of y (cid:55)→ Ψ( β, y ) := Φ T ( α, y ) + Φ S ( y, β ).We fix the variable α , and denote by y ( β ) the critical point of this function. Fromthe Implicit Function Theorem, we have D β y ( β ) = − ( D y,y Ψ( β, y ( β ))) − D y,β Ψ( β, y ( β )) . Here, we work in coordinates and our convention regarding the notation D is that D β y is the matrix of the differential of β (cid:55)→ y ( β ) (in the canonical basis) and that D y,β Ψ is such that its columns have the size of y and its lines the size of β .Differentiating the expression Φ T S ( α, β ) = Ψ( β, y ( β )), it appears that we have D β Φ T S ( α, β ) = D β Ψ( β, y ( β )), and then that D β,β Φ T S ( α, β ) = D β,β Ψ( β, y ( β )) + D β,y Ψ( β, y ( β )) D β y ( β )= D β,β Ψ( β, y ( β )) − D β,y Ψ( β, y ( β ))( D y,y Ψ( β, y ( β ))) − D y,β Ψ( β, y ( β )) . Then, differentiating Φ S ( β x , β ) = 0 and d y Φ S ( β x , β ) = β ξ , we find that for β = α ,and y = y ( α ) = α x , we have (using the decomposition β = ( β x , β ξ )) D β,β Ψ( α, α x ) = (cid:18) D y,y Φ S ( α x , α ) − I − I (cid:19) , D β,y Ψ( α, α x ) = (cid:18) − D y,y Φ S ( α x , α ) I (cid:19) , 14 2. FBI TRANSFORM ON COMPACT MANIFOLDS and D y,y Ψ( α, α x ) = 2 i Im D y,y Φ S ( α x , α ) (recall that Φ T = − Φ S ). Let us write thedecomposition into real and imaginary part: D y,y Φ S ( α x , α ) = A + iB . Recall that B is definite positive by assumption. Summing up, we find thatIm D β,β Φ T S ( α, α ) = 12 (cid:18) AB − A + B − AB − − B − A B − (cid:19) We compute (cid:16) B − / u B / v (cid:17) (cid:18) AB − A + B − AB − − B − A B − (cid:19) (cid:32) B − / uB / v (cid:33) = v + u + | B − / AB − / u | − (cid:104) B − / AB − / u, v (cid:105) = u + ( B − / AB − / u − v ) . This certainly is a positive definite quadratic form. Since A + iB is a symbol oforder 1, with B being elliptic, this is uniform. We deduceIm D β,β Φ( α, α ) ≥ C (cid:104)| α |(cid:105) g KN . (cid:3) Remark . We proved Lemma 2.13 by computing directly the Hessian ofthe phase Φ T S . However, there is also a geometric proof of this lemma, in the spiritof the “fundamental” Lemma 1.16 and of the estimate Lemma 2.12 that was crucialin the proof of Lemma 2.10.Let us explain how the geometric proof of Lemma 2.13 goes. We only need toconsider the case Λ = T ∗ M . When α, β ∈ T ∗ M , then the phase y (cid:55)→ Φ T ( α, y ) +Φ S ( y, β ) is real for real y ’s. The imaginary part of this phase is pluriharmonic andconsequently the critical point of the phase is a saddle point for its imaginary part.The critical value Φ T S ( α, β ) of the phase y (cid:55)→ Φ T ( α, y ) + Φ S ( y, β ) is attained onthe steepest descent contour. Since the imaginary part of the phase is non-negativefor real y ’s, we see that the real is roughly aligned with the steepest phase descent.It implies that Im Φ T S ( α, β ) is larger than the minimum of the imaginary part ofΦ T ( α, y ) + Φ S ( y, β ) for real y . This proves that Im Φ T S ( α, β ) is essentially largerthan (cid:104)| α |(cid:105) | α x − β x | .When (cid:12)(cid:12) α ξ − β ξ (cid:12)(cid:12) / (cid:104)| α |(cid:105) is much larger than | α x − β x | , then the real part of ∇ y (Φ T ( α, y ) + Φ S ( y, β )) must be large when y is real. It implies that the criticalpoint of the phase is away from the real (at a distance essentially larger than (cid:12)(cid:12) α ξ − β ξ (cid:12)(cid:12) / (cid:104)| α |(cid:105) ) and consequently, we can improve the previous estimate by thesquare of this distance multiplied by (cid:104)| α |(cid:105) (the size of the Hessian).We now deduce Propositions 2.4 and 2.5 from Lemma 2.13. Proof of Proposition 2.4. The proof is an application of the celebratedSchur’s test [ Mar02 , Lemma 2.8.4]. We start by observing that near the diagonal,using (2.10), | H ( β ) − H ( α ) | h ≤ Cτ (cid:18) (cid:104)| α |(cid:105) h (cid:19) /s d KN ( α, β ) . Together with (2.11), this proves that the factor exp(( H ( β ) − H ( α ) /h ) in (2.70)can be absorbed in the remainders from (2.71). It follows that, for τ small enough,the reduced kernel of T Λ P S Λ satisfies (here α and β are on Λ and e is a symbolsupported near the diagonal of Λ × Λ) K Λ T P S ( α, β ) = e i Φ TS ( α,β )+ iH ( α ) − iH ( β ) h e ( α, β ) + O (cid:18)(cid:18) h (cid:104)| α |(cid:105) + (cid:104)| β |(cid:105) (cid:19) ∞ (cid:19) . .2. LIFTING PSEUDO-DIFFERENTIAL OPERATORS 115 In order to understand the action of T Λ P S Λ from L k (Λ) to L k − m (Λ) we study thekernel K ( α, β ) := K Λ T P S ( α, β ) (cid:104)| α |(cid:105) k − m (cid:104)| β |(cid:105) k = e i Φ TS ( α,β )+ H ( β ) − H ( α ) h e ( α, β ) (cid:104)| α |(cid:105) k − m (cid:104)| β |(cid:105) k + O (cid:18)(cid:18) h (cid:104)| α |(cid:105) + (cid:104)| β |(cid:105) (cid:19) ∞ (cid:19) . (2.75)In order to apply Schur’s test, we need to find uniform bounds on (cid:90) Λ | K ( α, β ) | d β and (cid:90) Λ | K ( α, β ) | d α. We will only deal with the first integral, hence we fix α ∈ Λ. We will also ignore thecontribution of the error term in (2.75), that is easily dealt with. Hence, we want abound, uniform in α and h , on (cid:90) Λ e − Im Φ TS ( α,β ) − H ( β )+ H ( α ) h | e ( α, β ) | (cid:104)| α |(cid:105) k − m (cid:104)| β |(cid:105) k d β. (2.76)Recalling that e belongs to the symbol class h − n S mKN (Λ × Λ), since e is supportednear the diagonal we have that, for some C > | e ( α, β ) | (cid:104)| α |(cid:105) k − m (cid:104)| β |(cid:105) k ≤ Ch − n . Hence, we must estimate (the integral is over β ∈ Λ) h − n (cid:90) d KN ( α,β ) ≤ η e − Im Φ TS ( α,β ) − H ( β )+ H ( α ) h d β. Here, provided η has been chosen small enough when applying Lemmas 2.9 and2.10, so that Lemma 2.13 applies, it follows from the estimate on the Jacobian inLemma 2.2 that the integral (2.76) is controlled by h − n (cid:90) R n exp (cid:32) − (cid:104)| α |(cid:105) x + (cid:104)| α |(cid:105) − ξ h (cid:33) d x d ξ and the result follows from the changes of variable x (cid:48) = h − (cid:104)| α |(cid:105) x and ξ (cid:48) = h − (cid:104)| α |(cid:105) − ξ . (cid:3) Proof of Proposition 2.5. We will only prove the first estimate, the otherone is obtained similarly (replacing an α by a β in the computation below). Weuse the same notations as in the previous proof, and denote by ˜ e the symbol e thatappears in the particular case P = I . Hence, we see that the reduced kernel of T Λ P S Λ − p Λ Π Λ : L k (Λ) → L k − m + (Λ) writes (cid:101) K ( α, β ) := e H ( β ) − H ( α ) h ( K T P S ( α, β ) − p Λ ( α ) K T S ( α, β )) (cid:104)| α |(cid:105) k − m + (cid:104)| β |(cid:105) k = e i Φ TS ( α,β )+ H ( β ) − H ( α ) h ( e ( α, β ) − p Λ ( α )˜ e ( α, β )) (cid:104)| α |(cid:105) k − m + (cid:104)| β |(cid:105) k + O C ∞ (cid:18)(cid:18) h (cid:104)| α |(cid:105) + (cid:104)| β |(cid:105) (cid:19) ∞ (cid:19) . As in the previous proof, we want to apply Schur’s test [ Mar02 , Lemma 2.8.4], andwe let as an exercise to the reader to deal with the error term. In order to apply 16 2. FBI TRANSFORM ON COMPACT MANIFOLDS Schur’s test, we use the expansion (2.34) to find that | e ( α, β ) − p Λ ( α )˜ e ( α, β ) | ≤ | e ( α, α ) − p Λ ( α )˜ e ( α, β ) | + Ch − n (cid:104)| α |(cid:105) m d KN ( α, β ) ≤ Ch − n +1 (cid:104)| α |(cid:105) m − + Ch − n (cid:104)| α |(cid:105) m d KN ( α, β ) , and then | e ( α, β ) − p Λ ( α )˜ e ( α, β ) | (cid:104)| α |(cid:105) k − m + (cid:104)| β |(cid:105) k ≤ Ch − n +1 (cid:104)| α |(cid:105) − + Ch − n (cid:104)| α |(cid:105) d KN ( α, β ) . We must consequently estimate h − n +1 (cid:90) d KN ( α,β ) ≤ η (cid:104)| α |(cid:105) − e − Im Φ TS ( α,β ) − H ( β )+ H ( α ) h d β and h − n (cid:90) d KN ( α,β ) ≤ η (cid:104)| α |(cid:105) d KN ( α, β ) e − Im Φ TS ( α,β ) − H ( β )+ H ( α ) h d β. By the same argument as in the proof of Proposition 2.4, we see that these integralsare controlled respectively by h − n +1 (cid:104)| α |(cid:105) − (cid:90) R n exp (cid:32) − (cid:104)| α |(cid:105) x + (cid:104)| α |(cid:105) − ξ h (cid:33) d x d ξ and h − n (cid:104)| α |(cid:105) (cid:90) R n (cid:18) | x | + | ξ |(cid:104)| α |(cid:105) (cid:19) exp (cid:32) − (cid:104)| α |(cid:105) x + (cid:104)| α |(cid:105) − ξ h (cid:33) d x d ξ, and the result follows from the same change of variable as in the proof of Proposition2.4. (cid:3) As promised, we deduce Corollaries 2.3 and 2.4 from Proposition 2.4. Proof of Corollary 2.3. We have a regularizing procedure that is indepen-dent of the choice of Λ. We start by considering u ∈ ( E ,R ) (cid:48) with R > (cid:15) > u (cid:15) = S T ∗ M e − (cid:15) (cid:104) α (cid:105) T T ∗ M u, (2.77)is an element of E ,R . Given Λ a ( τ , τ > u (cid:15) = S Λ e − (cid:15) (cid:104) α (cid:105) T Λ u. Let us now assume that u ∈ H k Λ for some such Lagrangian Λ. Then, by dominatedconvergence, as (cid:15) tends to 0, we see that exp( − (cid:15) (cid:104) α (cid:105) ) T Λ u converges to T Λ u in L k (Λ). Then, since Π Λ is bounded on that space by Proposition 2.4, we see that T Λ u (cid:15) = Π Λ exp( − (cid:15) (cid:104) α (cid:105) ) T Λ u converges to Π Λ T Λ u = T Λ u also in that space, andthus in H k Λ , FBI as (cid:15) → T T ∗ M is an isometry between L ( M ) and L (cid:0) T ∗ M (cid:1) (because S T ∗ M is the adjoint of T T ∗ M and S T ∗ M T T ∗ M = I ). In particular, L ( M ) = H T ∗ M . (cid:3) Proof of Corollary 2.4. We denote by H k ( M ) the usual Sobolev space oforder k on M . Since S = T ∗ is a left inverse for T , we see that T is an isometry andconsequently L ( M ) = H T ∗ M . From the fact that S = T ∗ , it follows that H − kT ∗ M isthe dual of H kT ∗ M (this is in fact a general fact that is valid for any Λ, as we willsee it in Lemma 2.24). We can consequently assume that k > .2. LIFTING PSEUDO-DIFFERENTIAL OPERATORS 117 Let u ∈ H k Λ . It follows from Remark 2.8 that u is a distribution. According toProposition 1.11, there is a G pseudor A with principal symbol (cid:104) α (cid:105) k . By Proposition2.4, we see that Au ∈ H T ∗ M = L ( M ) (since T ∗ M is (0 , k > u ∈ L ( M ). By elliptic regularity (in the C ∞ category), it follows that u ∈ H k ( M ) with norm controlled by (cid:107) u (cid:107) L ( M ) + (cid:107) Au (cid:107) L ( M ) ≤ C (cid:107) u (cid:107) H k Λ . Reciprocally, assume that u ∈ H k ( M ). By Proposition 1.11, we find a G pseudor B whose principal symbol is (cid:104) α (cid:105) − k . We can use Theorem 4 to construct aparametrix C for B , this is a G pseudor of order k such that BC = CB = I for h small enough. Since C has order k , the distribution Cu is an element of L ( M )and using Proposition 2.4, we see that u = BCu is an element of H k Λ , with normcontrolled by the H k ( M ) norm of u . (cid:3) Fromthe results of § § Proposition . Let s ≥ and u ∈ U s ( M ) . Then u belongs to G s ( M ) if andonly if T T ∗ M u belongs to G s ( M ) . We leave as an exercise to the reader to give a local version of Proposition 2.6.These considerations suggest to give the following definition of wave front set. Definition . Let s ≥ u ∈ ( E ,R ) (cid:48) for R large enough. We definethe G s wave front set WF G s ( u ) ⊆ T ∗ M \ { } of u in the following way: a point α ∈ T ∗ M \ { } does not belong to WF G s ( u ) if and only if there is a conicalneighbourhood Γ of α in T ∗ M and C > T T ∗ M u ( β ) = | β |→ + ∞ β ∈ Γ O (cid:18) exp (cid:18) − C (cid:104) β (cid:105) s (cid:19)(cid:19) . (2.78)We define mutatis mutandis the semi-classical wave front set WF G s ,h ( u ) of u = u ( h ). Remark . In R n , the usual definition [ H¨or71 ] of the wave front set WF G s ( u )may be rephrased using the flat transform T R n , defined in (6). The condition (2.78)is then replaced by (cid:12)(cid:12) T R n u ( α x , α ξ ; h ) (cid:12)(cid:12) ≤ C exp (cid:18) − Ch s (cid:19) , as h → , where α remains in a compact set. With this definition, one has to take h → | x − α x | , instead of (cid:104)| ξ |(cid:105)| x − α x | .However the two notions are the same.Lemmas 2.9 and 2.10 allow us to control the wave front set of the elementsof H (Lemma 2.14 below will be the main tool in the proof of Proposition 3.2).Notice that Lemma 2.14 has to be understood with h fixed, so that G is assumedto be elliptic as a classical symbol. Lemma . Let s ≥ and G be a symbol on ( T ∗ M ) (cid:15) (for some (cid:15) > ) inthe class S /sKN . For τ > define the symbol G = h − /s τ G and then Λ by (2.4) .Let α ∈ T ∗ M \ { } be such that G is negative and classically elliptic of order /s on a conical neighbourhood of α . Then, if τ and h are small enough, for any u ∈ H we have that α / ∈ WF G s ( u ) . 18 2. FBI TRANSFORM ON COMPACT MANIFOLDS Observe how this is coherent with the heuristics that the space H behavesformally as the space “exp(Op( G ) /h ) L ( M )”. Proof. Notice first that for some C > 0, the Lagrangian Λ is ( Cτ, τ is small enough.Let Γ be a conical neighbourhood of α on which G is negative and elliptic oforder δ . By assumption, there is a constant c > β ∈ Γ large enoughwe have G ( β ) ≤ − (cid:104) β (cid:105) s C . Choose some small (cid:15) > β ∈ Γ and γ ∈ Λ such that the distancebetween β and γ for the Kohn–Nirenberg metric is less than (cid:15) . Then, we haveIm Φ T S ( β, γ ) = Im Φ T S (cid:16) β, e − H ωIG ( γ ) (cid:17) + Im (cid:16) Φ T S ( β, γ ) − Φ T S ( β, e − H ωIG ( γ )) (cid:17) . By Lemma 2.13 (with Λ = T ∗ M ), the first term in the right-hand side is non-negative,so this is ≥ d β Im Φ T S ( β, γ ) · ( H ω I G ( γ )) + O (cid:16) τ h ( − s ) (cid:104) β (cid:105) /s − (cid:17) , ≥ d β Im Φ T S ( γ, γ ) · ( H ω I G ( γ )) + O (cid:16) ( (cid:15) + τ h − s ) τ h − s (cid:104) β (cid:105) s (cid:17) , ≥ − Im θ ( H ω I G ( γ )) + O (cid:16) ( (cid:15) + τ h − s ) τ h − s (cid:104) β (cid:105) s (cid:17) , Using (2.10), this is ≥ H ( γ ) − τ h − s G ( γ ) + O (cid:16) ( (cid:15) + τ h − s ) τ h − s (cid:104) β (cid:105) s (cid:17) , ≥ H ( γ ) − τ h − s G ( β ) + O (cid:16) ( (cid:15) + τ h − s ) τ h − s (cid:104) β (cid:105) s (cid:17) , ≥ H ( γ ) + τ h − s C (cid:104) β (cid:105) s , provided τ, h and (cid:15) were small enough. Notice here that the quantification on (cid:15) doesnot depend on τ nor h . Now, if u ∈ H we write T T ∗ M u = T T ∗ M S Λ T Λ u = T T ∗ M S Λ e Hh (cid:16) e − Hh T Λ u (cid:17) . Then, from Lemma 2.9, we see that the kernel of T T ∗ M S Λ e Hh is exponentiallydecaying when d KN ( β, γ ) ≥ (cid:15) (provided that τ is small enough, that is why itwas important that (cid:15) does not depend on τ ), and for β ∈ Γ and γ ∈ Λ such that d KN ( β, γ ) ≤ (cid:15) we have T T ∗ M S Λ ( β, γ ) e H ( γ ) h = e i Φ TS ( β,γ )+ H ( γ ) h e ( β, γ ) + O (cid:18) exp (cid:18) − (cid:104)| β |(cid:105) + (cid:104)| γ |(cid:105) Ch (cid:19)(cid:19) , for some symbol e . Then, the result follows from the fact that e − Hh T Λ u is in L andthat, if β is large enough and with the assumption above, (cid:12)(cid:12)(cid:12)(cid:12) e i Φ TS ( β,γ )+ H ( γ ) h (cid:12)(cid:12)(cid:12)(cid:12) ≤ exp (cid:18) − τCh s (cid:104) β (cid:105) s (cid:19) . (cid:3) The G s wave front set interacts nicely with G s pseudors. Indeed, we have Proposition . Let A be a G s pseudor, and u ∈ U s ( M ) . Then WF G s ( Au ) ⊂ WF G s ( u ) . .2. LIFTING PSEUDO-DIFFERENTIAL OPERATORS 119 Proof. We consider α / ∈ WF G s ( u ), and seek to prove that α / ∈ WF G s ( Au ).Using Lemma 2.9 and 2.5, we find for η > C > β ∈ T ∗ M , T ( Au )( β ) = (cid:90) d KN ( γ,β ) ≤ η K T AS ( β, γ ) T u ( γ ) dγ + O (cid:32) exp (cid:32) − (cid:18) (cid:104) β (cid:105) Ch (cid:19) s (cid:33)(cid:33) . From the assumption that α / ∈ WF G s ( u ), we deduce that for some η > 0, thequantity | T u ( γ ) | is controlled by exp( − ( (cid:104) γ (cid:105) /Ch ) /s ) in the conical neighbourhoodof size 2 η of α . In particular for β in the conical neighbourhood of size η of α , wefind, using Lemma 2.13 to see that Im Φ T S ( β, γ ) is non-negative when β and γ arereal, that T ( Au )( β ) = O (cid:32) exp (cid:32) − (cid:18) (cid:104) β (cid:105) Ch (cid:19) s (cid:33)(cid:33) . (cid:3) We can also use I -Lagrangian spaces to prove elliptic regularity for G s pseudors. Proposition . Let m ∈ R . Let A be a G s pseudor on M , semi-classicallyelliptic of order m . Assume that h is small enough. Then, if u ∈ U s ( M ) is suchthat Au ∈ G s ( M ) , we have that u ∈ G s ( M ) . Proposition 2.8 is a result of elliptic regularity that extends previous results[ BDMK67, Zan85, Rod93 ]. The main improvement here is that we use our classof G s G s pseudors rather than the more common class of G G s pseudors. Remark . As will be clear from the proof, this tool is quite flexible.For example, instead of “semi-classically elliptic”, one could assume “elliptic andRe A ≥ A , i.e. toprove that the inverse for A is itself a G s pseudor (we know that this inverse existswhen h is small enough by the usual C ∞ pseudo-differential calculus). However, onecan see that Proposition 2.8 follows from the following result. Lemma . Under the assumptions of Proposition 2.8, if h is small enoughthen the inverse of A is continuous from G s ( M ) to itself and extends as a continuousoperator from U s ( M ) to itself. Proof. Since the formal adjoint of A is also a semi-classically elliptic G s pseudor of order m , according to Proposition 1.8, then we only need to prove that A − is bounded from G s ( M ) to itself. To do so, we consider the Lagrangian Λdefined by (2.4) with G ( α ) = − cτ h − s (cid:104)| α |(cid:105) s , where c > τ , s )-adapted Lagrangian). Provided that τ and h are smallenough, then we know by Proposition 2.4 that A is bounded on H k Λ to H k − m Λ forevery k ∈ R . We will start by proving that for τ and h small enough, then A has abounded inverse on H . To do so, apply Proposition 2.5 to write T Λ AS Λ = aπ Λ + O L m − (Λ) → L (Λ) (cid:16) h (cid:17) , (2.79)where a denotes an almost analytic extension for the symbol of a . It follows fromthe ellipticity of A that there is C > α ∈ Λ we have | a ( α ) | ≥ (cid:104)| α |(cid:105) m C . This is true by assumption for α ∈ T ∗ M and remains true after a small perturbation(we assume c (cid:28) a − defines a bounded operator 20 2. FBI TRANSFORM ON COMPACT MANIFOLDS from L (Λ) to L m (Λ). We define the operator B = S Λ a − T Λ , which is consequently bounded from H to H m Λ . It follows from (2.79) that (thesize of the remainder is measured both as an endomorphism of H and of H m Λ ) BA = I + O (cid:16) h (cid:17) . Hence, we find by Von Neumann’s argument that, for h small enough, A has aninverse A − bounded from H to H m Λ . Moreover, we see that this result is uniformin the parameter τ that appears in the definition of G , so that the inverse A − exists for h ≤ h where h does not depend on τ .Now let R > 0. If τ is small enough, it follows by Corollary 2.2 that E s,R ( M ) ⊆H with a continuous inclusion. However, the ellipticity of G with Proposition2.6 and Lemma 2.14 (or rather a short inspection of their proofs) gives that thereis R > H ⊆ E s,R ( M ). Hence, if h < h the inverse A − of A from H to H m Λ induces an inverse for A from E s,R ( M ) to E s,R ( M ). Since boththese spaces are included in C ∞ ( M ), this is nothing else that the inverse of A on C ∞ ( M ) restricted to E s,R ( M ). Thus, the inverse of A is bounded from E s,R ( M )to E s,R ( M ). Since R > A is bounded from G s ( M )to itself. (cid:3) We use the same notations as in the previous section: T is an analytic FBItransform given by Theorem 6, and S = T ∗ denotes its adjoint, Λ is a ( τ , τ small enough. The associated operators T Λ and S Λ have been defined in § h > G s pseudor P on the space H defined in § T Λ P S Λ acting on H , FBI . The kernel ofthis operator was already described in Lemmas 2.9 and 2.10. With Toeplitz calculusin mind, we would like to compare the operator T Λ P S Λ with a multiplicationoperator. We gave a first result in this direction, Lemma 2.5. However, the errorterm in this result is quite big, and we intend to improve it, giving a multiplicationformula valid at any order.It would be possible to find a symbol p on Λ such that T Λ P S Λ (cid:39) Π Λ p Π Λ , witha very small error. Here, we recall that Π Λ = T Λ S Λ is a bounded projector from L (Λ) to H , FBI . This is not the most useful form for T Λ P S Λ though. It is in factmuch more interesting to replace Π Λ by the orthogonal projector B Λ on H , FBI – itis a legitimate operator since H , FBI is closed. Indeed, if we can find a symbol σ onΛ such that B Λ T Λ P S Λ B Λ (cid:39) B Λ σB Λ (2.80)up to a very small error, then we can use the symbol σ to compute scalar productsin H : (cid:104) P u, u (cid:105) H (cid:39) (cid:104) σT Λ u, T Λ u (cid:105) L (Λ) . We can then use the Hilbert structure of H to try to invert P for instance. Onemay think of the difference between representations in the form B Λ σB Λ and Π Λ p Π Λ as the difference between the Weyl quantization and the left quantization. Both of .3. BERGMAN PROJECTOR AND SYMBOLIC CALCULUS 121 them are legitimate quantizations, however one of them is much more practical forthe handling of adjoint operators.For this whole idea to work, we need to understand the structure of B Λ . Guidedby the fact that B T ∗ M = Π T ∗ M , it is reasonable to conjecture that the kernel of B Λ takes in general a form similar to that of Π Λ . Recall that the latter, up to negligibleremainders, takes the form e i Φ TS ( α,β ) h σ Π Λ ( α, β ) . (2.81)where σ Π Λ is a symbol supported near the diagonal of Λ × Λ and the phase Φ T S satisfies the estimate Lemma 2.13.The method to prove it is indeed the case was first elaborated by Boutet deMonvel and Guillemin [ BG81 ], and improved upon by [ HS86 ] and [ Sj¨o96a ]; wewill explain its details. • The first step is to observe that there exists local pseudo-differentialoperators Z j , j = 1 . . . n , such that Z j Π Λ = O ( h ∞ ) as operators from L k to L − k for every k . The fact that the image of Π Λ is (almost) in the kernelof such operators stems from the decomposition Π Λ = T Λ S Λ . There is noreason for this to be true for a general operator whose kernel is in the form e i Φ( α,β ) /h a ( α, β ). • The second step is to consider operators A whose kernel essentially takethe form e i Φ( α,β ) /h a ( α, β ), and such that Π Λ A = A Π ∗ Λ = A – with sometechnical conditions. One can then use the Z j ’s to prove that the phase Φis completely determined, and that the amplitude is completely determinedby its values on the diagonal. • Finally, the observation that, given a symbol σ , the composition Π Λ σ Π ∗ Λ issuch an operator with oscillating kernel, leads to a parametrix constructionshowing that there exists f a symbol of order 0 such that B Λ (cid:39) Π Λ f Π ∗ Λ modulo a very small error.A Toeplitz representation (2.80) of G s pseudor will follow, that can be rephrasedas a multiplication formula – Proposition 2.11 – that will be essential in theapplications. Finally, we will also state basic results about a Toeplitz calculus thatwe will use in § H k Λ (it will also be very important in the applications).Let us point out that, now that Lemmas 2.9 and 2.10 allowed us to replace T ∗ M by Λ, we are working in the C ∞ category on Λ. We will use tools from [ MS75 ] thatrequire to choose an almost analytic neighbourhood (cid:101) Λ, that is an embedding Λ ⊆ (cid:101) Λof Λ in a complex analytic manifold of dimension 2 n that makes Λ totally real in (cid:101) Λ.We can take for instance (cid:101) Λ = T ∗ (cid:102) M , but any other choice is legitimate. We can alsotake C ∞ coordinates on Λ and use C n as an almost analytic neighbourhood. Noticealso that when considering complex conjugates, we will always mean it in thesecoordinates. For instance if f is an element of T ∗ α Λ ⊗ C (cid:39) T ∗ (cid:101) Λ for some α ∈ Λ, then¯ f is defined using the tensor product structure. This amounts to say that ¯ f is the C -linear form on T α Λ ⊗ C (cid:39) T α (cid:101) Λ whose restriction to T α Λ is obtained from f bycomposition with the usual complex conjugacy.Finally, let us mention that in this section we will identify an operator with itskernel, writing for instance Π Λ ( α, β ) for the kernel of Π Λ . Annihilating pseudors. In this section, we will build the Z j ’s evoked afew lines above. The existence of such operators vanishing on the image of T shouldbe seen as a generalization of the following relations satisfied by the flat transform 22 2. FBI TRANSFORM ON COMPACT MANIFOLDS defined in (6): (cid:18) ∂∂x j − i ∂∂ξ j − ih ξ j (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) = Z j, R n T R n = 0 . The collection of these equations forms a twisted ∂ equation. In our non-flat case,the Z j ’s will have to be pseudo-differential, but the fundamental idea is similar. Tobe prudent, we define beforehand what we mean by a pseudor on Λ: Definition . Let Λ be a ( τ , Q an operatoron Λ (maybe depending on h as usual) whose kernel is uniformly properly supported.Assume additionally that in local coordinates near a point α ∈ Λ where the Kohn–Nirenberg metric is uniformly close to the standard metric, the kernel of Q isthe kernel of a semi-classical pseudo-differential operator with Planck constant (cid:126) = h/ (cid:104)| α |(cid:105) , whose symbol is uniformly compactly supported in the impulsionvariable. Then we say that Q is an h -pseudor on Λ.Using the map T ∗ M → Λ given by exp H ω I G , and local coordinates ( u, η ) on T ∗ M , we have local coordinates α = exp H ω I G ( u, η ) on Λ. Taking into account therelevant scaling, Q is a h -pseudor on Λ if its kernel takes in these coordinates theform (near the diagonal)1(2 πh ) n (cid:90) e ih ( (cid:104) u − u (cid:48) ,u ∗ (cid:105) + (cid:104) η − η (cid:48) ,η ∗ (cid:105) ) q ( u, u (cid:48) , η, η (cid:48) , u ∗ , η ∗ )d u ∗ d η ∗ , where the symbol q is supported for | u ∗ | + | η − η (cid:48) | ≤ C (cid:104) η (cid:105) , | η ∗ | + | u − u (cid:48) | ≤ C andsatisfies for k, (cid:96) ∈ N n and k (cid:48) , (cid:96) (cid:48) ∈ N n (2.82) (cid:12)(cid:12)(cid:12) ( ∂ u,u (cid:48) ) k ( ∂ η,η (cid:48) ) (cid:96) ( ∂ u ∗ ) k (cid:48) ( ∂ η ∗ ) (cid:96) (cid:48) q ( u, u (cid:48) , η, η (cid:48) , u ∗ , η ∗ ) (cid:12)(cid:12)(cid:12) ≤ C k,(cid:96),k (cid:48) ,(cid:96) (cid:48) (cid:104) η (cid:105) m −| k (cid:48) |−| (cid:96) | . If a symbol q satisfies these conditions, we write q ∈ S mc ( T ∗ Λ), and say that Q isan h -pseudor of order m .We pick a cutoff χ supported near the diagonal of Λ × Λ. In a chart domain,given a symbol p ∈ S mc ( T ∗ Λ) depending only on u, η, u ∗ , η ∗ , we letOp( p ) f ( u, η ) = 1(2 πh ) n (cid:90) e ih ( (cid:104) u − u (cid:48) ,u ∗ (cid:105) + (cid:104) η − η (cid:48) ,η ∗ (cid:105) ) p ( u, η, u ∗ , η ∗ ) f ( u (cid:48) , η (cid:48) ) × χ ( u, u (cid:48) , η, η (cid:48) )d u ∗ d η ∗ d u (cid:48) d η (cid:48) . This defines an h -pseudor of order m (at least in a smaller chart domain). It followsfrom the results of [ MS75 ] (in particular, Formula 2.28) that if a function f on Λhas an expansion in the large chart domain in the form f = e iψ/h a + O ( h/ (cid:104) η (cid:105) ) ∞ ,with ψ and a symbols of respective order 1 and ˜ m in the Kohn–Nirenberg class onΛ such that Im ψ ≥ 0, and Im ψ ( α ) = 0, then near α , in the small chart domain,modulo O ( h/ (cid:104) η (cid:105) ) ∞ (all these error terms are in C ∞ ),(2.83) Op( p ) f ∼ e ih ψ (cid:88) k ≥ h k k ! b k . By the expansion (2.83), we mean that for every N ∈ N we haveOp( p ) f = e ih ψ N (cid:88) k =0 h k k ! b k mod h N +1 S m + ˜ m − N − KN (Λ) . The coefficients in this expansion are given by the expression(2.84) b k = ( ∇ u (cid:48) ,η (cid:48) · ∇ u ∗ ,η ∗ ) k (cid:2) p ( u, η ; ( u ∗ , η ∗ + ρ ( u, u (cid:48) ; η, η (cid:48) )) a ( u (cid:48) ) (cid:3) | u (cid:48) = u,η (cid:48) = η,u ∗ = η ∗ , .3. BERGMAN PROJECTOR AND SYMBOLIC CALCULUS 123 where ρ is such that ψ ( u (cid:48) , η (cid:48) ) − ψ ( u, η ) = (cid:104) ( u (cid:48) , η (cid:48) ) − ( u, η ) , ρ ( u, u (cid:48) ; η, η (cid:48) ) (cid:105) . Here, weuse the coordinates on Λ described above and we have identified p with one of itsalmost analytic extensions in the u ∗ , η ∗ variable. In particular, the leading term inthe expansion (2.84) is given by b = p ( u, η, d u,η ψ ) . We will be chiefly interested in the composition of a pseudors Q = Op( p ) andΠ Λ . However, since we are interested in the action of Π Λ on L ( e − H/h d α ), it isnatural to consider the kernel of the composition Qe − H/h Π Λ e H/h rather than Q Π Λ .According to our discussion, its kernel will have a semi-classical expansion, with aprincipal symbol in the form p ( u, η, d u,η Φ ◦ T S ) , where Φ ◦ T S is the phase of the reduced kernel of Π Λ :Φ ◦ T S ( α, β ) = iH ( α ) + Φ T S ( α, β ) − iH ( β ) . This suggest considering the set (for some small (cid:15) > J Λ = (cid:110) ( α, d α Φ T ( α, y ) − i Im θ ( α )) : α ∈ Λ , y ∈ (cid:102) M , d (cid:102) M ( α x , y ) < (cid:15) (cid:111) . This is a priori a submanifold of T ∗ Λ ⊗ C (the complexification of the cotangentspace). Indeed, Definition 1.7 of an admissible phase implies that if τ is smallenough then y (cid:55)→ d α Φ T ( α, y ) is a holomorphic immersion near α x . Recall that for β near α we have d α Φ T S ( α, β ) = d α Φ T ( α, y c ( α, β )) where y c ( α, β ) denotes the criticalpoint of y (cid:55)→ Φ T ( α, y ) + Φ S ( y, β ). It follows that for β near α in Λ, the“reduced”phase of Π Λ satisfies d α Φ ◦ T S ( α, β ) ∈ J Λ . We will show that Proposition . Let Λ be a ( τ , -adapted Lagrangian with τ small enough.Around any fiber T ∗ x M , there exist h -pseudors Z j , j = 1 . . . n , of order , such that(i) Z j = Op( ζ j ) , with ζ j ∼ (cid:80) h k ζ j,k a symbol of order ;(ii) the kernel of Z j e − H/h Π Λ e H/h is O ( h/ (cid:104) η (cid:105) ) ∞ in C ∞ ;(iii) each ζ j,k is holomorphic in u ∗ , η ∗ near J Λ ∩ T ∗ Λ and ζ j, vanishes on J Λ ;(iv) in a uniform neighbourhood of J Λ ∩ T ∗ Λ , the d ζ j, form a uniformly freefamily. The asymptotic expansion in (i) is thought in the following sense: for every N ∈ N , we have ζ j = N (cid:88) k =0 h k ζ j,k mod h N +1 S − N − c (cid:0) T ∗ Λ (cid:1) . When we write “around any fiber T ∗ x M ”, we mean that the points (i)-(iv) only holdnear points α ∈ Λ such that in the coordinates ( u, η ) described above, u is close to x . We can then cover Λ by a finite number of domains on which the pseudors fromProposition 2.9 are available.The difficulty in the proof of Proposition 2.9 is that we have to use the decom-position Π Λ = T Λ S Λ , but we are not able to obtain an expansion for Op( p ) e − H/h T ,because the imaginary part of the phase Φ T + iH ( α ) is not positive. We propose herea solution inspired by the proof of [ HS86 , Proposition 6.7]. We will approximatethe Z j ’s by differential operators of increasingly large order. It is morally very closeto the argument based on formal applications of the stationary phase method from 24 2. FBI TRANSFORM ON COMPACT MANIFOLDS [ HS86 ], but we privileged this method because it does not rely on the results from[ Sj¨o82 ], with which the reader may not be familiar. Proof of Proposition 2.9. We will first build the ζ j,k ’s as “formal” solu-tions, and then check that they indeed solve the problem. As we said before, wecannot prove an expansion for Op( p ) e − H/h T . However, we can still compute theterms that appear in the right hand side of (2.83). As a formal series in powers of h whose coefficients are functions, it is a well defined object, so that for p ∈ S mc ( T ∗ Λ),we define(2.86) (cid:20) e − i Φ T ( · ,y )+ Hh Op( p ) (cid:21) formal ( e − H/h T ) := (cid:88) k ≥ h k k ! b k , with, as in (2.84), b k defined by the expression( ∇ u (cid:48) ,η (cid:48) · ∇ u ∗ ,η ∗ ) k (cid:2) p ( u, η ; ( u ∗ , η ∗ ) + ρ ( u, u (cid:48) , η, η (cid:48) )) a T ( u (cid:48) , η (cid:48) , y ) (cid:3) | u (cid:48) = u,η = η (cid:48) ,u ∗ = η ∗ =0 , and ρ defined byΦ T ( u (cid:48) , η (cid:48) , y ) − Φ T ( u, η, y ) + i ( H ( u (cid:48) , η (cid:48) ) − H ( u, η )) = (cid:104) ( u (cid:48) , η (cid:48) ) − ( u, η ) , ρ ( u, u (cid:48) , η, η (cid:48) ) (cid:105) . Here as usual, we identified p with one of its almost analytic extension in thecoordinates ( u, η ). Taking p = ζ j itself a formal sum of symbols ζ j = (cid:80) h k ζ j,k , wetry to solve (cid:20) e − i Φ T ( · ,y )+ Hh Op( ζ j ) (cid:21) formal ( e − H/h T )( α, y ) = 0 . If this were an actual composition, since Λ and the symbol a of T themselvesmay depend on h , there would be many ways to expand this sum in powers of h .However, we are here using non-ambiguously the expansion given by (2.83), leadingto equations of the form(2.87) ζ j, ( u, η, d u,η (Φ T + iH )) = 0 , and(2.88) ζ j,k ( u, η, d u,η (Φ T + iH )) = Q k ( ζ j, , ζ j, , . . . , ζ j,k − )( u, η, y ) ,Q k being some differential operator whose coefficients depend on a , Φ T and Λ.However this dependence is symbolic, so its comes with uniform estimates. Sincethe map ( α, y ) (cid:55)→ ( α, d α (Φ T ( α, y ) + iH ( α )))is a local diffeomorphism onto J Λ , there is no obstruction to the equations abovehaving a solution. Quite on the contrary, there are many solutions, because thesymbols ζ j,k are recursively prescribed only on J Λ .We start with ζ j, . The proof of the fact that J Λ is an analytic manifold of T ∗ Λ ⊗ C , of codimension n uses the local inversion theorem, and thus comes withuniform estimates. In particular, we can choose the ζ j, so that • each ζ j, is a symbol of order 0, holomorphic in the u ∗ , η ∗ variable, near J Λ ∩ T ∗ Λ – the real points of J Λ ; • each ζ j, vanishes on J Λ ; • near J Λ ∩ T ∗ Λ, the d ζ j, form a uniformly free family; • the ζ j, ’s are supported for | u ∗ | ≤ C (cid:104) η (cid:105) , | η ∗ | ≤ C .Using the symplectic structure of J Λ , we could additionally assume that { ζ j, , ζ (cid:96), } =0, but this will not be necessary for us.Let us now explain how to construct the higher order symbols. Since thedifferential of y (cid:55)→ d α ξ Φ T ( α, y ) .3. BERGMAN PROJECTOR AND SYMBOLIC CALCULUS 125 at y = α x is the identity (under natural identifications) by assumption, and sinceexp H ω I G is close to the identity, J Λ is uniformly transverse to the foliation tangent to ∂/∂u ∗ . It follows that if a function has its values determined on J Λ and that functiondoes not depend on u ∗ , that function is completely determined in a neighbourhoodof J Λ . In particular, we can solve the equations (2.87) and (2.88) near the realpoints of J Λ , with symbols ζ j,k , holomorphic in η ∗ in a neighbourhood of J Λ ∩ T ∗ Λ,not depending on u ∗ .Now, we can build ζ j some C ∞ symbols of order 0 so that ζ j ∼ (cid:88) h k ζ j,k . We have to prove that for β ∈ Λ in the chosen chart, and α close to β ,(2.89) Op( ζ j ) e − Hh Π Λ e Hh ( α, β ) = O C ∞ (( h/ (cid:104)| β |(cid:105) ) ∞ ) . (the proposition also requires an estimate far from the diagonal, but that estimate istrivial at this point). In order to prove (2.89), we introduce the following truncationsof ζ j : ζ ≤ Nj = N (cid:88) k =0 (cid:0) ζ j,k (cid:1) (2 N ) , where, ( ζ j,k ) (2 N ) is obtained from ζ j,k by taking its 2 N Taylor expansion in u ∗ and η ∗ at Re θ ( u, η ). We also write ζ >Nj = ζ j − ζ ≥ Nj , and split Op( ζ j ) into Op( ζ ≤ Nj )and Op( ζ >Nj ).We deal first with Op( ζ >Nj ), to do let us consider a symbol p on T ∗ Λ (as definedabove) such that p ( u, η, u (cid:48) , η (cid:48) , u ∗ , η ∗ ) = O (cid:16) (cid:104) η (cid:105) − N (cid:12)(cid:12) ( u ∗ , η ∗ ) − Re θ ( u, η ) (cid:12)(cid:12) N (cid:17) + O (cid:16) h N (cid:104) η (cid:105) − N (cid:17) . (2.90)Here, the O ’s must be understood in a space of functions satisfying (2.82). Applyingthe method of [ MS75 ] again, we find thatOp( p ) e − Hh Π Λ e Hh ( α, β ) = e ih Φ ◦ TS q ( α, β ) + O ( h/ (cid:104)| β |(cid:105) ) ∞ , and we have an asymptotic expansion for the symbol qq ∼ (cid:88) k ≥ h k q k . The q k ’s write in the coordinates ( u, η ): q k ( u, η, u (cid:48) , η (cid:48) ) = ( ∇ u (cid:48)(cid:48) ,η (cid:48)(cid:48) · ∇ u ∗ ,η ∗ ) k (cid:104) p (cid:0) u, η ; ( u ∗ , η ∗ ) + ρ ( u, u (cid:48) , u (cid:48)(cid:48) , η, η (cid:48) , η (cid:48)(cid:48) ) (cid:1) × σ Π Λ ( u (cid:48)(cid:48) , η (cid:48)(cid:48) , u (cid:48) , η (cid:48) ) (cid:105) | u (cid:48)(cid:48) = u,η (cid:48)(cid:48) = ηu ∗ = η ∗ =0 , (2.91)where ρ satisfiesΦ ◦ T S ( u (cid:48)(cid:48) , η (cid:48)(cid:48) , u (cid:48) , η (cid:48) ) − Φ ◦ T S ( u, η, u (cid:48) , η (cid:48) ) = (cid:104) ( u (cid:48)(cid:48) , η (cid:48)(cid:48) ) − ( u, η ) , ρ ( u, u (cid:48) , u (cid:48)(cid:48) , η, η (cid:48) , η (cid:48)(cid:48) ) (cid:105) , and σ Π Λ is from (2.81). Here, we recall that the phase Φ T S satisfies the estimateLemma 2.13, so that the application of the stationary phase method is legitimate.Consequently, we must have ρ ( u, u (cid:48) , u (cid:48)(cid:48) , η, η (cid:48) , η (cid:48)(cid:48) ) = d u,η Φ ◦ T S ( u, η, u (cid:48) , η (cid:48) ) + O ( (cid:12)(cid:12) u (cid:48)(cid:48) − u (cid:12)(cid:12) + (cid:104) η (cid:105) − (cid:12)(cid:12) η (cid:48)(cid:48) − η (cid:12)(cid:12) )= Re θ ( u, η ) + O ( (cid:12)(cid:12) u (cid:48)(cid:48) − u (cid:12)(cid:12) + (cid:12)(cid:12) u − u (cid:48) (cid:12)(cid:12) + (cid:104) η (cid:105) − (cid:12)(cid:12) η (cid:48)(cid:48) − η (cid:12)(cid:12) + (cid:104) η (cid:105) − (cid:12)(cid:12) η − η (cid:48) (cid:12)(cid:12) ) . 26 2. FBI TRANSFORM ON COMPACT MANIFOLDS It follows then from our assumption (2.90) that in (2.91) we differentiate at most2 k times a term of the form O (cid:16) (cid:104) η (cid:105) − N (cid:0) (cid:12)(cid:12) u (cid:48)(cid:48) − u (cid:12)(cid:12) + (cid:12)(cid:12) u − u (cid:48) (cid:12)(cid:12) + (cid:104) η (cid:105) − (cid:12)(cid:12) η (cid:48)(cid:48) − η (cid:12)(cid:12) + (cid:104) η (cid:105) − (cid:12)(cid:12) η − η (cid:48) (cid:12)(cid:12) + (cid:104) η (cid:105) − (cid:12)(cid:12) u ∗ (cid:12)(cid:12) + (cid:12)(cid:12) η ∗ (cid:12)(cid:12) (cid:1) N (cid:17) + O (cid:16) (cid:104) η (cid:105) − N h N (cid:17) . Hence, for k ≤ N , q k ( u, n, u (cid:48) , η (cid:48) ) is a O (cid:16) (cid:104) η (cid:105) − N (cid:0) (cid:12)(cid:12) u − u (cid:48) (cid:12)(cid:12) N − k ) + (cid:104)| η |(cid:105) k − N ) (cid:12)(cid:12) η − η (cid:48) (cid:12)(cid:12) N − k ) (cid:1) + (cid:104) η (cid:105) − N h N (cid:17) . Gathering these estimates, we deduce that q ( α, β ) = O (cid:18)(cid:16) (cid:104)| β |(cid:105) − d KN ( α, β ) + (cid:104)| β |(cid:105) − h (cid:17) N (cid:19) . and it follows from the fact that Im Φ ◦ T S ≥ (cid:104)| β |(cid:105) d ( α, β ) thatOp( p ) e − Hh Π Λ e Hh ( α, β ) = O ( h N (cid:104)| β |(cid:105) − N ) . Now, we would like to take p = ζ >Nj and apply this estimate. We cannot workdirectly like that because ζ >Nj does not satisfy the support condition that weassumed for p . However, introducing a cutoff function in impulsion, we can write ζ >Nj = p + r where p satisfies (2.90) and satisfies the support condition that werequired for symbol on T ∗ Λ. The symbol r is supported away from the graph ofRe θ , so that non-stationary phase method proves thatOp( r ) e − Hh Π Λ e Hh ( α, β ) = O ( h ∞ (cid:104)| β |(cid:105) −∞ ) . We have thus proved thatOp( ζ >Nj ) e − Hh Π Λ e Hh ( α, β ) = O ( h N (cid:104)| β |(cid:105) − N ) . We consider now the other parts ζ ≤ Nj of ζ j . To do so, we will use the factthat Π Λ = T Λ S Λ . It is crucial here that Op( ζ ≤ Nj ) is a differential operator with afinite expansion in powers of h . Actually, we are slightly abusing notations because ζ ≤ Nj is not compactly supported in u ∗ , η ∗ , but this is not a problem because it ispolynomial. We can certainly compute the action of a differential operator on T and we have Op( ζ Nj ) e − H/h Π Λ e H/h = (Op( ζ Nj ) e − H/h T Λ ) S Λ e H/h . Since the series (defined by the procedure above) in (cid:20) e − i Φ T ( · ,y )+ Hh Op( ζ Nj ) (cid:21) formal ( e − H/h T )( α, y )is finite, it coincides with e − i Φ T + Hh Op( ζ Nj ) e − H/h T Λ However, using an argument similar to the one above, it also coincides with (cid:20) e − i Φ T ( · ,y )+ Hh Op( ζ j ) (cid:21) formal ( e − H/h T )( α, y ) ≡ . up to the order ( (cid:104)| α |(cid:105) − d ( α x , y ) + (cid:104)| α |(cid:105) − h ) N (because ζ j and ζ ≤ Nj coincides up tothe order N in h and 2 N in (cid:12)(cid:12) ( u ∗ , η ∗ ) − Re θ ( u, η ) (cid:12)(cid:12) ). Notice that the holomorphy in y is preserved by this procedure, so that we can use the Holomorphic Stationary PhaseMethod Proposition 1.6 as in the proof of Lemma 2.10 to compute the kernel of .3. BERGMAN PROJECTOR AND SYMBOLIC CALCULUS 127 Op( ζ ≤ Nj ) e − H/h Π Λ e H/h . Making explicit the terms in the Stationary Phase Method,we find thatOp( ζ ≤ Nj ) e − H/h Π Λ e H/h ( α, β ) = e i Φ ◦ TS ( α,β ) h O (cid:18)(cid:16) (cid:104)| α |(cid:105) − h + (cid:104)| α |(cid:105) − d KN ( α, β ) (cid:17) N (cid:19) = O (cid:16) h N (cid:104)| α |(cid:105) − N (cid:17) . Here, we used the coercivity of Im Φ ◦ T S again. Summing back ζ ≤ Nj and ζ >Nj , wefind that Op( ζ j ) e − H/h Π Λ ( α, β ) e H/h = O (cid:32)(cid:18) h (cid:104)| α |(cid:105) (cid:19) N (cid:33) . Since here N can be taken arbitrarily large, the proof is complete (derivatives aredealt with similarly). (cid:3) A class of FIOs. Let us consider an operator A whose kernel is essentiallysupported near the diagonal on Λ, and whose kernel has an expansion as in (2.81).Borrowing the terminology of [ MS75 ], this means that A is a Fourier IntegralOperator – FIO – with complex phase, associated with the Lagrangian manifold L Φ := (cid:8) ( α, d α Φ( α, β ); β, d β Φ( α, β )) | α, β ∈ Λ (cid:9) ⊂ T ∗ (cid:16)(cid:101) Λ × (cid:101) Λ (cid:17) . From the condition that the imaginary part of Φ is coercive away from the diagonal,we deduce that the real points of L Φ are exactly( L Φ ) R = (cid:8) ( α, d α Φ( α, α ); α, d β Φ( α, α )) | α ∈ Λ (cid:9) . In the case of Φ ◦ T S ( α, β ) := iH ( α ) + Φ T S ( α, β ) − iH ( β ), from Lemma 2.13 and itsproof, we deduce that( L Φ ◦ TS ) R = ∆ Λ := { ( α, Re θ ( α ); α, − Re θ ( α )) | α ∈ Λ } . The condition of coercivity also implies that L Φ is a strictly positive Lagrangianmanifold – see Definition 3.3 in [ MS75 ]. It will turn out that every relevant operatorin our setting shares these properties with Φ ◦ T S .We will study systematically such operators that are left invariant by Π Λ andΠ ∗ Λ . Let us start with a definition. During all this section, we fix a ( τ , τ and h small enough. Since we will be working on Λ, theresults and estimates will depend on Λ. However, the constants appearing will onlydepend on C k estimates on Λ, and in this sense, we will say that is valid uniformlyin Λ. When a G s pseudor P appears, implicitly, we assume that Λ is ( τ , s )-adapted. Definition . Let m ∈ R . A complex FIO associated with ∆ Λ – of order m – A on Λ is an operator whose reduced kernel is of the form A ( α, β ) e H ( β ) − H ( α ) h = e i Φ A ( α,β ) h σ A ( α, β ) + O C ∞ (cid:18)(cid:18) h (cid:104)| α |(cid:105) + (cid:104)| β |(cid:105) (cid:19) ∞ (cid:19) . (2.92)Here, Φ A ∈ S KN (Λ × Λ) and σ A ∈ h − n S mKN (Λ × Λ) are symbols on Λ × Λ inKohn–Nirenberg classes. Moreover, we assume that for α ∈ Λ we have(2.93) Φ( α, α ) = 0 and d α Φ( α, α ) = Re θ ( α ) , and that there are C, η > σ A is supported in { ( α, β ) ∈ Λ × Λ : d KN ( α, β ) ≤ η } , and for every α, β ∈ Λ such that d KN ( α, β ) ≤ η we have(2.94) 1 C (cid:104)| α |(cid:105) d KN ( α, β ) ≤ Im Φ A ( α, β ) ≤ C (cid:104)| α |(cid:105) d KN ( α, β ) . We say that Φ A is the (reduced) phase of A and that σ A is its symbol. 28 2. FBI TRANSFORM ON COMPACT MANIFOLDS When studying FIO, we will often need to consider remainders that are negligiblein the following sense. Definition . An operator A on Λ is said to be negligible, or to have negligiblekernel, if it is a FIO with symbol 0, that is if its reduced kernel satisfies A ( α, β ) e H ( β ) − H ( α ) h = O C ∞ (cid:18)(cid:18) h (cid:104)| α |(cid:105) + (cid:104)| β |(cid:105) (cid:19) ∞ (cid:19) . The main goal of this section is to build the necessary tools to prove that theorthogonal projector B Λ is a complex FIOs associated with ∆ Λ . Insofar as we willnot be making use of any other type of FIOs in this section, and will be workingwith just one Lagrangian Λ, we will just write “FIO” for “complex FIO associatedwith ∆ Λ ”. Remark . Let us mention that the arguments in the proof of Proposition2.4 imply that an FIO of order m is bounded from L k (Λ) to L k − m (Λ) for every k ∈ R . In particular, it makes sense to compose these operators. Notice also that,due to the coercivity condition (2.94), the constant η in Definition 2.5 can be takenarbitrarily small (just multiply σ A by a cutoff function supported near the diagonaland put the remainder in the error term in (2.92)).It follows from Lemmas 2.9, 2.10 and 2.13 that Π Λ is an FIO of order 0. Moregenerally, if P is a G s pseudor of order m , then the same lemmas imply that if τ and h are small enough then T Λ P S Λ is an FIO of order m (this fact has been provenin the proof of Proposition 2.4). The phase associated to both of these operators isΦ ◦ T S , and it satisfies the coercivity condition (2.94) due to Lemma 2.13.Notice also that a negligible operator is a O ( h N ) as on operator from L − N (Λ)to L N (Λ) for every N ∈ N . Lemma . Let A and B be FIOs, whose phases are respectively Φ A and Φ B .Let f be a symbol in Kohn–Nirenberg class on Λ . Then A ∗ , AB , f A and Af alsoare FIOs. The phase of Af B is given by Φ AB ( α, β ) = v.c. γ (Φ A ( α, γ ) + Φ B ( γ, β )) . (2.95) Here, v.c. stands for critical value, and this critical value is defined in the sense ofalmost analytic extension (see [ MS75 ] in particular Lemma 2.1). Moreover, froman expansion for f and the symbols of A and B , we deduce an asymptotic expansionfor the symbol of Af B given by the stationary phase method. It will be convenient to give a name to the reduced phase of Π ∗ Λ , which isΦ ∗ T S ( α, β ) := − Φ ◦ T S ( β, α ) = − iH ( α ) − Φ T S ( β, α ) + iH ( β ). Recall that, in order tostudy the orthogonal projector B Λ , we want to study operators of the form Π Λ f Π ∗ Λ where f is a symbol on Λ. From Lemma 2.16, we already know that Π Λ f Π ∗ Λ is acomplex FIO adapted with ∆ Λ . Remark . If A is an FIO of order m , we say that its symbol satisfies anasymptotic expansion σ A (cid:39) πh ) n (cid:88) k ≥ h k a k , if, for every k ∈ N , we have a k ∈ S m − kKN (Λ × Λ), and, for every N ∈ N , we have that σ A − πh ) n N − (cid:88) k =0 h k a k h N − n S m − NKN (Λ × Λ) . .3. BERGMAN PROJECTOR AND SYMBOLIC CALCULUS 129 Proof of Lemma 2.16. The statement for Af and f A elementary. For theadjoint, it suffices to observe that if Φ satisfies (2.93) and (2.94), then so doesΦ ∗ ( α, β ) := − Φ( β, α ) . Thus, we only need to consider the composition AB . Since the kernels of A and B are rapidly decaying away from the diagonal and grow at most polynomially nearthe diagonal, the kernel of AB is rapidly decaying (in C ∞ ) at any fixed distance ofthe diagonal in Λ × Λ. Let us turn to the reduced kernel of AB near the diagonal.The error term in (2.92) is proved to be negligible as in the non-diagonal case.Consequently, we may assume that the reduced kernel of AB is given by the integral (cid:90) Λ e i (cid:104)| α |(cid:105) h Ψ α,β ( γ ) σ A ( α, γ ) σ B ( γ, β ) d γ, (2.96)where the phase Ψ α,β is defined byΨ α,β ( γ ) = Φ A ( α, γ ) + Φ β ( γ, β ) (cid:104)| α |(cid:105) . Thanks to the condition (2.94) that we imposed on the phases Φ A and Φ B , wesee that this phase has non-negative imaginary part. From (2.93), we see thatwhen α = β the phase Ψ α,α has a critical point at γ = α , which is non-degeneratesince (2.94) imposes that the Hessian of the phase has a definite positive imaginarypart. Melin–Sj¨ostrand’s C ∞ version of the stationary phase method with complexphases [ MS75 , Theorem 2.3] thus applies. We apply these results after a properrescaling (in α ξ , β ξ , γ ξ ) that replaces the estimates using the Kohn–Nirenberg metricby uniform C ∞ estimates. Notice that when doing so there is a Jacobian thatappears in (2.96) which is a symbol of order n . This is because the volume form d γ in (2.96) is not the volume form associated to the Kohn–Nirenberg metric but tothe symplectic form ω R . In particular, one can check that the determinant of theHessian of γ (cid:55)→ Φ A ( α, γ ) + Φ β ( γ, β ) associated to the volume form d γ is a symbol oforder 0, and not 2 n as we could expect from the coercivity condition (2.94). Thus,we find that for α, β near the diagonal, the kernel of AB is given by e i Φ AB ( α,β ) h σ AB ( α, β ) , where σ AB is a symbol in the class h − n S mKN (Λ × Λ), where m is the sum of theorders of A and B (recall that the dimension of Λ is 2 n ). The phase Φ AB is givenby (2.95) in the sense of almost analytic extension. Since we already understandthe reduced kernel of AB off diagonal, it only remains to prove that the phase Φ AB satisfies the conditions (2.93) and (2.94) near the diagonal. For (2.93), it is not toohard, since we already identified the critical point when α = β .For (2.94), we pick α ∈ Λ and consider Ψ α,β ( γ ) as a function of β and γ near α .We want to apply the Fundamental Lemma 1.16. We deduce from (2.94), satisfiedby Φ A and Φ B , that Im Ψ α,α ( α ) = 0 and that, for β and γ near α , we haveIm Ψ α,β ( γ ) ≥ d KN ( α, β ) + d KN ( α, γ ) . Consequently, ( β, γ ) (cid:55)→ Im Ψ α,β ( γ ) has a critical point at β = γ = α , and itsHessian is (uniformly) definite positive. Recalling thatΦ AB ( α, β ) = (cid:104)| α |(cid:105) v.c γ Ψ α,β ( γ ) , we find using Lemma 1.16 (with a proper rescaling) that β (cid:55)→ Im Φ AB ( α, β ) has acritical point at β = α whose Hessian is greater than C − (cid:104)| α |(cid:105) g KN . The condition(2.94) follows then by Taylor’s formula using the symbolic estimates on Φ AB .Notice that we applied here Lemma 1.16 to the phase Ψ α,β which is not analytic.However, there is no problem here since the proof of Lemma 1.16 is based on a 30 2. FBI TRANSFORM ON COMPACT MANIFOLDS direct computation on the Hessian of the imaginary part of the phase at β = α .The result of this computation is still valid here since when β = α the critical pointof the phase is real and consequently all the almost analytic errors vanish. (cid:3) Remark . As in the more regular cases described in § MS75 , Theorem 2.3] comes with an asymptotic expansion withsemi-explicit coefficients. Actually, the expression for these coefficients is the sameas in Remarks 1.20 and 1.24. However, when defining the differential operator P m ,all the objects must be replaced by their almost analytic extensions, in particularthe Morse coordinates are now defined in the sense of [ MS75 , (2.13)] Remark . From now on, the notion of jets and of equation holding atinfinite order will be essential. Let N be a C ∞ manifold. Let x be a point of N and f be a C ∞ function defined on a neighbourhood of x in N . In general, wecannot define the Taylor expansion of f at x : the first term in the expansion, givenby the derivative d f ( x ) of f , is the only one that is always intrinsically defined.However, when d f ( x ) = 0, then the second derivative d f ( x ) is well-defined. Moregenerally, if the k first derivatives d f ( x ) , . . . , d k f ( x ) vanish, then one may defineintrinsically the k + 1th derivative d k +1 f ( x ) of f at x (this is a k + 1th symmetriclinear form on T x M ). Hence, while we cannot define the Taylor expansion of f at x , it makes sense to say that f vanishes to order k ∈ N ∪ {∞} at x . Notice that if d is a distance on M induced by a smooth Riemannian metric then it is equivalentto say that f vanishes to order k at x and that f ( x ) = x → x O (cid:16) d ( x, x ) k (cid:17) .Now, if P is a submanifold of N and f and g are two C ∞ functions defined ona neighbourhood of P , we say that the equation f = g holds at infinite order on P ,or in the sense of jets on P , if the function f − g vanishes to infinite order at everypoint x ∈ P . The space of jets may then be defined as the quotient C ∞ ( N ) / ∼ ,where the relation ∼ is defined by: f ∼ g if and only if f = g to infinite order on P .In the analysis, below, we will often have N = Λ × Λ and P the diagonal ofΛ × Λ. Then, if f : N → C is a symbol of order m that vanishes to all orders on P ,for every M > k ∈ N there is a constant C k,M such that for every ( α, β ) ∈ N such that d KN ( α, β ) ≤ M we have | f ( α, β ) | ≤ C k,M (cid:104)| α |(cid:105) m d KN ( α, β ) k , where d KN denotes the distance induced by the Kohn–Nirenberg metric. Here,the constant C k,M depends on the symbolic estimates on f . In particular, even ifthe vanishing to infinite order of f on the diagonal looks like a merely algebraiccondition, it can be used to derive effective estimates on f since we already have acontrol on the derivatives of f (in the Kohn–Nirenberg metric).Finally, if N , N are two C ∞ manifolds, P , P are submanifolds respectivelyof N and N and f is a C ∞ map from N to N , we say that f is valued in P –or loosely f ∈ P – at infinite order on P if, for every C ∞ function g : N → C identically equal to zero on P , the function g ◦ f vanishes at infinite order on P .2.3.1.3. “Eikonal” condition on the phase. From this section, we will consider A , an FIO in the sense of the previous section, and assume that Π Λ A = A Π ∗ Λ = A .We will start by finding some constraints on the phase of A . In this direction, thefirst step is to observe that Lemma . Let A and B be FIOs. Denote respectively by Φ A and Φ B thephases of A and B . Assume that Φ A satisfies d α Φ A ∈ J Λ (2.97) .3. BERGMAN PROJECTOR AND SYMBOLIC CALCULUS 131 at infinite order on the diagonal of Λ × Λ . Then, if Φ AB denotes the phase of AB provided by Lemma 2.16, the phase Φ AB also satisfies (2.97) at infinite order onthe diagonal. Proof. We will use the same stratagem that we used to deduce the differentialof Φ T S from the differential of Φ T , taking into account the use of almost analyticextensions. Recall that Φ AB is defined by (2.95) using almost analytic extensions.More precisely, if (cid:101) Φ A and (cid:101) Φ B denote almost analytic extensions respectively for Φ A and Φ B , then, when α is near β , there is a unique point γ c ( α, β ) near α and β inan almost analytic neighbourhood of Λ such that ∂ β (cid:101) Φ A ( α, γ c ( α, β )) + ∂ α (cid:101) Φ B ( γ c ( α, β ) , β ) = 0 , where ∂ denotes the holomorphic part of the exterior derivative. Then we haveΦ AB ( α, β ) = (cid:101) Φ A ( α, γ c ( α, β )) + (cid:101) Φ B ( γ c ( α, β ) , β ) . Differentiating this expression with respect to α , we find that, for α near β ,d α Φ AB ( α, β ) = d α (cid:101) Φ A ( α, γ c ( α, β ))+ (cid:16) ¯ ∂ β (cid:101) Φ A ( α, γ c ( α, β )) + ¯ ∂ α (cid:101) Φ B ( γ c ( α, β ) , β ) (cid:17) ◦ d α γ c ( α, β ) . (2.98)Since (cid:101) Φ A and (cid:101) Φ B are almost analytic and γ c ( α, α ) = α is real, we see that thesecond term in the right-hand side of (2.98) vanish at infinite order on the diagonalof Λ × Λ. Thus, we see that the equation (2.97) satisfied at infinite order on thediagonal by d α Φ A is also satisfied by d α Φ AB . (cid:3) Since Π Λ A = A , we can replace (if necessary) Φ A and σ A by the phase andsymbol obtained by the application of the methods of [ MS75 ] as in the proof ofLemma 2.16, so as to assume that Φ A satisfies the assumptions of Lemma 2.17.We can consider adjoints, and for this introduce the new submanifolds of T ∗ Λ ⊗ C J Λ := (cid:110)(cid:16) α, α ∗ (cid:17) : (cid:0) α, α ∗ (cid:1) ∈ J Λ (cid:111) and J ∗ Λ := (cid:110)(cid:16) α, − α ∗ (cid:17) : (cid:0) α, α ∗ (cid:1) ∈ J Λ (cid:111) . (2.99)Here, the complex conjugacy is the one induced by the structure of tensor producton T ∗ Λ × C . The proof that led to Lemma 2.17 also gives that: Lemma . Let A and B be FIOs. Denote respectively by Φ A and Φ B thephases of A and B . Assume that Φ B satisfies d β Φ B ∈ J ∗ Λ (2.100) at infinite order on the diagonal of Λ × Λ . Then, if Φ AB denotes the phase of AB provided by Lemma 2.16, then Φ AB also satisfies (2.100) at infinite order on thediagonal. Here, it is important to notice that the manifold J ∗ Λ has been defined so thatthe phase Φ ∗ T S of Π ∗ Λ satisfies (2.100). Consequently, possibly changing again (andfor the last time) the phase Φ A as before, we can assume that Φ A also satisfies both(2.97) and (2.100). We will see that this is actually sufficient to determine Φ A . Tounderstand this, we start by observing that J Λ is really not a real manifold, as Lemma . Assume that Λ is a ( τ , -adapted Lagrangian with τ smallenough. The intersection of J Λ ∩ J Λ = Σ Λ is transverse in each fiber T α Λ ⊗ C ,where Σ Λ denotes the graph of Re θ in T ∗ Λ . 32 2. FBI TRANSFORM ON COMPACT MANIFOLDS Proof. If we prove that J T ∗ M and J T ∗ M are uniformly transverse, then theresult will follow by a perturbative argument. Additionally, it suffices to prove thetransversality in each fiber of T ∗ (cid:0) T ∗ M (cid:1) ⊗ C . Recall that for α ∈ T ∗ M we have J T ∗ M ∩ T α ( T ∗ M ) = { d α Φ T ( α, y ) | y ∈ (cid:102) M , d ( y, α x ) < (cid:15) } . Letting L denote the tangent space to J T ∗ M ∩ T α ( T ∗ M ) at Re θ ( α ), we want tocheck that L and L are uniformly transverse. The space L is the image of T α x (cid:102) M bythe differential at y = α x of the application y (cid:55)→ d α Φ T ( α, y ) . Since the phase Φ T is an admissible phase (recall Definition 1.7), we can write L inthe form (with local coordinates) (cid:26)(cid:18) Au + iBuu (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) u ∈ C n (cid:27) , where A , B are real and B is invertible. An element of the intersection L ∩ L corresponds to vectors u , v such that u = v and Au + iBu = Av − iBv , so that u = v = 0. Since the invertibility of B comes with uniform estimates, this comeswith a uniform estimate on the transversality of L and L . (cid:3) Lemma . Assume that Λ is a ( τ , -adapted Lagrangian with τ smallenough. There is a unique jet Φ on the diagonal of Λ × Λ that satisfies (2.97) , (2.100) and Φ( α, α ) = 0 , d α Φ( α, α ) = − d β Φ( α, α ) = Re θ ( α )(2.101) for every α ∈ Λ . Remark . Notice that it follows from Lemmas 2.17 and 2.18 that Φ fromLemma 2.20 is the jet on the diagonal of the phase of Π Λ Π ∗ Λ . We will denote by Φ Λ the phase of Π Λ Π ∗ Λ . Now, notice that if A is a FIO such that the jet of its phaseon the diagonal is Φ (which will happen by application of Lemma 2.20), we mayalways assume that its phase is exactly Φ Λ . Indeed, it follows from the coercivitycondition (2.94) that the error that occurs when replacing the phase of A by Φ Λ may be considered as part of the remainder term in (2.92).Notice also that the phase ( α, β ) (cid:55)→ − Φ Λ ( β, α ) satisfies the conditions fromLemma 2.20 too. Hence, we may replace Φ Λ by ( α, β ) (cid:55)→ (Φ Λ ( α, β ) − Φ Λ ( β, α )) / Λ satisfies Φ Λ ( α, β ) = − Φ Λ ( β, α ).The proof of Lemma 2.20 is based on an almost analytic version of the followingelementary observation. Let Γ be a Lagrangian submanifold of T ∗ R m , and f : T ∗ R m → R a smooth function. Assume that f vanishes on Γ. Denoting H f thecorresponding Hamiltonian vector field, we pick u a vector tangent to Γ and observethat ω ( u, H f ) = d f ( u ) = 0 , so that H f has to be tangent to Γ. If we have many independent functions thatvanish on Γ, its tangent space is thus determined. We already have the functions ζ j, (from Proposition 2.9) that vanish on J Λ . It will be useful to introduce thefunctions ζ ∗ j, ( α, α ∗ ) = ζ j, ( α, − α ∗ ) . They vanish on J ∗ Λ . .3. BERGMAN PROJECTOR AND SYMBOLIC CALCULUS 133 Proof of Lemma 2.20. The existence is immediate since the phase of Φ Λ from Remark 2.19 satisfies the conditions from the lemma. Let us then focus on theuniqueness.We start by picking an almost analytic extension of Φ and consider the graphΓ Φ = (cid:110) ( α, ∂ α Φ; β, ∂ β Φ) (cid:12)(cid:12)(cid:12) α, β ∈ (cid:101) Λ (cid:111) ⊂ T ∗ (cid:101) Λ , where ∂ α denotes the C -linear part of the exterior derivative, that is d α = ∂ α + ¯ ∂ α .This is an almost analytic Lagrangian manifold, in the sense that the complexanalytic symplectic form vanishes at infinite order near the real points of Γ Φ ,which are exactly the points of ∆ Λ (for more details on almost analytic Lagrangianmanifolds, see § MS75 ]).We can also pick almost analytic extensions of the ζ ( ∗ ) j, ’s. From the conditions(2.97) and (2.100), we deduce that ζ j, ( α, ∂ α Φ) and ζ ∗ j, ( β, ∂ β Φ) are O ( | α − β | + | Im α | + | Im β | ) ∞ . We identify ζ j, (resp. ζ ∗ j, ) with ( α, α ∗ , β, β ∗ ) (cid:55)→ ζ j, ( α, α ∗ ) (resp. ζ ∗ j, ( β, β ∗ )).Denoting by ω the complex symplectic form of T ∗ (cid:101) Λ × (cid:101) Λ, we have for u a tangentvector to Γ Φ , and f one of the functions ζ j, , ζ ∗ j, , j = 1 . . . nω ( u, H f ) = ∂f ( u ) = d f ( u ) + O ( | α − β | ) ∞ = O ( | α − β | + | Im α | + | Im β | ) ∞ . In particular, H f is tangent to Γ Φ at all orders on the diagonal. Since Γ Φ has(almost analytic complex-)dimension 4 n , contains ∆ Λ , which has dimension 2 n , andwe have 2 n vector fields tangent to Φ, it suffices to check that these vector fieldsform a free family, generating a space transverse to T ∆ Λ ⊗ C .The fact that the d ζ j, form a free family on J Λ near its real points implies thatthe H ζ j, , j = 1 . . . n are free, and likewise for the H ζ ∗ j, , j = 1 . . . n . Since they acteither on α, α ∗ or on β, β ∗ , the H f ’s form a free family. We now have to considerthe transversality with ∆ Λ . We consider a, b ∈ C n such that n (cid:88) j =1 a j H ζ j, + n (cid:88) j =1 b j H ζ ∗ j, ∈ T ∆ Λ ⊗ C . From the structure of ∆ Λ , this implies (cid:88) a j d α ∗ ζ j, ( α, Re θ ( α )) − (cid:88) b j d α ∗ ζ ∗ j, ( α, − Re θ ( α )) = 0 . From the definition of ζ ∗ j, , this means (here the complex conjugacy makes sensedue to the tensor product structure on T ∗ Λ ⊗ C ) (cid:88) a j d α ∗ ζ j, ( α, Re θ ( α )) − (cid:88) b j d α ∗ ζ j, ( α, Re θ )( α )) = 0In the fiber T ∗ α Λ, recall that the ζ j, , j = 1 . . . n , form a system of equations for J Λ , sothat the linear forms d α ∗ ζ j, generate the orthogonal dual of T Re θ ( α ) J Λ ∩ T ∗ α Λ. Theircomplex conjugates thus generate the dual of T Re θ ( α ) J Λ ∩ T ∗ α Λ. The transversalitylemma 2.19 thus implies that equation above only admits 0 as solution, so a = b = 0and the proof is complete. (cid:3) Transport equations on the symbol. Let us now come back to our originaldatum of A , an FIO such that Π Λ A = A Π ∗ Λ = A . We assume, as we may, that Φ A satisfies both (2.97) and (2.100), so that Φ A can be replaced by the phase Φ Λ fromRemark 2.19. Under these conditions, A is determined up to a negligible operatorby the value of its symbol on the diagonal of Λ × Λ, as we prove now. 34 2. FBI TRANSFORM ON COMPACT MANIFOLDS Lemma . Assume that Λ is a ( τ , -adapted Lagrangian with τ smallenough. Let A be an FIO with phase Φ Λ . Assume that Π Λ A = A and A Π ∗ Λ = A .Assume in addition that the symbol σ A of A is an O ( h/ (cid:104) α (cid:105) ) ∞ on the diagonal, then A is a negligible operator in the sense of Definition 2.6. The first idea may be to compute directly the following product and try toidentify the terms:Π Λ A ( α, β ) e H ( β ) − H ( α ) h ∼ e i ΦΠΛ A ( α,β ) h (cid:88) k ≥ h k P k (cid:16) σ Π Λ ( α, γ ) σ A ( γ, β ) (cid:17) | γ = γ c ( α,β ) . Here, the operators P k are differential operators of order 2 k in the variable γ . Inparticular, for the term of order h k , the coefficient involves 2 k derivatives of σ A . Atleast formally, the equation Π Λ A = A thus gives a collection of PDEs on the symbolof A , that are a priori not easy to interpret. This is why we have to take a differentapproach, relying crucially on the Z j ’s constructed in § Proof. The proof of this lemma occupies the rest of this section. Far from thediagonal, the kernel of A is very small by definition, so we can concentrate on aneighbourhood of fixed size of the diagonal. From the coercivity condition (2.94)satisfied by Φ A , we see that we only need to prove that all the derivatives of σ A onthe diagonal of Λ × Λ are O (( h/ (cid:104)| α |(cid:105) ) ∞ ).To do so, recall the operators Op( ζ j ) from Proposition 2.9 (we can work “arounda fiber” in the sense of Proposition 2.9) and notice thatOp( ζ j ) e − Hh Ae Hh ( α, β ) = (cid:16) Op( ζ j ) e − H/h Π Λ e H/h (cid:17) e − H/h Ae H/h ( α, β )= O (cid:18)(cid:18) h (cid:104)| α |(cid:105) + (cid:104)| β |(cid:105) (cid:19) ∞ (cid:19) . We want to deduce some information about the symbol of A . Certainly, applying[ MS75 ] once again,Op( ζ j ) e − Hh A ( α, β ) e Hh = e ih Φ A ( α,β ) c ( α, β ) + O (( h/ (cid:104)| β |(cid:105) ) ∞ ) , where c is supported near the diagonal, and has a semi-classical expansion involvingderivatives of ζ j and σ A . Since e i Φ A /h is a Gaussian term centered at the diagonal,we deduce that for d ( α, β ) ≤ Ch / , c ( α, β ) = O (( h/ (cid:104)| α |(cid:105) ) ∞ ) . This is a priori a L ∞ estimate. However, we know that c is a symbol, so that usingfinite differences and Taylor’s Formula, we deduce that this estimate actually holdsin C ∞ . We want to deduce that σ A = O ( h/ (cid:104)| α |(cid:105) ) ∞ at infinite order on the diagonal.We consider now the semi-classical expansion for c , which starts as (we use thesame coordinates ( u, η ) as in § ζ j, ( u, η, d u,η Φ A ( u, η, u (cid:48) , η (cid:48) ) + i d u,η H ( u, η )) σ A ( u, η, u (cid:48) , η (cid:48) ) + O ( h ) . From the choice of ζ j, and Lemma 2.17, the first term here vanishes at infiniteorder on the diagonal. Next, we consider the term of order h in the expansion. Forthis we recall the usual expansion (see for example (2.28) in [ MS75 ]). (cid:90) e ih (cid:104) x − y,ξ (cid:105) p ( x, ξ ) e ih S ( y ) a ( y )d y d ξ = e ih S ( x ) (cid:20) p ( x, d x S ) a ( x ) + hi ∇ ξ p · ∇ x a + O ( h a ) (cid:21) + O ( h ∞ ) . .3. BERGMAN PROJECTOR AND SYMBOLIC CALCULUS 135 (here, O ( h a ) is a bit of an abuse of notations since it involves derivatives of a ). Itfollows that c ( u, η, u (cid:48) , η (cid:48) ) = hi ∇ u ∗ ,η ∗ ζ j, ( u, η ; d u,η Φ ◦ T S ( u, η, u (cid:48) , η (cid:48) )) · ∇ u,η σ A ( u, η, u (cid:48) , η (cid:48) )+ hζ j, ( u, η, d u,η Φ ◦ T S ( u, η, u (cid:48) , η (cid:48) )) σ A ( u, η, u (cid:48) , η (cid:48) )+ h (cid:104) η (cid:105) − P j σ A ( u, η, u (cid:48) , η (cid:48) ) + O C ∞ ( h ∞ ) . Here P j comes from the application of the stationary phase method, and in particularwe know that if we ca prove that the derivatives of the symbol σ A on the diagonalare O ( (cid:104) η (cid:105) − N h N ) then the derivatives of the symbol P j σ ( u, η, u, η ) satisfy the sameestimates. Writing this in a more intrinsic fashion, we deduce that,d α ∗ ζ j, (cid:0) α, d α Φ ◦ T S ( α, β ) (cid:1) · d α σ A ( α, β )+ iζ j, (cid:0) α, d α Φ ◦ T S ( α, β ) (cid:1) σ A ( α, β )+ h (cid:104)| α |(cid:105) − P j σ A ( α, β ) = O C ∞ (cid:18)(cid:18) h (cid:104)| α |(cid:105) (cid:19) ∞ (cid:19) (2.102)Here, it makes sense to apply d α ∗ ζ j, to d α σ A since there are elements respectivelyof ( T ∗ α Λ) ∗ and T ∗ α Λ.Now we will find n others equations. Indeed, since A Π ∗ Λ = A , we have Π Λ A ∗ = A ∗ , so that (2.102) is also satisfied by σ A ∗ ( α, β ) = σ A ( β, α ). We can rewrite ourequations in the form ( j = 1 . . . n ) X j σ A + F j σ A + h (cid:104)| α |(cid:105) − P j σ A = O (cid:18)(cid:18) h (cid:104)| α |(cid:105) (cid:19) ∞ (cid:19) . The X j ’s are complex vector fields on Λ × Λ (defined near the diagonal), i.e. sectionsof T (Λ × Λ) ⊗ C . Using again Lemma 2.19, as in the proof of Lemma 2.20, wededuce that they generate a space which is supplementary to the complexifiedtangent space to the diagonal. The Q j ’s have the same general properties than the P j ’s. Denoting γ = α − β and δ = α + β in local coordinates, we deduce that(2.103) ∇ γ σ A = L ( γ, δ ) ∇ δ σ A ( α, β ) + σ A ( γ, δ ) N ( γ, δ ) + h (cid:104)| α |(cid:105) − Qσ A ( γ, δ ) + O ( h ∞ ) . Here, L and N are respectively a matrix and a vector that satisfies symbolic estimatesin γ, δ . The operator Q satisfies the same kind of estimates as the P j ’s or the Q j ’s.Our assumption on σ A implies that all its derivatives (of any order) with respectto δ are O (( h/ (cid:104)| α |(cid:105) ) ∞ ) on the diagonal of Λ × Λ. Hence, differentiating (2.103) anynumber of times and then restricting to the diagonal, we find by induction that thederivatives of all orders (with respect to both γ and δ ) of σ A are O ( h/ (cid:104)| α |(cid:105) ) on thediagonal.However, from this newly acquired knowledge, we find that the derivatives of Qσ A ( α, β ) are in fact O ( h/ (cid:104)| α |(cid:105) ) on the diagonal. We can consequently play thesame game to find that the derivatives of all orders (with respect to both γ and δ ) of σ A are in fact O (( h/ (cid:104)| α |(cid:105) ) ) on the diagonal. Then, we keep iterating theseprocedures and we find by induction that all the derivatives of σ A are O (( h/ (cid:104)| α |(cid:105) ) ∞ )on the diagonal of Λ × Λ. This ends the proof of the lemma. (cid:3) We are now readyto deduce useful consequences from the analysis of § B Λ on H , FBI in L (Λ). We start by showing that: Lemma . There is a real-valued symbol f ∈ S KN (Λ) positive and ellipticof order such that Π Λ f Π ∗ Λ and (cid:0) Π Λ f Π ∗ Λ (cid:1) differ by a negligible operators. 36 2. FBI TRANSFORM ON COMPACT MANIFOLDS Proof. From Lemmas 2.17, 2.18 and 2.20, we know that the phases of Π Λ f Π ∗ Λ and its square coincide with the phase Φ Λ from Remark 2.19 at infinite order onthe diagonal. Hence, we may assume that the phases of Π Λ f Π ∗ Λ and its squareare exactly Φ Λ . We will be in position to apply Lemma 2.21 if we can find f suchthat the symbols of Π Λ f Π ∗ Λ and its square coincides on the diagonal up to an O (( h/ (cid:104)| α |(cid:105) ) ∞ ). We want to construct f with an asymptotic expansion f ∼ (cid:88) k ≥ h k f k . Then, the symbols c and ˜ c of Π Λ f Π ∗ Λ and (cid:0) Π Λ f Π ∗ Λ (cid:1) respectively have asymptoticexpansions (in the sense of Remark 2.16) c ∼ πh ) n (cid:88) k ≥ h k c k , ˜ c ∼ πh ) n (cid:88) k ≥ h k ˜ c k (2.104)that are deduced from the expansion for f (recall Lemma 2.16). In order to applyLemma 2.21, we want to choose f such that the c k and ˜ c k coincide on the diagonalof Λ × Λ for every k ≥ c ( α, α ) = f ( α ) g ( α, α ), where g ( α, β ) a symbol of order 0, whoserestriction to the diagonal is positive and elliptic (we recall that the order of g isdiscussed in the proof of Lemma 2.16). In general, c k ( α, α ) = g ( α, α ) f k ( α ) + A k ( f , . . . , f k − )( α, α ) , (2.105)where A k ( f , . . . , f k − ) is a symbol of order − k that only depends on f , . . . , f k − .On the other hand, we also get (˜ g is a symbol of order 0 with properties that aresimilar to those of g ) (cid:101) c ( α, α ) = ˜ g ( α, α ) c ( α, α ) , and for k ≥ (cid:101) c k ( α, α ) = 2˜ g ( α, α ) c ( α, α ) c k ( α, α ) + B k ( c , . . . , c k − )( α, α ) , where B k ( c , . . . , c k − )( α, α ) is a symbol of order − k depending only on c , . . . , c k − .Consequently by choosing f ( α ) = (˜ g ( α, α ) g ( α, α )) − , we ensure that c and ˜ c coincide on the diagonal (as well as the ellipticity and positivity of f , provided that h is small enough). This imposes for k ≥ c k ( α, α ) = − B k ( c , . . . , c k − )( α, α ) . In turn, we get(2.106) f k ( α ) = − g ( α, α ) − ( A k ( f , . . . , f k − )( α, α ) + B k ( c , . . . , c k − )( α, α )) . Now, by induction, we prove that the f k ’s may be chosen real valued. We gavean explicit expression for f that ensures that it is real-valued (and positive).Assume that f , . . . , f k are all real-valued, then Π ∗ Λ ( f + · · · + h k f k )Π Λ is self-adjoint. In particular, if K denotes the reduced kernel of this operator we have K ( α, β ) = K ( β, α ). Recall that the phase of Π ∗ Λ ( f + · · · + h k f k )Π Λ is Φ Λ anddenote by σ its symbol. Then we have, up to negligible terms, K ( α, β ) = K ( α, β ) + K ( β, α )2 = e i Φ Λ ( α,β ) σ ( α, β ) + σ ( β, α )2 . Here, we recall that Φ Λ ( α, β ) = − Φ Λ ( β, α ). We may consequently assume that σ ( α, β ) = − σ ( β, α ). This implies in particular that A k +1 ( f , . . . , f k )( α, α ) is real.Now, (Π ∗ Λ ( f + · · · + h k f k )Π Λ ) will also be self-adjoint, and we can use the same .3. BERGMAN PROJECTOR AND SYMBOLIC CALCULUS 137 argument to ensure that B k +1 ( c , . . . , c k )( α, α ) is also real. Using then (2.106) todefine f k +1 , we find that it is real-valued. (cid:3) We can now make explicit the structure of the orthogonal projector B Λ . Lemma . The orthogonal projector B Λ is a FIO. More precisely, if f is asin Lemma 2.22, then B Λ and Π Λ f Π ∗ Λ differ by a negligible operator. Proof. Let f be as in Lemma 2.22 and set B ◦ Λ = Π λ f Π ∗ Λ . Notice that, since f is real valued, the operator B ◦ Λ is self-adjoint. Moreover, since f is positive andelliptic, for h small enough we have f ≥ C − for some C > 0, and hence, if h issmall enough, we have for u ∈ H , FBI (the scalar product is in L (Λ)) (cid:104) B ◦ Λ u, u (cid:105) = (cid:104) f Π ∗ Λ u, Π ∗ Λ u (cid:105) ≥ C (cid:13)(cid:13) Π ∗ Λ u (cid:13)(cid:13) ≥ C (cid:32) (cid:13)(cid:13) Π ∗ Λ u (cid:13)(cid:13) (cid:107) u (cid:107)(cid:107) u (cid:107) (cid:33) ≥ C (cid:104) Π ∗ Λ u, u (cid:105) (cid:107) u (cid:107) ≥ C (cid:18) (cid:104) u, Π Λ u (cid:105)(cid:107) u (cid:107) (cid:19) = 1 C (cid:107) u (cid:107) . (2.107)Hence, the spectrum of B ◦ Λ consists of 0 and a bounded subset E of R ∗ + , boundedfrom below by C − . This is because the image of B ◦ Λ is contained in H , FBI and B ◦ Λ vanishes on the orthogonal of H , FBI . Then, we may choose a loop γ around E but not around 0, and define the spectral projector (cid:101) B Λ = 12 iπ (cid:90) γ (cid:0) z − B ◦ Λ (cid:1) − d z. We start by showing that B ◦ Λ approximates (cid:101) B Λ . To do so, notice that if z ∈ γ thenwe have (cid:0) z − B ◦ Λ (cid:1) (cid:18) z + 1 z ( z − B ◦ Λ (cid:19) = 1 + 1 z ( z − (cid:16) B ◦ Λ − ( B ◦ Λ ) (cid:17) . When h is small enough, applying Lemma 2.22, we may invert the operator 1 +( z ( z − − (cid:16) B ◦ Λ − ( B ◦ Λ ) (cid:17) by mean of Von Neumann series (uniformly for z ∈ γ ).Thus, we have (cid:18) z ( z − (cid:16) B ◦ Λ − ( B ◦ Λ ) (cid:17)(cid:19) − = 1 + R ( z ) , where R ( z ) is a negligible operator (uniformly in z ∈ γ ). Moreover, R ( z ) is valuedin H , FBI and commute with B ◦ Λ . Consequently, we may write (cid:101) B Λ = 12 iπ (cid:90) γ (cid:18) z + 1 z ( z − B ◦ Λ (cid:19) (1 + R ( z )) d z = B ◦ Λ + 12 iπ (cid:90) γ R ( z ) (cid:18) z + 1 z ( z − B ◦ Λ (cid:19) d z. (2.108)Since R ( z ) is negligible, if we prove that (cid:101) B Λ = B Λ , we will be done. However, (cid:101) B Λ is an orthogonal projector by construction since B ◦ Λ is self-adjoint. Additionally,since it is the projection on the non-zero part of the spectrum of B ◦ Λ , the spectraltheorem ensures that B ◦ Λ u = 0 if and only if (cid:101) B Λ u = 0. It follows that ker (cid:101) B Λ is theorthogonal of H , FBI (by (2.107)), and thus that (cid:101) B Λ = B Λ . (cid:3) Remark . Recalling Remark 2.15, we see that for every k ∈ R the operator B Λ is bounded from L k (Λ) to H k Λ , FBI . Consequently, if σ is a symbol of order m on Λ and k ∈ R , the operator B Λ σB Λ is bounded from L k (Λ) to H k − m Λ , FBI . 38 2. FBI TRANSFORM ON COMPACT MANIFOLDS Before going further into the study of Toeplitz calculus, let us mention thatLemma 2.23 allows to identify the dual of the spaces H k Λ . Lemma . Assume that τ and h are small enough and let k ∈ R . Then, if u ∈ H − k Λ and v ∈ H k Λ the pairing (cid:104) u, v (cid:105) := (cid:104) T Λ u, T Λ v (cid:105) L (Λ) (2.109) is well-defined and induces an (anti-linear) identification between H − k Λ and the dualof H k Λ (inducing equivalent, but a priori not equal norms). Proof. It is clear from the definition of the spaces H k Λ and H − k Λ that thepairing (2.109) is well-defined and induces a bounded anti-linear map from H − k Λ to the dual of H k Λ . Let us denote this map by A . We start by proving that A issurjective. Let (cid:96) be a continuous linear form on H k Λ and let ˜ (cid:96) be the linear form on L k (Λ) defined by ˜ (cid:96) ( u ) = (cid:96) ( S Λ u ) . Recall that Π Λ = T Λ S Λ is bounded on L k (Λ) by Proposition 2.4, and hence that S Λ is bounded from L k (Λ) to H k Λ . Consequently, ˜ (cid:96) is a well-defined and continuouslinear form on L k (Λ), and thus there exists v (cid:96) ∈ L − k (Λ) such that for every u inthe space L k (Λ) we have ˜ (cid:96) ( u ) = (cid:104) v (cid:96) , u (cid:105) L (Λ) . Hence, if u ∈ H k Λ we have (cid:96) ( u ) = ˜ (cid:96) ( T Λ u ) = (cid:104) v (cid:96) , T Λ u (cid:105) L (Λ) = (cid:104) T Λ S Λ B Λ v (cid:96) , T Λ u (cid:105) L (Λ) . Thus, we have A ( S Λ B Λ v (cid:96) ) = (cid:96) and A is surjective. Here, we used the fact that B Λ is bounded on H − k Λ , FBI (see Remark 2.20).The injectivity of A follows from its surjectivity by duality (the result is symmet-ric in k and − k ), and thus A is an isomorphism by the Open Mapping Theorem. (cid:3) We want now to prove the “multiplication formula” (Proposition 2.11), that willbe the main tool to study the spectral properties of an Anosov vector field actingon a space of anisotropic ultradistributions. To do so we first prove a “Toeplitzproperty” for G s pseudo-differential operators. Proposition . Let s ≥ . Let P be a G s pseudo-differential operator of order m . Assume that Λ is a ( τ , s ) -Gevrey adapted La-grangian with τ small enough. Let f ∈ S ˜ mKN (Λ) be a symbol of order ˜ m on Λ withuniform estimates when h → . Then there is a symbol σ ∈ S m + ˜ mKN (Λ) of order m + m (cid:48) on Λ such that, for h small enough, B Λ f T Λ P S Λ B Λ and B Λ σB Λ differ by anegligible operator.Moreover, σ is given at first order by σ ( α ) = f ( α ) p Λ ( α ) mod hS m + ˜ m − KN (Λ) , where p Λ denotes the restriction to Λ of an almost analytic extension of the principalsymbol of p (see Remark 1.10). Proof of Proposition 2.10. It follows from Lemma 2.16 that the operators B Λ f T Λ P S Λ B Λ and B Λ σB Λ both are FIOs. Moreover, according to Lemmas 2.17,2.18 and 2.20, they share the same phase and it is Φ Λ (from Remark 2.19). Finally, B Λ f T Λ P S Λ B Λ and B Λ σB Λ inherit from B Λ the left invariance by Π Λ and the right .3. BERGMAN PROJECTOR AND SYMBOLIC CALCULUS 139 invariance by Π ∗ Λ . Consequently, according from Lemma 2.21, we only need to choose σ so that the symbols of B Λ f T Λ P S Λ B Λ and B Λ σB Λ agree up to O (( h/ (cid:104)| α |(cid:105) ) ∞ ) onthe diagonal of Λ × Λ.As in the proof of Lemma 2.22, we will construct σ ∈ S m + ˜ mKN (Λ) with anasymptotic expansion σ ∼ (cid:88) k ≥ h k σ k . The σ k ∈ S m + ˜ m − kKN (Λ)’s will be constructed by induction. It follows from Lemma2.16 that the symbol ˜ σ ∈ h − n S m + ˜ mKN (Λ × Λ) of B Λ σB Λ has an asymptotic expansion(in the sense of Remark 2.16) ˜ σ ∼ πh ) n (cid:88) k ≥ h k ˜ σ k . Here, we see that we are using the same stationary phase argument than whencomputing B = B Λ , the only difference being the multiplication by σ in between.Consequently, we find that ˜ σ is given on the diagonal by˜ σ ( α, α ) = c ( α, α ) σ ( α ) , where c is the first term in the expansion (2.104) for the symbol c of B Λ . Moregenerally, we have˜ σ k ( α, α ) = c ( α, α ) σ k ( α ) + C k ( σ , . . . , σ k − )( α, α )with C k ( σ , . . . , σ k − ) a symbol of order m + ˜ m − k . Hence, we only need to take σ ( α ) = (2 πh ) n e ( α, α ) c ( α, α ) , where e denotes the symbol of B Λ f T Λ P S Λ B Λ , and for k ≥ σ k ( α ) = − C k ( σ , . . . , σ k − )( α, α ) c ( α, α ) . This is possible due to the ellipticity of c ( α, α ).It remains to check the first order asymptotic for σ . To do so, we only needto see that e is given at first order on the diagonal by c ( α, α ) f ( α ) p Λ ( α ). Recallthat Π Λ and T Λ P S Λ share the same phase. Moreover,it follows from Lemma 2.10that the symbol of T Λ P S Λ differ from the symbol of Π Λ by a factor p Λ ( α ), on thediagonal and in first order approximation. Hence, the symbols of B Λ = B Λ Π Λ B Λ and B Λ f T Λ P S Λ B Λ differ in first order approximation on the diagonal by a factor f p Λ . (cid:3) We state now what will be the key tool in the spectral analysis of P in thenext chapter. Notice that the multiplication formula, Proposition 2.11, is a strongerstatement than Theorem 7. Proposition . Let s ≥ and P be a G s pseudo-differential operator of order m . Assume that Λ is a ( τ , s ) -Gevrey adapted La-grangian with τ small enough. Let p Λ denotes the restriction to Λ of an almostanalytic extension of the principal symbol of P . Assume that h is small enough. Let f ∈ S ˜ mKN (Λ) be a symbol of order ˜ m on Λ , uniformly in h . Then, if m , m ∈ R are such that m + m = m + ˜ m − , there is a constant C > such that for any u, v ∈ H ∞ Λ , we have (the scalar product is in L (Λ) ) |(cid:104) f T Λ P u, T Λ v (cid:105) − (cid:104) f p Λ T Λ u, T Λ v (cid:105)| ≤ Ch (cid:107) u (cid:107) H m (cid:107) v (cid:107) H m . (2.110) 40 2. FBI TRANSFORM ON COMPACT MANIFOLDS Proof. First, notice that from Proposition 2.4 we know that P u belongs to H ∞ Λ , so that it makes sense to write (cid:104) f T Λ P u, T Λ v (cid:105) = (cid:104) B Λ f T Λ P S Λ B Λ T Λ u, T Λ v (cid:105) . Then from Proposition 2.10, we can find σ a symbol on Λ such that B Λ f T Λ P S Λ B Λ = B Λ σB Λ + O H − N Λ , FBI →H N Λ , FBI ( h N ) , Let σ = f p Λ denote the first order approximation of σ given by Proposition 2.10.Since σ − σ = O ( h (cid:104)| α |(cid:105) m + ˜ m − ), it is clear that B Λ ( σ − σ ) B Λ is a O ( h ) as abounded operator from H m Λ , FBI to H − m Λ , FBI , recalling that m − m − ˜ m + 1 = − m .Hence, we see that B Λ f T Λ P S Λ B Λ = B Λ f p Λ B Λ + O H m , FBI → H − m , FBI ( h ) , and the result readily follows. (cid:3) Remark . In the applications, we will not use the asymptotic h → α large, with h > h to be small enough to make themachinery work.Using the same kind of arguments as in the proof of Proposition 2.10, we mayprove the following statements about composition of Toeplitz operators. This is leftas an exercise to the reader. Proposition . Let σ ∈ S m KN (Λ) and σ ∈ S m KN (Λ) . Then there is a symbol σ σ ∈ S m + m KN (Λ) such that ( B Λ σ B Λ ) ◦ ( B Λ σ B Λ ) and B Λ σ σ B Λ differ by a negligible operator. Moreover, σ σ coincides with the product σ σ up to hS m + m − KN (Λ) . In prevision of the next section, we state and prove a last result about thisToeplitz calculus. Lemma . Let σ ∈ S mKN (Λ) be a symbol of order m < − n on Λ . Then theoperator B Λ σB Λ from H , FBI to itself is trace class, with trace class norm controlledby a semi-norm of σ (in the class of symbols of order m ). Moreover, the trace of B Λ σB Λ is the integral of its kernel on the diagonal of Λ × Λ . Proof. The first observation is that if an operator R has a kernel satisfying(2.111) R ( α, β ) e H ( β ) − H ( α ) h = O C N ( L ( (cid:104)| α |(cid:105) + (cid:104)| β |(cid:105) ) − N ) , for N large enough, then it is trace class, with trace norm O ( L ). This follows fromusual techniques. For example, see [ DS99 , Section 9].Let σ be the symbol σ ( α ) = (cid:104)| α |(cid:105) m on Λ. By a parametrix construction ( σ is elliptic), we find a symbol σ of order m/ B Λ σB Λ = ( B Λ σ B Λ ) ◦ ( B Λ σ B Λ ) + R, where R is an operator with a smooth kernel that satisfies (2.111). We only proceedat a finite number of steps in the parametrix construction in order to be able tocontrol the derivatives of the kernel of R by some semi-norm of σ . However, byproceeding at a large enough number of steps in the parametrix construction, wemay obtain an estimate of the form (2.111) for an arbitrary large N . In particular,we control the trace class norm of R by some semi-norm of σ .In order to prove that B Λ σB Λ is trace class, we want now to prove that B Λ σ B Λ and B Λ σ B Λ are Hilbert–Schmidt. To do so, we only need to prove that their reducedkernels are square integrable. When investigating the kernel of B Λ σ B Λ by mean of .3. BERGMAN PROJECTOR AND SYMBOLIC CALCULUS 141 the non-stationary phase method (as in the proof of Lemma 2.16), one only needsestimates on a finite number of derivatives of σ to obtain an estimate of the form B Λ σ B Λ ( α, β ) e H ( β ) − H ( α ) h = e i ΦΛ( α,β ) h c ( α, β ) + O C N (cid:0) ( (cid:104)| α |(cid:105) + (cid:104)| β |(cid:105) ) − N (cid:1) , (2.112)where N may be taken as large as we want. In particular, the remainder in (2.112)is square integrable. Here, c ( α, β ) is a symbol of order m/ B Λ σ B Λ is square integrable (using that Φ Λ satisfies the coercivity condition (2.94) and working as in the proof of Proposition2.4). Hence, the kernel of B Λ σ B Λ is square integrable (with norm controlled by asemi-norm of σ and hence of σ ). The operator B Λ σ B Λ is Hilbert–Schmidt too forthe same reason.We just proved that B Λ σB Λ is trace class. To prove that its trace is the integralof its kernel on the diagonal of Λ × Λ, just approximate σ by compactly supportedkernel. (cid:3) The aim of this section is to prove thefollowing proposition. It will be used in the applications to deduce the Schattenproperty of the resolvent from the ellipticity of the generator of the flow (see Lemma3.3 and Theorem 10). Proposition . Assume that Λ is a ( τ , -adapted Lagrangian with τ smallenough. Assume that h is small enough. Let m > and q ∈ R , then the inclusionof H m + q Λ into H q Λ is compact. In addition, if ( µ k ) k ∈ N denotes the singular values ofthis inclusion then we have µ k = k → + ∞ O (cid:18) k mn (cid:19) . In particular, it belongs to the Schatten class S p for any p > n/m . Notice that we get an equivalent statement if we replace H q Λ and H m + q Λ inProposition 2.13 respectively by H q Λ , FBI and H m + q Λ , FBI . We will rather consider thiscase in the following. That the inclusion is compact is an easy consequence of thefact that it may be written as the restriction of the operator with smooth kernel Π Λ to H m + q Λ , FBI , the proof is left as an exercise to the reader. The following result will beuseful in the proof of Proposition 2.13. Lemma . Assume that Λ is a ( τ , -adapted Lagrangian with τ smallenough. Assume that h is small enough. Let m > . Define the unbounded operator A = B Λ (cid:104)| α |(cid:105) m B Λ on H , FBI with domain H m Λ , FBI . The operator A is closed, self-adjoint, positive, and has compact resolvent. Moreover, does not belong to thespectrum of A , and, if ( λ k ) k ∈ N denotes the sequence of eigenvalues of A (orderedincreasingly), then there is a constant C such that for all k ∈ N we have µ k ≤ Cλ − k – where the µ k ’s are from Proposition 2.13. Proof. Let ( u (cid:96) ) (cid:96) ∈ N be a sequence in H m Λ , FBI such that ( u (cid:96) ) (cid:96) ∈ N converges tosome u in H , FBI and ( Au (cid:96) ) (cid:96) ∈ N converges to some v in H , FBI . Then, since A isbounded from H , FBI to H − m Λ , FBI , the sequence ( Au (cid:96) ) (cid:96) ∈ N converges to Au in H − m Λ , FBI .Since the convergence in H − m Λ , FBI or in H , FBI implies pointwise convergence (it isa consequence of the structure of Π Λ ), then Au = v . Using Proposition 2.12 toconstruct a parametrix for A , we see that since Au = v belongs to H , FBI , we havethat u ∈ H m Λ , FBI , and hence A is closed.Let us prove that A is self-adjoint. For this, we observe that for u ∈ L − m (Λ)and v ∈ L m (Λ), we have(2.113) (cid:104) B Λ u, v (cid:105) = (cid:104) u, B Λ v (cid:105) . 42 2. FBI TRANSFORM ON COMPACT MANIFOLDS This follows from the fact that B Λ is bounded on every space L k (Λ), and the densityof these spaces in one another. Now, let u ∈ H , FBI and notice that Au ∈ H − m Λ , FBI .In particular, if v ∈ H m Λ , FBI then (cid:104) Au, v (cid:105) makes sense, and (cid:104) u, Av (cid:105) = (cid:104) u, B Λ (cid:104)| α |(cid:105) m B Λ v (cid:105) = (cid:104)(cid:104)| α |(cid:105) m u, v (cid:105) = (cid:104) B Λ (cid:104)| α |(cid:105) m B Λ u, v (cid:105) = (cid:104) Au, v (cid:105) . From this, we see that Dom( A ) ⊆ Dom( A ∗ ), and in general, that if u ∈ Dom( A ∗ ), Au − A ∗ u is orthogonal to all v ∈ H m Λ , FBI . But since H m Λ , FBI is dense in H − m Λ , FBI , wededuce that in that case, Au = A ∗ u , so that u ∈ Dom( A ).To see that A is positive, just notice that for u ∈ H m Λ , FBI (Λ) we have(2.114) (cid:104) Au, u (cid:105) ≥ (cid:107) u (cid:107) H , FBI since (cid:104)| α |(cid:105) m ≥ 1. It also follows that 0 does not belong to the spectrum of A . Usinga parametrix construction, we see that the resolvent of A sends H , FBI continuouslyinto H m Λ , FBI , and is hence compact as an operator from H , FBI to itself.Let ( u k ) k ∈ N be an orthonormal sequence of eigenvectors of A , that is Au k = λ k u k .Then if u ∈ H m Λ , FBI , we have using Plancherel’s formula (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) u − N − (cid:88) k =0 (cid:104) u, u k (cid:105) u k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H , FBI = + ∞ (cid:88) k = N |(cid:104) u, u k (cid:105)| = + ∞ (cid:88) k = N |(cid:104) u, Au k (cid:105)| λ k ≤ λ − N + ∞ (cid:88) k = N |(cid:104) Au, u k (cid:105)| ≤ λ − N (cid:107) Au (cid:107) H , FBI ≤ C λ − N (cid:107) u (cid:107) H m Λ , FBI . Then the lemma follows from [ GGK00 , Theorem IV.2.5]. (cid:3) Before starting the proof of Proposition 2.13, we need to do two reductions. Reduction . Let m > and assume that, for this value of m , Proposition2.13 holds for q = 0 . Then, for this particular value of m , Proposition 2.13 holdsfor any q ∈ R . Reduction . Assume that Proposition 2.13 holds for all m ∈ ]0 , and for q = 0 , then Proposition 2.13 holds. Proof of Reduction 1. Let q ∈ R . Let us deal first with the case q > A = B Λ (cid:104)| α |(cid:105) q B Λ on H , FBI with domain H q Λ , FBI . We knowfrom Lemma 2.26 that A is a closed self-adjoint operator and that 0 does not belongto its spectrum. Using Proposition 2.12 to construct a parametrix for A , we seethat the operator A − is bounded from H , FBI to H q Λ , FBI . Hence, if j denotes theinclusion of H m + q Λ , FBI into H q Λ , FBI and j (cid:48) denotes the inclusion of H m Λ , FBI into H , FBI ,then we have j = A ◦ j (cid:48) ◦ A − , and the estimate on the singular values of j (cid:48) that weassumed carries on to the singular values of j .We deal now with the case q < 0. We set A = B Λ (cid:104)| α |(cid:105) − q B Λ . As above, weconstruct an inverse A − for A (the operator A − is a priori defined on H , FBI ) anda parametrix E for A (the parametrix E is defined in particular on H m + q Λ , FBI ). Then .3. BERGMAN PROJECTOR AND SYMBOLIC CALCULUS 143 we have AE = I − R where R is a negligible operator and hence A − = E + A − R may be extended to an operator from H m + q Λ , FBI to H m Λ , FBI , which is still an inverse for A by a density argument. From this point, the proof in this case q < q > 0, interchanging A and A − . (cid:3) Proof of Reduction 2. Let m > 0. By Reduction 1, we only need to dealwith the case q = 0. Choose N large enough so that mN < 1, and for (cid:96) = 0 , . . . , N − j (cid:96) the inclusion of H ( (cid:96) +1) mN Λ , FBI into H (cid:96)mN Λ , FBI . Then if j denotes the inclusionof H m Λ , FBI into H , FBI we have j = j N − ◦ · · · ◦ j . But our assumption, Reduction 1 and [ GGK00 , Theorem IV.2.5] imply that forany k ∈ N and (cid:96) ∈ { , . . . , N − } there is an operator L k,(cid:96) from H ( (cid:96) +1) mN Λ , FBI to H (cid:96)mN Λ , FBI of rank at most k such that the operator norm j (cid:96) − L k,(cid:96) is less than C ( k + 1) − n/Nm (for some C > (cid:0) j N − − L k,N − (cid:1) ◦ · · · ◦ (cid:0) j − L k, (cid:1) (2.115)is less than C N ( k + 1) − n/m . However, expanding we see that the operator (2.115)differs from j by an operator of rank at most (cid:16) N − (cid:17) k . Hence, using [ GGK00 ,Theorem IV.2.5] again, we have µ ( N − ) k ≤ C N ( k + 1) nm . And the result follows since the sequence ( µ k ) k ∈ N is decreasing. (cid:3) We can now start the proof of Proposition 2.13. According to the reductionsabove, we may assume m ∈ ]0 , 1[ and q = 0. Introduce then the unbounded operator A = B Λ (cid:104)| α |(cid:105) m B Λ on H , FBI with domain H m Λ , FBI . By Lemma 2.26, the proof ofProposition 2.13 reduces to the following lemma. Lemma A ) . Recall that m ∈ ]0 , and let N ( r ) denotes the number of eigenvalues less than r of A . Then N ( r ) = r → + ∞ O ( r n/m ) . Proof. First, take an integer N > n +1 > N m > n (this is possiblethanks to our assumption on m ). By Proposition 2.12, we may write A N = B Λ σB Λ + R, where σ is a symbol of order N m with leading term (cid:104)| α |(cid:105) Nm and R is a negligibleoperator. For r ≥ 0, define the symbol a r by a r ( α ) = ( r + (cid:104)| α |(cid:105) Nm ) − . Then, usinga finite number of terms only in Proposition 2.12, we find that A N B Λ a r B Λ = B Λ (cid:104)| α |(cid:105) Nm a r B Λ + B Λ σ r B Λ + R r . We will detail later the properties of σ r and R r . Then we can write( A N + r ) − = B Λ a r B Λ − ( A N + r ) − B Λ σ r B Λ − (cid:16) A N + r (cid:17) − R r . Notice that the a r ’s form a family of symbols of order − N m with uniform estimateson any semi-norm for this symbol class. But if we consider now the a r ’s like symbolsof order − N m + 1 then any semi-norm of a r in this symbol class is dominated by r − /Nm . It is clear that the symbol σ + r of A N + r grows like r . Hence, by taking anexpansion with a large enough number of terms, we may ensure that the trace classnorm of R r acting on H , FBI is dominated by r /Nm − ( R r is a continuous function 44 2. FBI TRANSFORM ON COMPACT MANIFOLDS of a r and σ r ). Now, since the quadratic form associated with A N is positive, astandard argument ensures that the norm of ( A N + r ) − acting on H , FBI is lessthan 1 /r . Hence, the trace class operator norm of ( A N + r ) − R r is dominated by r /Nm .In order to control the term ( A N + r ) − B Λ σ r B Λ , we notice that σ r is a continuousfunction of σ and a r . As above, if we see σ like a symbol of order N m and a r likea symbol of order − N m + 1 then σ r , as a symbol of order 0, does not grow fasterthan some r − /Nm . Hence, as a bounded operator on H , FBI , the Toeplitz B Λ σ r B Λ is dominated by r − /Nm . By a parametrix construction and Lemma 2.25, one seesthat the trace class norm of ( A N + r ) − is uniformly bounded when r tends to + ∞ .Hence, we haveTr (cid:18)(cid:16) A N + r (cid:17) − (cid:19) = r → + ∞ Tr ( B Λ a r B Λ ) + O (cid:16) r − Nm (cid:17) . By Lemma 2.25, since the symbol a r is of order − N m < − n , we retrieve that B Λ a r B Λ is trace class and moreover, its trace is the integral of its kernel on thediagonal. By the usual stationary phase argument, we find that the kernel of B Λ a r B Λ is of the form e i ΦΛ( α,β ) h ( c r ( α, β ) + d r ( α, β )) + R r ( α, β ) . Here, c r ( α, β ) is a symbol of order − N m such that c r ( α, α ) = ( r + (cid:104)| α |(cid:105) Nm ) − . Thesymbol d r is of order − N m − 1, but if we see a r as a symbol of order − N m + 1,then we see that, as a symbol of order − N m , this d r is dominated by r − /Nm . Webound the remainder term R r in the same way. Hence, we find thatTr (cid:18)(cid:16) A N + r (cid:17) − (cid:19) = r → + ∞ (cid:90) Λ r + (cid:104)| α |(cid:105) Nm d α + O (cid:16) r − Nm (cid:17) . (2.116)And since N m < n + 1 we have − Nm < nNm − o of r nNm − . In order to estimate the integral in the right handside of (2.116), we may split it in the integral over {(cid:104)| α |(cid:105) ≤ r /Nm } and the integralover {(cid:104)| α |(cid:105) > r /Nm } . To bound the integral over {(cid:104)| α |(cid:105) ≤ r /Nm } , just bound theintegrand by r − and notice that the volume of this set is dominated by r n/Nm (thiscan be done using the fact that the Jacobian of exp( − H ω I G ) is very close to be 1, seeLemma 2.2, and that exp( − H ω I G ) does not change scale, see Lemma 2.1, where G isas in Definition 2.2). Hence, this first integral is dominated by r nNm − . To boundthe integral over {(cid:104)| α |(cid:105) > r /Nm } , split it into integrals over annulus where (cid:104)| α |(cid:105) isroughly equal to an integer and compare it to a Riemann series to find that it isalso dominated by r n/Nm − . Finally, we findTr (cid:18)(cid:16) A N + r (cid:17) − (cid:19) = r → + ∞ O (cid:16) r nNm − (cid:17) . Then using Lidskii’s Trace Theorem, we have for large λ and r , and some constant C > N ( λ ) λ N + r ≤ (cid:88) k ≥ λ Nk + r = Tr (cid:16) A N + r (cid:17) − ≤ Cr nNm − . Hence N ( λ ) ≤ C ( λ N + r ) r nNm − , and the result follows by taking r = λ N . (cid:3) HAPTER 3 Ruelle–Pollicott resonances and Gevrey Anosovflows This chapter is devoted to the application of the tools that we developed inChapter 2 to the study of the spectral theory of Gevrey Anosov flows. After giving anoverview of our results, we will in § G s Anosov vector field X , constructa I-Lagrangian space adapted to the study of its Ruelle–Pollicott spectrum. In § § Results: Ruelle–Pollicott resonances and dynamical determinant forGevrey Anosov flows. We start by recalling some well-known facts in the C ∞ case. Let M be a compact C ∞ manifold and X be a C ∞ vector field on M that doesnot vanish. We denote by ( φ t ) t ∈ R the flow generated by X and assume that φ t isAnosov. Recall that it means that there is a decomposition T M = E ⊕ E u ⊕ E s of the tangent bundle with the following properties:(i) the decomposition T M = E ⊕ E u ⊕ E s is invariant by d φ t for every t ∈ R ;(ii) E is the span of the vector field X ;(iii) there are constants C, θ > v ∈ E s and t > | d φ t v | ≤ Ce − θt | v | ;(iv) there are constants C, θ > v ∈ E u and t > | d φ − t v | ≤ Ce − θt | v | .Here |·| denotes any Riemannian metric on T M . We also choose a C ∞ weight V : M → C (which is real-valued in most of the applications) and define, for t ≥ Koopman operator L t u = exp (cid:18)(cid:90) t V ◦ φ τ d τ (cid:19) u ◦ φ t . (3.1)Notice that this definition makes sense when u is a smooth function, but also when u is a distribution D (cid:48) ( M ). In fact, when M , φ t and V are G s for some s ≥ L t may be defined as an operator on U s ( M ). Let us introduce thedifferential operator P := X + V , so thatdd t (cid:48) (cid:0) L t (cid:48) u (cid:1) | t (cid:48) = t = P L t u. In the case of C ∞ flows, the anisotropic Banach spaces of distributions – see[ BL07, BL13, FS11 ] – that we mentioned in the introduction always form a scale( H r ) r> of Banach spaces with the following properties:(i) C ∞ ( M ) ⊆ H r ⊆ D (cid:48) ( M ), both inclusions are continuous and the first onehas dense image;(ii) ( L t ) t ≥ is a strongly continuous semi-group on H r and its generator is P ; (iii) the intersection of the spectrum of P acting on H r with the half-plane { z ∈ C : Re z > − r } contains only isolated eigenvalues of finite multiplici-ties.The property (i) is just a non-triviality assumption: it implies that the elements of H r are objects that live on the manifold M . This property can easily be softenedwithout any harm in the theory. For instance, working with a G s flow, one couldreplaced C ∞ ( M ) by G s ( M ) and D (cid:48) ( M ) by U s ( M ). The point (ii) is of utmostimportance: it is the property that ensures that the spectrum of P on H r has adynamical interpretation and allows to describe the asymptotic of L t when t tendsto + ∞ .If the spaces H r are highly non-canonical, the spectrum of P acting on H r isintrinsically defined by V and the vector field X . This may be shown using thefollowing argument from [ FS11 ] (we will refer to this argument in the proof ofTheorem 10): when Re z (cid:29) 1, the resolvent ( z − P ) − of P , as an operator on L ( M ) for example, is the Laplace transform of L t , that is R ( z ) : u (cid:55)→ (cid:90) + ∞ e − zt L t u d t. (3.2)Now, the point (ii) above implies that, when Re z (cid:29) 1, the resolvent of P acting on H r is also given by (3.2). Since H r is intermediate between C ∞ ( M ) and D (cid:48) ( M ),the point (iii) implies that the family of operator R ( z ) : C ∞ ( M ) → D (cid:48) ( M ) admitsa meromorphic continuation to { z ∈ C : Re z > − r } with residues of finite rank. Bythe analytic continuation principle, this meromorphic continuation does not dependon the choice of the space H r . Moreover, letting r tends to + ∞ , we see that R ( z )has a meromorphic continuation to the whole complex plane.As in the case without potential, the poles of the complex continuation of R ( z )are called the Ruelle–Pollicott resonances of P . Moreover, if λ ∈ C is a Ruelle–Pollicott resonance, then the residue of R ( z ) is (up to some injections) a spectralprojector π λ of finite rank for P . The rank of π λ is the multiplicity of λ as a Ruelleresonance and the elements of the image of π λ are called resonant states associatedwith λ for P . Notice that the resonant states are not necessarily eigenvectors for P (there may be Jordan blocks as shown in [ CP19 ]).We can introduce a generalization of ζ X ( z ) associated to P : for Re z (cid:29) 1, welet ζ X,V ( z ) = exp (cid:32) − (cid:88) γ T γ T γ e (cid:82) γ V e − zT γ (cid:12)(cid:12) det(1 − P γ ) (cid:12)(cid:12) (cid:33) , (3.3)where the sum γ runs over the periodic orbits of φ t . If γ is a periodic orbit then T γ denotes its length, T γ its primitive length (i.e. the length of the smallest periodicorbits with the same image), the integral (cid:82) γ V is defined by (cid:82) γ V = (cid:82) T γ V ( φ t ( x )) d t for any x in the image of γ , and P γ is the linearized Poincar´e map associated with γ , i.e. P γ = d φ T γ ( x ) | Eu ( x ) ⊕ Es ( x ) . The map P γ depends on the point x in the imageof γ , but its conjugacy class is well-defined. It follows from elementary estimateson the number of periodic orbits for φ t (see [ DZ16 , Lemma 2.2] for instance) that ζ X,V ( z ) is well-defined for Re z (cid:29) 1. It is proven in [ GLP13 ] (see also [ DZ16 ] foran alternative proof) that ζ X,V extends to a holomorphic function on C whose zerosare the Ruelle resonances (counted with multiplicity).Let us now state our main result. It extends Theorem 1 to the case withpotentials. Theorem . Let s ∈ [1 , + ∞ [ . Let M be a n -dimensional G s compact manifold.Let X be a G s vector field that generates an Anosov flow ( φ t ) t ∈ R . Let V : M → C . RUELLE–POLLICOTT RESONANCES AND GEVREY ANOSOV FLOWS 147 be a G s potential. Then there is a constant C > such that for every z ∈ C we have (cid:12)(cid:12) ζ X,V ( z ) (cid:12)(cid:12) ≤ C exp ( C | z | ns ) . In particular, the order of ζ X,V is less than ns . As a by-product of the proof of Theorem 8, we deduce the following bound onthe number of Ruelle resonances. Proposition . Under the assumptions of Theorem 8 and if N ( r ) denotesthe number of Ruelle resonances of P of modulus less than r , then we have N ( r ) = r → + ∞ O ( r ns ) . Remark . Proposition 3.1 is an immediate consequence of Theorem 8 andJensen’s formula [ Boa54 , Theorem 1.2.1]. However, we will need to prove Proposi-tion 3.1 before Theorem 8. Notice also that Theorem 8 and Hadamard’s FactorizationTheorem (see [ Boa54 , § Remark . It is well-known that the finiteness of the order of the dynam-ical determinant implies that a trace formula associated to P holds (see [ J´ez19 ,Proposition 1.5] for a precise statement). In particular, we prove here that the traceformula associated with P holds when the data are Gevrey. However, this is nota new result since [ J´ez19 , Corollary 1.8] states that the trace formula holds for alarger class of regularity than Gevrey.We will also give a statement about the G s wave front sets (in the sense of § T ∗ M = E ∗ ⊕ E ∗ s ⊕ E ∗ u of the cotangent bundle of M . We define the space E ∗ as theannihilator of E u ⊕ E s , the space E ∗ s as the annihilator of E ⊕ E s and the space E ∗ u as the annihilator of E ⊕ E u . Then we have the following statement: Proposition . Under the assumption of Theorem 8, if f is a resonant statefor P = X + V then the Gevrey wave front set WF G s ( f ) (see Definition 2.3) of f iscontained in the stable codirection E ∗ s . Remark . Proposition 3.2 may seem surprising to readers familiar withpapers on Ruelle resonances from the microlocal community (for instance [ FS11,DZ16, DZ17 ]). Indeed, in these papers resonant states have their wave front setsin E ∗ u . The difference is due to the fact that we study resonant states associated withthe Koopman operator (the operator (3.1) for t ≥ 0) while [ FS11, DZ16, DZ17 ]study resonant states associated with the transfer operator (the operator (3.1) for t ≤ FRS08 ], while studying the Koopman operator, resonant states havetheir wave front set contained in E ∗ u . This is just because [ FRS08 ] uses a differentconvention for the definition of E ∗ u and E ∗ s than [ FS11, DZ16, DZ17 ]. Finally,it is well-known in the dynamical community that resonant states are “smooth inthe stable direction” (if we study the Koopman operator). This is precisely whathappens here: in practical terms, having its wave front set in E ∗ s means “beingsmooth in the stable direction”. This may seem strange and is due to the conventionfor the definition of E ∗ u and E ∗ s that we borrowed from [ FS11 ] (we have in particularthat dim E ∗ u = dim E s ).While it may seem obvious to specialists, let us notice that our results extend tothe case of vector bundles. Let F → M be a complex G s vector bundle. Assume that φ t lifts to a G s one parameter subgroup of vector bundle automorphisms ( L t ) t ∈ R 48 3. RUELLE–POLLICOTT RESONANCES AND GEVREY ANOSOV FLOWS of F . In this context, we may define the Koopman operator L t for t ∈ R and u asmooth section of F by L t u ( x ) = L − t ( u ◦ φ t ( x )) (this is the natural analogue of(3.1) in the scalar case). Then, we may replace in the results above the operator P = X + V by the operator P define on smooth sections of F by P u = dd t ( L t u ) | t =0 . To highlight the similarity with the scalar case, we may choose a G s connection ∇ on F . Then the operator P writes P = ∇ X + A where A is a G s section of the bundleof automorphisms of F . We see here that the principal symbol of P is scalar, hencethe eventual functional analytic complications that could arise when introducingvector bundles are restricted to the sub-principal level and are consequently dealtwith easily.Concerning the algebra, the traces and determinants computations in the bundlecase are carried out using the bundle version of Guillemin’s Trace Formula [ Gui77 ].Hence, in this context the dynamical determinant writes (for Re z (cid:29) ζ F ( z ) = exp (cid:32) − (cid:88) γ T γ T γ Tr ( L γ ) e − zT γ (cid:12)(cid:12) det (cid:0) I − P γ (cid:1)(cid:12)(cid:12) (cid:33) . Here, if γ is a periodic orbit of φ t and x a point of the image of γ , the endomorphism L γ is the restriction of L T γ to F x (the conjugacy class of L γ is well-defined).Using a trick due to Ruelle to write zeta functions as alternate products ofdynamical determinants (as in [ GLP13, DZ16 ]), the bundle version of Theorem 1implies that the Ruelle zeta function , i.e. ζ R ( z ) := exp (cid:32) − (cid:88) γ T γ T γ e − zT γ (cid:33) , associated with a G s Anosov flow, has order less than ns .Returning back to the scalar case, we will also use the tools that we developedin order to study stochastic and deterministic perturbations of Gevrey Anosov flows,respectively in § § § P = X + V of the form P (cid:15) = P + (cid:15) ∆ , where (cid:15) ≥ m > G s pseudor. The operator P (cid:15) acting on L (with its natural domain)has its real part Re V + (cid:15) ∆ bounded from above, hence its resolvent set is non-empty(and it generates a strongly continuous semi-group). When (cid:15) > 0, the operator P (cid:15) is elliptic, so that its resolvent is compact and P (cid:15) has discrete spectrum σ L ( P (cid:15) )on L ( M ). We know from [ DZ15 , Theorem 1] that σ L ( P (cid:15) ) converges locally tothe Ruelle spectrum of P . We will prove a global version of this result. To do so,we need to introduce a new distance to compare spectrum. If z ∈ C , we define thedistance d z on C ∪ {∞} \ z by d z ( x, y ) = (cid:12)(cid:12)(cid:12)(cid:12) z − x − z − y (cid:12)(cid:12)(cid:12)(cid:12) . We will prove in § DZ15 , Theorem 1]. Theorem . Under the assumption of Theorem 8, if h is small enough, thenfor every p > ns and z ∈ R + large enough, there is a constant C > such that forevery (cid:15) > small enough, we have d z,H (cid:0) σ Ruelle ( P ) ∪ {∞} , σ L ( P (cid:15) ) ∪ {∞} (cid:1) ≤ C | ln (cid:15) | − p . .1. I-LAGRANGIAN SPACES ADAPTED TO A GEVREY ANOSOV FLOW 149 Here, d z,H denotes the Hausdorff distance associated to the distance d z and σ Ruelle ( P ) the Ruelle spectrum of P . Remark . Theorem 9 also applies if ∆ is a classical differential operatorwith G s coefficients (for instance, the Laplacian associated to a G s metric). Indeed,one only needs to replace ∆ by h m ∆ and (cid:15) by h − m (cid:15) in Theorem 9.In the proof of Theorem 9, we will use ideas similar to those of [ GZ19a ], withthe necessary modifications in our different context (in particular, we need to dealwith the flow direction and the fact that we are dealing with operators of order 1).The convergence in Theorem 9 seems to be very weak. However, we think that itis not reasonable to expect too fast a convergence in such a global result. Indeed,when we add the pseudo-differential operator ∆ to P in order to form P (cid:15) , since∆ has higher order, we can expect that the spectrum of P (cid:15) looks globally like thespectrum of ∆, rather than like the Ruelle spectrum of P . Indeed, the higher orderoperator will be predominant at higher frequencies. Furthermore, the spectrum of∆ is contained in R − while we expect some kind of vertical structure for the Ruellespectrum of P (see for instance [ JZ17, FT13 ]). Hence, we may expect σ L ( P (cid:15) )to be some kind of “flattened” version of the Ruelle spectrum of P , and its globalstructure is thus very different from the actual Ruelle spectrum of P .Finally, § BL07, BL13 ], and we will focus on the particularity of our highlyregular context. The main results of § § From now on, s ∈ [1 , + ∞ [ is fixed, X is a G s vector field on a n -dimensional G s manifold M that generates an Anosov flow ( φ t ) t ∈ R , and V : M → C is a G s function.We define the differential operator P = X + V , and the associated Koopman operatoris given by (3.1). Without loss of generality, we may assume that M is endowed witha structure of real-analytic Riemannian manifold (coherent with its G s structure,see Remark 1.1).The machinery from § (cid:102) M a complex neighbourhood for M . According to Theorem 6, there is an analyticFBI transform T on M such that T ∗ T = I . As above we set S = T ∗ . In order toapply the results from the previous chapter to the operator P , we need first to finda suitable ( τ , s )-adapted Lagrangian Λ. The Lagrangian Λ will be defined by (2.4)where the symbol G is defined by G = τ G where τ (cid:28) h − /s and G is a so-called escape function . Section § G (see Lemma3.1). In section 3.1.2, we will then describe the spectral theory of P on the relatedI-Lagrangian space. Recall that the decomposition T M = E ⊕ E u ⊕ E s of the tangent bundle induces a dual decomposition T ∗ M = E ∗ ⊕ E ∗ u ⊕ E ∗ s of the cotangent bundle. Here, E ∗ = ( E u ⊕ E s ) ⊥ , the space E ∗ u =( E ⊕ E u ) ⊥ and E ∗ s = ( E ⊕ E s ) ⊥ . We denote by p : T ∗ M → C the principal symbolof the semi-classical differential operator hP . We recall for α = ( α x , α ξ ) ∈ T ∗ M , p ( α ) = iα ξ ( X ( α x )) . (3.4) 50 3. RUELLE–POLLICOTT RESONANCES AND GEVREY ANOSOV FLOWS To apply the machinery presented in the previous part, we will need an almostanalytic extension for p . We construct it in the following way: we take a G s almostanalytic extension (cid:101) X for X , given by Lemma 1.1 if s > (cid:101) X = X if s = 1), and then we set for α ∈ (cid:0) T ∗ M (cid:1) (cid:15) (for some small (cid:15) > p ( α ) = iα ξ (cid:16) (cid:101) X ( α x ) (cid:17) . (3.5)It will be important when constructing the escape function G that this almostanalytic extension is linear in α ξ . We are now ready to construct G . Lemma . Let C and C s be conical neighbourhoods respectively of E ∗ and of E ∗ s in T ∗ M . Let δ ≥ . Then there are arbitrarily small (cid:15) > (cid:15) > and a symbol G of order δ on ( T ∗ M ) (cid:15) , supported in ( T ∗ M ) (cid:15) with the following properties:(i) the restriction of G to T ∗ M is negative and classically elliptic of order δ outside of C s ;(ii) the restriction of { G , Re ˜ p } to T ∗ M is negative and classically elliptic oforder δ outside of C ;(iii) if s = 1 , there are C, (cid:15) > such that { G , Re ˜ p } ≤ C on ( T ∗ M ) (cid:15) . If s > , there is (cid:15) such that for every N > there is C N > such that forall α ∈ ( T ∗ M ) (cid:15) we have { G , Re ˜ p } ( α ) ≤ C N (1 + | Im α x | N (cid:104)| α |(cid:105) δ ) .Here, the Poisson Bracket { G , Re ˜ p } is the one associated with the real symplecticform ω I on ( T ∗ M ) (cid:15) . Before proving Lemma 3.1, let us explain why we need our escape function tosatisfy these properties. We recall that a symbol is said to be “classically ellipticof order δ ” if it is greater than (cid:104)| α |(cid:105) δ when α is large enough. The point (i) ishere to control the G s wave front set of elements of the set H , in particular ofresonant states, using Lemma 2.14. The point (ii) will be used in the proof ofLemma 3.3 to deduce the hypoellipticity of the operator P acting on H from themultiplication formula, Proposition 2.11. Finally, the point (iii) will be used inthe proof of Proposition 3.3 to show that the Koopman operator (3.1) defines acontinuous semi-group on H . This property is what ensures that the spectrumof P on our spaces has a dynamical meaning. Even though point (ii) seems tobe the most crucial one, since it is the one that allows us to enter the world ofSchatten operators and eventually prove Theorem 8, the importance of (iii) couldnot be overestimated. Finally, notice that, in order to apply Lemma 2.10 in themost favorable case for us, it will be natural in the following to choose δ = 1 /s .The proof of Lemma 3.1 will take the rest of this section. Proof of Lemma 3.1. We want to understand { G , Re ˜ p } in order to controlhow the real part of ˜ p evolves under the flow of H ω I G . However, since { G , Re ˜ p } = − { Re ˜ p, G } , we may understand { G , Re ˜ p } by controlling how G evolves underthe flow of − H ω I Re ˜ p . Hence, we need to understand the dynamics of this flow. To doso, we may multiply (cid:101) X by a bump function identically equals to 1 near M (sincewe only claim properties for G near T ∗ M and we will not use the high regularityof (cid:101) X in this proof). Then, it follows from the formula (2.3) that the flow of − H ω I Re ˜ p is complete. We denote this flow by (Θ t ) t ∈ R and writeΘ t ( α ) = (cid:0) Θ t,x ( α ) , Θ t,ξ ( α ) (cid:1) . .1. I-LAGRANGIAN SPACES ADAPTED TO A GEVREY ANOSOV FLOW 151 Using (2.3), we see that H ω I Re ˜ p is given in coordinates ( x + iy, ξ + iη ) with (cid:101) X =( (cid:101) X , . . . , (cid:102) X n ) by − H ω I Re ˜ p = n (cid:88) j =1 Re (cid:102) X j ∂∂x j + Im (cid:102) X j ∂∂y j − (cid:32)(cid:42) ξ, ∂ Im (cid:101) X∂y j (cid:43) + (cid:42) η, ∂ Re (cid:101) X∂y j (cid:43)(cid:33) ∂∂ξ j − (cid:32)(cid:42) ξ, ∂ Im (cid:101) X∂x j (cid:43) + (cid:42) η, ∂ Re (cid:101) X∂x j (cid:43)(cid:33) ∂∂η j . (3.6)Indeed, it follows from (3.4) that in such coordinates we haveRe ˜ p = − (cid:68) ξ, Im (cid:101) X (cid:69) − (cid:68) η, Re (cid:101) X (cid:69) . From (3.6), we see that the projection Θ t,x of Θ t is in fact given by the formulaΘ t,x ( α ) = ˜ φ t ( α x ) , where ( ˜ φ t ) t ∈ R denotes the flow of (cid:101) X (in particular, the restriction of ˜ φ t to M is φ t ).Then, we notice that in (3.6), the component of − H ω I Re ˜ p along ∂/∂ξ and ∂/∂η islinear in ( ξ, η ). It implies thatΘ t,ξ ( α ) = L t ( α x )( α ξ ) , where L t ( α x ) is a R -linear application from T ∗ α x (cid:102) M to T ∗ ˜ φ t ( α x ) (cid:102) M (that dependssmoothly on t and α x ). Now, since (cid:101) X satisfies the Cauchy–Riemann equations andis tangent to M on M , we find that, for y = 0, in the same system of coordinatesthan (3.6), we have ∂ Re (cid:101) X∂y = − ∂ Im (cid:101) X∂x = 0 and ∂ Im (cid:101) X∂y = ∂ Re (cid:101) X∂x = ∂X∂x . (3.7)By uniqueness in the Cauchy–Lipschitz Theorem, we find by plugging (3.7) in (3.6)that, for x ∈ M and t ∈ R we have L t ( x ) = T (cid:16) d φ t ( x ) − (cid:17) . (3.8)Hence, the hyperbolicity of φ t will have important consequences on the dynamics ofΘ t . Let us “complexify” the bundles E ∗ ,u,s . For x ∈ M , we denote by E C , ∗ , E C , ∗ u and E C , ∗ s the complexification of E ∗ , E ∗ u and E ∗ s , considering linear forms valued in C instead of R . For instance, for x ∈ M , we write E C , ∗ ,x for the subspace of T ∗ x M ⊗ C consisting of R -linear maps from T x M to C that vanish on E u ⊕ E s (or, under anatural identification, of C -linear forms on T ∗ x (cid:102) M that vanish on E u ⊕ E s ). From thefact that T x M = E ,x ⊕ E u,x ⊕ E s,x is a totally real subspace of maximal dimensionof T x (cid:102) M , we deduce that T ∗ x (cid:102) M = E C , ∗ ,x ⊕ E C , ∗ u,x ⊕ E C , ∗ s,x . Since E ,x , E u,x and E s,x depends in a H¨older-continuous fashion on x ∈ M , so does E C , ∗ ,x , E C , ∗ u,x and E C , ∗ s,x .Consequently, we may extend continuously E C , ∗ , E C , ∗ u and E C , ∗ s to (cid:102) M . Then, if (cid:102) M is small enough, we have T x (cid:102) M = E C , ∗ ,x ⊕ E C , ∗ u,x ⊕ E C , ∗ s,x for all x ∈ (cid:102) M . A priori , thisdecomposition is only invariant under L t for t ∈ R and x ∈ M . If σ ∈ T ∗ x (cid:102) M then wewrite σ = σ + σ u + σ s for the decomposition of σ under T x (cid:102) M = E C , ∗ ,x ⊕ E C , ∗ u,x ⊕ E C , ∗ s,x .Then, we put a real Riemannian metric on (cid:102) M and define for x ∈ (cid:102) M and γ > C γu ( x ) = (cid:110) σ ∈ T ∗ x (cid:102) M : | σ | + | σ s | ≤ γ | σ u | (cid:111) and C γs ( x ) = (cid:110) σ ∈ T ∗ x (cid:102) M : | σ | + | σ u | ≤ γ | σ s | (cid:111) . 52 3. RUELLE–POLLICOTT RESONANCES AND GEVREY ANOSOV FLOWS Without loss of generality, we may assume that C is closed and does not intersect E ∗ u ⊕ E ∗ s \ { } . We may also assume that there is a closed conic neighbourhood C C , of E C , ∗ in T ∗ (cid:102) M such that C C , ∩ T ∗ M = C . Choose then a small closed conicneighbourhood C s of E ∗ s ⊕ E ∗ in T ∗ M .From (3.8) and standard arguments in hyperbolic dynamics, there are a large T > 0, small 0 < γ < γ , some λ > C > α = ( α x , α ξ ) ∈ T ∗ M ⊗ C , and T ≥ T then:a. either Θ T ,ξ ( α ) ∈ C γ u ( ˜ φ T ( α x )) or Θ − T ,ξ ( α ) ∈ C γ s ( ˜ φ − T ( α x )) or we have α ∈ C C , ;b. if Θ − T ,ξ ( α ) ∈ C γu ( ˜ φ − T ( α x )) then Θ T ,ξ ( α ) ∈ C γ u ( ˜ φ T ( α x )) and we have (cid:12)(cid:12) Θ T ,ξ ( α ) (cid:12)(cid:12) ≥ λ (cid:12)(cid:12) Θ − T ,ξ ( α ) (cid:12)(cid:12) ;c. if Θ T ,ξ ( α ) ∈ C γs ( ˜ φ T ( α x )) then Θ − T ,ξ ( α ) ∈ C γ s ( ˜ φ − T ( α x )) and we have (cid:12)(cid:12) Θ T ,ξ ( α ) (cid:12)(cid:12) ≤ λ − (cid:12)(cid:12) Θ − T ,ξ ( α ) (cid:12)(cid:12) ;d. if α ∈ T ∗ M does not belong to C s , then, for t ≥ 0, we have thatΘ t,ξ ( α ) / ∈ C γs ( ˜ φ t ( α x )), and, for t ≥ T , we have Θ t,ξ ( α ) ∈ C γ u ( ˜ φ t ( α x )) and (cid:12)(cid:12) Θ t,ξ ( α ) (cid:12)(cid:12) ≥ C − (cid:12)(cid:12) α ξ (cid:12)(cid:12) .Since we ask here for α ∈ T ∗ M ⊗ C , these are consequences of the hyperbolicity of φ t , that is it only relies on the dynamic on M . We want to apply a perturbationargument to show that b and c remain true on a small complex neighbourhood of M , but we need first to fix the value of T . Hence, we fix the value of T , largeenough such that we havesup α ∈ T ∗ M \{ } (cid:12)(cid:12) α ξ (cid:12)(cid:12) δ (cid:90) − T (cid:12)(cid:12) Θ t,ξ ( α ) (cid:12)(cid:12) δ d t < T − T C δ . (3.9)Now that T is fixed, it follows from a perturbation argument that, up totaking a smaller λ , a smaller γ , a larger γ and a larger C , the properties b andc above remain true when ( x, σ ) ∈ T ∗ ( M ) (cid:15) , for some small (cid:15) > m ∈ S KN ( T ∗ ( M ) (cid:15) ) of order 0 on T ∗ ( M ) (cid:15) , valued in [ − , • if x ∈ ( M ) (cid:15) and σ ∈ T ∗ x (cid:102) M \ ( C γu ( x ) ∪ C γs ( x )) or σ is near 0 then m ( x, σ ) =0; • there is C > x ∈ ( M ) (cid:15) and σ ∈ C γ s ( x ) satisfies | σ | ≥ C then m ( x, σ ) = 1; • there is C > x ∈ ( M ) (cid:15) and σ ∈ C γ u ( x ) satisfies | σ | ≥ C then m ( x, σ ) = − • m is non-positive on C γu and non-negative on C γs .We also choose a C ∞ function χ : R → [0 , 1] vanishing on ] −∞ , 1] and taking value1 on [2 , + ∞ [. Then, we may define G near T ∗ M ⊗ C by the formula G ( α ) = (cid:90) T T m (Θ t ( α )) (cid:12)(cid:12) Θ t,ξ ( α ) (cid:12)(cid:12) δ d t − Aχ ( (cid:12)(cid:12) α ξ (cid:12)(cid:12) ) (cid:12)(cid:12)(cid:12) Re (cid:16) α ξ (cid:16) (cid:101) X ( α x ) (cid:17)(cid:17)(cid:12)(cid:12)(cid:12) δ , (3.10)where A is a large constant to be chosen later. Then we multiply G by a bumpfunction to satisfy the claim on the support. It does not change the value of G near T ∗ M and hence it will not interfere with the properties (i),(ii) and (iii). Wemay consequently use the formula (3.10) to prove (i),(ii) and (iii). .1. I-LAGRANGIAN SPACES ADAPTED TO A GEVREY ANOSOV FLOW 153 We start by proving (i). Take α ∈ T ∗ M \ C s and write G ( α ) = (cid:90) T m (Θ t ( α )) (cid:12)(cid:12) Θ t,ξ ( α ) (cid:12)(cid:12) δ d t + (cid:90) T m (Θ t ( α )) (cid:12)(cid:12) Θ t,ξ ( α ) (cid:12)(cid:12) δ d t − Aχ ( (cid:12)(cid:12) α ξ (cid:12)(cid:12) ) (cid:12)(cid:12)(cid:12) α ξ (cid:16) (cid:101) X ( α x ) (cid:17)(cid:12)(cid:12)(cid:12) δ . (3.11)The first two terms in (3.11) are symbols of order δ that do not depend on A . Thelast term is elliptic of order δ on a conical neighbourhood of C (since we assumedthat C is closed and does not intersect E ∗ u ⊕ E ∗ s \ { } ). Hence, if A is large enough, G is classically elliptic and negative on a conical neighbourhood of C in T ∗ M .In fact, we see that this conical neighbourhood may be chosen arbitrarily large assoon as it does not intersect E ∗ u ⊕ E ∗ s \ { } . Then, if α does not belong to thisneighbourhood of C , since it does not belong to C s either, we see that α / ∈ C s (provided that C s has been chosen narrow enough), and hence property d. abovegives that, provided α ξ is large enough, (cid:90) T m (Θ t ( α )) (cid:12)(cid:12) Θ t,ξ ( α ) (cid:12)(cid:12) δ d t ≤ (cid:90) T T m (Θ t ( α )) (cid:12)(cid:12) Θ t,ξ ( α ) (cid:12)(cid:12) δ d t ≤ − (cid:90) T T (cid:12)(cid:12) Θ t,ξ ( α ) (cid:12)(cid:12) δ d t ≤ − T − T C δ (cid:12)(cid:12) α ξ (cid:12)(cid:12) δ . Hence our choice of T (see (3.9)) ensures that G is negative and elliptic of order δ outside of C s (the last term in (3.11) is always non-positive). We just proved (i).Now, we prove (ii). To do so we compute { G , Re ˜ p } = − { Re ˜ p, G } = − H ω I Re ˜ p G = m (Θ T ( α )) (cid:12)(cid:12) Θ T ,ξ ( α ) (cid:12)(cid:12) δ − m (Θ − T ( α )) (cid:12)(cid:12) Θ − T ,ξ ( α ) (cid:12)(cid:12) δ + AH ω I Re ˜ p (cid:18) χ ( (cid:12)(cid:12) α ξ (cid:12)(cid:12) ) (cid:12)(cid:12)(cid:12) Re (cid:16) α ξ (cid:16) (cid:101) X ( α x ) (cid:17)(cid:17)(cid:12)(cid:12)(cid:12) δ (cid:19) . (3.12)We start by considering the term m (Θ T ( α )) (cid:12)(cid:12) Θ T ,ξ ( α ) (cid:12)(cid:12) δ − m (Θ − T ( α )) (cid:12)(cid:12) Θ − T ,ξ ( α ) (cid:12)(cid:12) δ . (3.13)Assuming that α does not belong to C , we know that Θ T ,ξ ( α ) belongs to C γ u ( ˜ φ T ( α x )) or Θ − T ,ξ ( α ) belongs to C γ s ( ˜ φ − T ( α x )). Let us assume for instancethat Θ T ,ξ ( α ) belongs to C γ u ( ˜ φ T ( α x )) (the other case is symmetric). Then againthere are two possibilities: either Θ − T ,ξ ( α ) belongs to C γu ( ˜ φ − T ( α x )) or it does not.If it does then (for (cid:12)(cid:12) α ξ (cid:12)(cid:12) large enough) m (Θ T ( α )) (cid:12)(cid:12) Θ T ,ξ ( α ) (cid:12)(cid:12) δ − m (Θ − T ( α )) (cid:12)(cid:12) Θ − T ,ξ ( α ) (cid:12)(cid:12) δ ≤ − (cid:12)(cid:12) Θ T ,ξ ( α ) (cid:12)(cid:12) δ + (cid:12)(cid:12) Θ − T ,ξ ( α ) (cid:12)(cid:12) δ ≤ − (cid:16) λ δ − (cid:17) (cid:12)(cid:12) Θ − T ,ξ ( α ) (cid:12)(cid:12) δ ≤ − C − (cid:12)(cid:12) α ξ (cid:12)(cid:12) δ , for some C > 0. If Θ − T ,ξ ( α ) does not belong to C γu , then the situation is evensimpler since the term − m (Θ − T ,ξ ( α )) (cid:12)(cid:12) Θ − T ,ξ ( α ) (cid:12)(cid:12) δ is non-positive. Hence, the term(3.13) is negative and elliptic of order δ outside of C . We focus now on the other 54 3. RUELLE–POLLICOTT RESONANCES AND GEVREY ANOSOV FLOWS term in (3.12). To do so, we introduce the Hamiltonian vector field H ω R Im ˜ p of Im ˜ p with respect to the real symplectic form ω R and write: H ω I Re ˜ p (cid:18) χ ( (cid:12)(cid:12) α ξ (cid:12)(cid:12) ) (cid:12)(cid:12)(cid:12) Re (cid:16) α ξ (cid:16) (cid:101) X ( α x ) (cid:17)(cid:17)(cid:12)(cid:12)(cid:12) δ (cid:19) = H ω I Re ˜ p (cid:0) χ (cid:0)(cid:12)(cid:12) α ξ (cid:12)(cid:12)(cid:1)(cid:1) (cid:12)(cid:12)(cid:12) Re (cid:16) α ξ (cid:16) (cid:101) X ( α x ) (cid:17)(cid:17)(cid:12)(cid:12)(cid:12) δ − χ (cid:0)(cid:12)(cid:12) α ξ (cid:12)(cid:12)(cid:1) H ω R Im ˜ p (cid:18)(cid:12)(cid:12)(cid:12) Re (cid:16) α ξ (cid:16) (cid:101) X ( α x ) (cid:17)(cid:17)(cid:12)(cid:12)(cid:12) δ (cid:19) + (cid:16) H ω R Im ˜ p + H ω I Re ˜ p (cid:17) (cid:18)(cid:12)(cid:12)(cid:12) Re (cid:16) α ξ (cid:16) (cid:101) X ( α x ) (cid:17)(cid:17)(cid:12)(cid:12)(cid:12) δ (cid:19) . (3.14)The first term in the right hand side of (3.14) is compactly supported and mayconsequently be ignored. Since Re( α ξ ( (cid:101) X ( α x ))) = Im ˜ p ( α ), the second term is equalto 0. To estimate the last term in (3.14), we work in local coordinates ( x + iy, ξ + iη ).In these coordinates, we may compute H ω R Im ˜ p as we did for H ω I Re ˜ p in (3.6), and wefind that H ω R Im ˜ p + H ω I Re ˜ p = n (cid:88) j =1 (cid:32)(cid:42) ξ, ∂ Im (cid:101) X∂y j − ∂ Re (cid:101) X∂x j (cid:43) + (cid:42) η, ∂ Re (cid:101) X∂y j + ∂ Im (cid:101) X∂x j (cid:43)(cid:33) ∂∂ξ j + (cid:32)(cid:42) ξ, ∂ Re (cid:101) X∂y j + ∂ Im (cid:101) X∂x j (cid:43) + (cid:42) η, ∂ Re (cid:101) X∂x j − ∂ Im (cid:101) X∂y j (cid:43)(cid:33) ∂∂η j . (3.15)Since (cid:101) X satisfies the Cauchy–Riemann equation on M , we find that the vector field H ω R Im ˜ p + H ω I Re ˜ p vanishes on T ∗ M (even on T ∗ M ⊗ C in fact). It follows that outsideof a compact set, the only non-zero term in the right hand side of (3.12) is (3.13).The property (ii) follows.It remains to prove (iii). The analysis is based on (3.12) again. We start bystudying the term (3.13). Let α ∈ (cid:0) T ∗ M (cid:1) (cid:15) . If Θ T ,ξ ( α ) ∈ C γ u or Θ − T ,ξ ( α ) ∈ C γ s ,then the analysis from the proof of (ii) applies, and we see that (3.13) is non-positivefor α ξ large enough. Otherwise, Θ − T ,ξ ( α ) / ∈ C γu and Θ T ,ξ ( α ) / ∈ C γs and both termsin (3.13) are non-positive. Thus, the term (3.13) is always non-positive when (cid:12)(cid:12) α ξ (cid:12)(cid:12) is large enough. Hence, this term is bounded from above and we may focus onthe other term in (3.12), which is given by(3.14). The two first terms in the righthand side of (3.14) are dealt with as in the proof of (ii): the first is compactlysupported and the second is identically equal to 0. To deal with the last one, wenotice that, since (cid:101) X satisfies the Cauchy–Riemann equation at infinite order on M , the expression (3.7) for H ω R Im ˜ p + H ω I Re ˜ p in coordinates imply that the last termin (3.14) is a O ( | Im α x | ∞ (cid:104) α (cid:105) δ ). This settles the case s > 1. To deal with thecase s = 1, just notice that in that case (cid:101) X is holomorphic on (cid:102) M and consequently H ω R Im ˜ p + H ω I Re ˜ p = 0. (cid:3) Now, that we are equipped with a good escape function, we are in position toapply the tools from the previous chapter to study the spectral theory of P = X + V . With the notationsof the previous section, we set δ = 1 /s and let G be an escape function given byLemma 3.1 (for arbitrary C and C u , we only assume that C is closed an doesnot intersect E ∗ u ⊕ E ∗ s \ { } ). Then we define G = τ G and Λ = e H ωIG T ∗ M with τ = cτ h − /s , where c and τ are small. Notice that if c is small enough, then Λ isa ( τ , s )-adapted Lagrangian (in the sense of Definition 2.2), and hence the resultsfrom the previous chapter apply to the G s semi-classical pseudor hP . We will not .1. I-LAGRANGIAN SPACES ADAPTED TO A GEVREY ANOSOV FLOW 155 consider the asymptotic h → 0, and hence we will assume that h is fixed, smallenough so that the results from the previous chapter apply. We will also assumethat τ is small enough for the same reason.We want now to study the spectral theory of the operator P = X + V on thespace H defined by (2.15). Notice that, according to Proposition 2.4, if h and τ are small enough, then the operator hP (and hence P ) is bounded from H k Λ to H k − for every k ∈ R .We will start by proving: Lemma . P defines a closed operator on H with domain D ( P ) = (cid:110) u ∈ H : P u ∈ H (cid:111) . In order to ensure that the spectral theory of P on H has a dynamical meaning,we will then prove the following lemma. Proposition . The operator P is the generator of a strongly continuoussemi-group ( L t ) t ∈ R on H . Moreover, if t ≥ and u ∈ H ∩ L ( M ) then L t u isgiven by the expression (3.1) . We will then prove the following key lemma that will be used with Proposition2.13 to prove that the resolvent of P is in a Schatten class. Lemma P ) . There is a constant C > such that forevery u ∈ D ( P ) we have u ∈ H δ Λ and (cid:107) u (cid:107) H δ Λ ≤ C (cid:16) (cid:107) u (cid:107) H + (cid:107) P u (cid:107) H (cid:17) , where we recall that we set δ = 1 /s . Notice that in the case s = 1, in which we have δ = 1, the operator P is in factelliptic on H . With Proposition 2.13, we deduce then from Lemma 3.3 that P hasa good spectral theory on H . More precisely, we have: Theorem . If z is any element in the resolvent set of P , then the resolvent ( z − P ) − : H → H is compact and if ( σ k ) k ≥ denotes the sequence of its singularvalues, we have σ k = k → + ∞ O (cid:16) k − ns (cid:17) . In particular, the operator ( z − P ) − is in the Schatten class S p for any p > ns .Consequently, P has discrete spectrum on H , and this spectrum is the Ruellespectrum of P . The eigenvectors of P acting on H are also the resonant states for P . If N ( R ) denotes the number of Ruelle resonances of modulus less than R , wehave N ( R ) = R → + ∞ O ( R ns ) . Notice that Theorem 10 implies Proposition 3.1. We will see that it also impliesProposition 3.2. Theorem 8 will be proved in the following section as a corollary ofTheorem 10. The key tool in the proof of these results will be the multiplicationformula Proposition 2.11. We start by proving Lemma 3.2. Proof of Lemma 3.2. Let ( u m ) m ∈ N be a sequence of elements of D ( P ) suchthat ( u m ) m ∈ N converges to some u ∈ H and ( P u m ) m ∈ N converges to some v ∈ H .According to Proposition 2.4, P is bounded from H to H − so that ( P u m ) m ∈ N tends to P u in H − . Since H is continuously included in H − , it follows that P u = v ∈ H , and hence that P is closed. (cid:3) 56 3. RUELLE–POLLICOTT RESONANCES AND GEVREY ANOSOV FLOWS Then we prove an approximation lemma that will sometimes be useful (recallthat H ∞ Λ is defined by (2.18)). Lemma . Let R be large enough. Then, thedomains D ( P ) and D ( P ∗ ) of P and its adjoint on H both contain E ,R . Moreover,for every u ∈ D ( P ) there is a sequence of ( u n ) n ∈ N of elements of H ∞ Λ that tendsto u in H and such that ( P u n ) n ∈ N tends to P u in H . The same is true if wereplace P by P ∗ . Proof. First of all, D ( P ) contains H , and hence E ,R for R large enoughaccording to Corollary 2.2.Let χ be a C ∞ function from R to [0 , 1] compactly supported and identicallyequals to 1 near 0. Then define for (cid:15) > χ (cid:15) on Λ by χ (cid:15) ( α ) = χ ( (cid:15) (cid:104)| α |(cid:105) ). Then the χ (cid:15) ’s form a family of uniform symbols of order 0 on Λ. Hence,the operators I (cid:15) = S Λ B Λ χ (cid:15) B Λ T Λ are uniformly bounded on H . One easily checks that I (cid:15) converges to the identityin strong operator topology when (cid:15) tends to 0. In particular, if u ∈ D ( P ), we havethat I (cid:15) u tends to u in H when (cid:15) tends to 0. Moreover, we have P I (cid:15) u = I (cid:15) P u + [ P, I (cid:15) ] u (3.16)and I (cid:15) P u tends to P u in H when (cid:15) tends to 0. Then, we may write[ P, I (cid:15) ] = S Λ [ B Λ T Λ P S Λ B Λ , B Λ χ (cid:15) B Λ ] T Λ . Then using Propositions 2.10 and 2.12, we see that the [ P, I (cid:15) ]’s are uniformlybounded on H . But if v ∈ E ,R , then T Λ u is rapidly decaying and thus I (cid:15) u tendsto u in H and since P is bounded from H to H , we see that [ P, I (cid:15) ] v tends to0 in H . By a standard argument, this implies that [ P, I (cid:15) ] tends to 0 in strongoperator topology on H . Hence, from (3.16), we see that P I (cid:15) u tends to P u in H .Finally, thanks to Remark 2.20, the I (cid:15) u ’s belong to H ∞ Λ .We turn now to the study of the adjoint P ∗ of P . To do so, we apply Proposition2.10 to find a symbol σ of order 1 on Λ and a negligible operator R such that B Λ T Λ P S Λ B Λ = B Λ ( σ + R ) B Λ . Then define the operator (cid:101) P by (cid:101) P = S Λ B Λ (cid:0) ¯ σ + R ∗ (cid:1) B Λ T Λ , where R ∗ is the formal adjoint of R . Since ¯ σ is a symbol of order 1, we find that (cid:101) P is bounded from H k Λ to H k − for every k ∈ R . Now, if u, v ∈ H ∞ Λ then we have (thescalar product is in H ) (cid:104) P u, v (cid:105) = (cid:68) u, (cid:101) P v (cid:69) . (3.17)Using the approximation property for P that we just proved, we find that (3.17)remains true for u ∈ D ( P ) and v ∈ H ∞ Λ . Hence, H ∞ Λ (and in particular E ,R ) iscontained in the domain of P ∗ and if v ∈ H ∞ Λ then P ∗ v = (cid:101) P v . Now, if v belongs tothe domain of P ∗ , we find that, for u ∈ H ∞ Λ , (cid:10) u, P ∗ v (cid:11) = (cid:104) P u, v (cid:105) = (cid:68) u, (cid:101) P v (cid:69) . (3.18)Here, the last bracket makes sense because u ∈ H and (cid:101) P v ∈ H − . Since E ,R (and hence H ∞ Λ ) is dense in H , Lemma 2.24 implies with (3.18) that P ∗ v = (cid:101) P v .1. I-LAGRANGIAN SPACES ADAPTED TO A GEVREY ANOSOV FLOW 157 (in particular (cid:101) P v belongs to H ). Now, since (cid:101) P has the same structure than P , wemay prove the approximation property for P ∗ as we did for P . (cid:3) Proof of Proposition 3.3. We will apply Hille–Yosida Theorem to provethat P is the generator of a strongly continuous semi-group. We denote by p Λ therestriction to Λ of the almost analytic extension ˜ p of the principal symbol of hP given by (3.5). It follows from (iii) in Lemma 3.1 that there is a constant C > p Λ ≤ C . Indeed, since Re ˜ p vanishes on T ∗ M , the value of Re p Λ isobtained by integrating { G , Re ˜ p } on an orbit of time τ of the flow of H ω I G . Thisgives the upper bound on Re p Λ in the case s = 1. In the case s > 1, we need inaddition to notice that, as in Lemma 2.1, we remain at distance at most (cid:104)| α |(cid:105) δ − of the real. Hence, Re p Λ is less than C N (cid:104)| α |(cid:105) δ + N ( δ − (for any N > C N > δ < s > 1, the bound on Re p Λ follows bytaking N large enough.We see by Proposition 2.11 that, up to making C larger (depending on h , thatwe recall is fixed), we have for u ∈ H ∞ Λ Re (cid:104)− P u, u (cid:105) H ≥ − h (cid:104) Re p Λ T Λ P u, T Λ u (cid:105) L (Λ) − C (cid:107) u (cid:107) H ≥ − C (cid:107) u (cid:107) H . Hence, if z ∈ C , we haveRe (cid:104) ( z − P ) u, u (cid:105) H ≥ (Re z − C ) (cid:107) u (cid:107) H . By Cauchy–Schwarz, we find that (cid:107) ( z − P ) u (cid:107) H = (cid:107) ( z − P ) u (cid:107) H (cid:107) u (cid:107) H (cid:107) u (cid:107) H ≥ (cid:12)(cid:12)(cid:12) (cid:104) ( z − P ) u, u (cid:105) H (cid:12)(cid:12)(cid:12) (cid:107) u (cid:107) H ≥ Re (cid:104) ( z − P ) u, u (cid:105) H (cid:107) u (cid:107) H ≥ (Re z − C ) (cid:107) u (cid:107) H , (3.19)for u ∈ H ∞ Λ . By Lemma 3.4, this estimate remains true when u ∈ D ( P ). Thisproves that if Re z > C , then the operator z − P is injective and its image is closed.To prove that the image of z − P is dense, notice that if u ∈ H ∞ Λ thenRe (cid:10) ( z − P ) ∗ u, u (cid:11) H = Re (cid:104) ( z − P ) u, u (cid:105) H , and consequently (3.19) still holds when z − P is replaced by ( z − P ) ∗ (for u ∈ H ∞ Λ ,but it implies the same result for u ∈ D ( P ∗ ) by Lemma 3.4). Hence, ( z − P ) ∗ isinjective, and thus the image of z − P is closed. Consequently, z − P is invertibleand from (3.19), we see that (cid:13)(cid:13)(cid:13) ( z − P ) − (cid:13)(cid:13)(cid:13) ≤ z − C , for the operator norm on H . Hence, the Hille–Yosida Theorem applies (the domainof P is dense since it contains E ,R ), and we know that P is the generator of astrongly continuous semi-group.Denote by ( (cid:101) L t ) t ≥ the semi-group generated by P on H and ( L t ) t ≥ thesemi-group on L ( M ) defined by (3.1). We want to prove that for t ≥ u ∈ H ∩ L ( M ) we have L t u = (cid:101) L t u . Thanks to the semi-group property, weonly need to prove it for t ∈ [0 , t ] for some small t > 0. Then, since elementsof L ( M ) ∩ H may be simultaneously approximated in L ( M ) and in H byelements of E ,R (according to Corollary 2.3), we only need to prove the equalityfor u ∈ E ,R . Now, there is a t > R > u ∈ E ,R , the 58 3. RUELLE–POLLICOTT RESONANCES AND GEVREY ANOSOV FLOWS curve γ : [0 , t ] (cid:51) t (cid:55)→ L t u is C in E s,R with γ (cid:48) ( t ) = P γ ( t ). Provided that τ issmall enough, E s,R is continuously included in H (see Corollary 2.2) and hencethe curve γ has the same property in H . Consequently, we have γ ( t ) = (cid:101) L t u for t ∈ [0 , t ], according to [ ABHN11 , Proposition 3.1.11], ending the proof of theproposition. (cid:3) We turn now to the proof of the hypo-ellipticity of the operator P . Proof of Lemma 3.3. Assume first that u ∈ H ∞ Λ . Let χ + , χ − and χ be C ∞ functions from R → [0 , 1] such that χ + + χ + χ − = 1, and, for some small η > χ is supported in [ − η, η ], the function χ − is supported in ] −∞ , − η/ χ + is supported in [ η/ , + ∞ [. Then define for σ ∈ { + , − , } thesymbol f σ on Λ by f σ ( α ) = χ σ − i p (cid:16) e − τH ωIG ( α ) (cid:17) (cid:104)| α |(cid:105) . Then notice that if α is in the support of f + then we haveIm p Λ ( α ) = Im p (cid:16) e − τH ωIG ( α ) (cid:17) + O ( τ (cid:104)| α |(cid:105) ) ≥ η (cid:104)| α |(cid:105) , (3.20)provided that τ is small enough (depending on η ). And similarly, if α belongs tothe support of f − we have Im p Λ ( α ) ≤ − η (cid:104)| α |(cid:105) . (3.21)If α belongs to the support of f then we have (cid:12)(cid:12)(cid:12) p (cid:16) e − τH G ( α ) (cid:17)(cid:12)(cid:12)(cid:12) ≤ η (cid:104)| α |(cid:105) ≤ Cη (cid:68)(cid:12)(cid:12)(cid:12) e − τH G ( α ) (cid:12)(cid:12)(cid:12)(cid:69) . Hence, either e − τH G ( α ) is small, either it does not belong to C (provided that η is small enough, we use here the assumption that C does not intersect E ∗ u ⊕ E ∗ s ).In the second case, we may apply (ii) in Lemma 3.1 to find thatRe p Λ ( α ) = Re p (cid:16) e − τH ωIG ( α ) (cid:17) + τ { G , Re ˜ p } + O (cid:16) τ (cid:104)| α |(cid:105) δ − (cid:17) ≤ − C (cid:104)| α |(cid:105) δ + C, (3.22)provided that τ is small enough. Here, we added the constant C so that (3.22)remains true for any α in the support f .Now, Proposition 2.11 and (3.22) give that (the constant C may vary from oneline to another) − Re (cid:104)(cid:104)| α |(cid:105) δ f T Λ P u, T Λ u (cid:105)≥ − h (cid:90) Λ (cid:104)| α |(cid:105) δ f ( α ) Re p Λ ( α ) | T Λ u ( α ) | d α − C (cid:107) u (cid:107) H (cid:107) u (cid:107) H δ Λ ≥ C (cid:90) Λ f ( α ) (cid:104)| α |(cid:105) δ | T Λ u ( α ) | d α − C (cid:107) u (cid:107) H (cid:107) u (cid:107) H δ Λ . Applying Cauchy–Schwarz formula, we find then that (cid:90) Λ f ( α ) (cid:104)| α |(cid:105) δ | T Λ u ( α ) | d α ≤ C (cid:107) u (cid:107) H δ Λ (cid:16) (cid:107) P u (cid:107) H + (cid:107) u (cid:107) H (cid:17) . (3.23) .1. I-LAGRANGIAN SPACES ADAPTED TO A GEVREY ANOSOV FLOW 159 Working similarly with (3.22) replaced by the better estimates (3.20) and (3.21), wefind that (3.23) still holds when f is replaced by f + or f − . Summing these threeestimates, we get (cid:107) u (cid:107) H δ Λ ≤ C (cid:107) u (cid:107) H δ Λ (cid:16) (cid:107) P u (cid:107) H + (cid:107) u (cid:107) H (cid:17) . (3.24)Since the result is trivial when u = 0, we may divide by (cid:107) u (cid:107) H δ Λ in (3.24) to end theproof of the lemma when u ∈ H ∞ Λ .We deal now with the general case u ∈ D ( P ). Let ( u n ) n ∈ N be a sequence ofelements of H ∞ Λ as in Lemma 3.4. Since we already dealt with the case of elementsof H ∞ Λ we know that for some C > n ∈ N we have (cid:107) u n (cid:107) H δ Λ ≤ C (cid:16) (cid:107) u (cid:107) H + (cid:107) P u (cid:107) H (cid:17) . In addition, T Λ u n converges pointwise to T Λ u and hence the result follows by Fatou’sLemma. (cid:3) We can now prove Theorem 10. Proof of Theorem 10. Let z be any element of the resolvent set of P (theresolvent set of P is non-empty according to Proposition 3.3). If u ∈ H then wehave that P ( z − P ) − u = z ( z − P ) − u − u. Hence, ( z − P ) − and P ( z − P ) − are both bounded from H to itself and, con-sequently, Lemma 3.3 implies that ( z − P ) − is bounded from H to H δ Λ . Hence,Proposition 2.13 implies that ( z − P ) − , as an operator from H to itself, is compact,with the announced estimates on its singular values.We prove the estimates on the number of eigenvalues of P before proving thatthese eigenvalues are indeed the Ruelle resonances. Let z ∈ C be any point in theresolvent set of P and denote by (cid:101) N ( R ) the number of eigenvalues of ( z − P ) − ofmodulus larger than R − . Then let ( µ k ) k ∈ N denote the sequence of eigenvaluesof ( z − P ) − , ordered so that ( µ k ) k ∈ N is decreasing, and ( σ k ) k ∈ N the sequence ofits singular values, and choose p > δp/n < 1. According to [ GGK00 ,Corollary IV.3.4], we have then for every R > (cid:101) N ( R ) R p ≤ (cid:101) N ( R ) − (cid:88) k =0 | µ k | p ≤ (cid:101) N ( R ) − (cid:88) k =0 σ pk ≤ C (cid:101) N ( R ) − (cid:88) k =0 (1 + k ) − δpn ≤ C (cid:101) N ( R ) − δpn . Here, we applied the estimates on singular values that we just proved, and C mayvary from one line to another. It follows that (cid:101) N ( R ) ≤ C n/δp R n/δ . The estimateson N ( R ) follows since, if ( λ k ) k ∈ N denotes the sequence of eigenvalues of P , we havethe relation µ k = ( z − λ k ) − (up to reordering, and recall that δ = 1 /s ).It remains to prove that the eigenvalues of P acting on H are indeed the Ruelleresonances of P . To do so let R > τ is smallenough, so that (2.27) holds with s = 1 and that E ,R is dense in H (see Corollary2.3). We also assume that R is large enough so that E ,R is dense in C ∞ ( M ) (seeCorollary 2.1). Denote by i the inclusion of E ,R in H and by i the inclusion of H in ( E ,R ) (cid:48) . Then, we define (cid:101) R ( z ) = i ◦ ( z − P ) − ◦ i : E ,R → (cid:16) E ,R (cid:17) (cid:48) . 60 3. RUELLE–POLLICOTT RESONANCES AND GEVREY ANOSOV FLOWS Then, if we denote by i the inclusion of E ,R into C ∞ ( M ) and by j the inclusionof D (cid:48) ( M ) into ( E ,R ) (cid:48) , we see that (cid:101) R ( z ) = j ◦ R ( z ) ◦ i, (3.25)where R ( z ) is defined by (3.2) and seen as an operator from C ∞ ( M ) to D (cid:48) ( M ).Indeed, when Re z is large (3.25) follows from Proposition 3.3 since (cid:101) R ( z ) and R ( z )are both obtained as the Laplace transform of the family of operators (3.1). Theequality (3.25) then follows for any z by analytic continuation. Integrating on smallcircles, we see that (3.25) is also satisfied by the residues of (cid:101) R ( z ) and R ( z ). Sincethese residues have finite rank and since E ,R is dense in C ∞ ( M ), it follows thatthe eigenvalues of P on H (the poles of (cid:101) R ( z )) are the Ruelle resonances of P (thepoles of R ( z )) counted with multiplicity (the rank of the associated residues). Forthe same reason, the resonant spaces (the images of the residues) also coincide. (cid:3) As announced, we can now give the proof of Proposition 3.2. Proof of Proposition 3.2. Since the eigenvectors of P acting on H are theresonant states of P , they do not depend on G (as soon as G is as in Lemma 3.1)nor on τ (provided that τ is small enough). Hence, if C s is a conic neighbourhoodof E ∗ s in T ∗ M , we may choose G negative and elliptic of order δ = 1 /s outsideof C s . Then, taking τ small enough, we may apply Lemma 2.14 to see that if u is a resonant state for P then WF G s ( u ) ⊆ C s . Since C s is an arbitrary conicneighbourhood of E ∗ s , it follows that WF G s ( u ) ⊆ E ∗ s . (cid:3) At the heart of many “microlocal” results lie a trace formula, that links (purely)spectral information to geometric or dynamical data. Theorems 1 and 8 are noexception. Their proof will be given in this section, and this will give a glimpse ofthe relation between I-Lagrangian spaces and traces.In order to show that ζ X,V has finite order (under our Gevrey assumption), wewill relate it to a regularized determinant associated with the resolvent of P . Thiswill be based on the following version of Guillemin’s trace formula: Lemma . If the real part of z is positive and large enough and m is an integersuch that m > sn , then the operator ( z − P ) − m acting on H is trace class and Tr (cid:16) ( z − P ) − m (cid:17) = 1( m − (cid:88) γ T γ e (cid:82) γ V (cid:12)(cid:12) det (cid:0) I − P γ (cid:1)(cid:12)(cid:12) T m − γ e − zT γ . (3.26) Here, we use the notations defined in the introduction of the chapter (after (3.3) ). The fundamental reason for which this result holds is that whenever we cangive a reasonable meaning to Tr |H ( z − P ) − m , it formally does not depend on Λ.Indeed, this trace should coincide with the “flat trace” of ( z − P ) − m given by formalintegration of it Schwartz kernel on the diagonal of M × M . See [ DZ16 ] for anextensive discussion of the notion of flat trace in the context of Anosov flows. Proof. That ( z − P ) − m is trace class results from Theorem 10. For Re z largeenough, the convergence of the right hand side of (3.26) is provided by Margulis’bound [ Mar04 ] on the number of closed geodesics for an Anosov flow (see also[ DZ16 , Lemma 2.2]).First, recall the semi-group ( L t ) t ≥ generated by P (see Proposition 3.3), givenby the formula (3.1) on H ∩ L ( M ). It follows from the semi-group property and .2. TRACES AND I-LAGRANGIAN SPACES 161 from [ GGK00 , Theorem IV.5.5] thatTr H ( z − P ) − m = lim ρ → Tr H L ρ ( z − P ) − m . Thus, (3.26) will follow if we can prove that for ρ > H L ρ ( z − P ) − m = 1( m − (cid:88) γ T γ e (cid:82) γ V (cid:12)(cid:12) det (cid:0) I − P γ (cid:1)(cid:12)(cid:12) (cid:0) T γ − ρ (cid:1) m − e − z ( T γ − ρ ) , where x + = max( x, 0) denotes the positive part of a real number x . Indeed, theconvergence of the right hand side when ρ tends to zero is obtained by dominatedconvergence.Let then ρ > I (cid:15) = S exp( − (cid:15) (cid:104) α (cid:105) ) T for (cid:15) > 0. These are regularizing operators thatmap ( E ,R ) (cid:48) to E ,R for R > I (cid:15) L ρ ( z − P ) − m I (cid:15) isof trace class on L and H (actually on any reasonable space where L ρ ( z − P ) − m is bounded). Here, it makes sense to discuss the operator I (cid:15) L ρ ( z − P ) − m I (cid:15) actingon L or H because I (cid:15) is valued in E ,R . Moreover, thanks to Proposition 3.3, wemay use the expression (3.1) for L t and then L ρ ( z − P ) − m writes for Re z (cid:29) L ρ ( z − P ) − m = ( − m − ( m − (cid:90) + ∞ e − z ( t − ρ ) ( t − ρ ) + L t d t. (3.27)On the other hand, ( I (cid:15) ) (cid:15)> is a family of G pseudors, uniformly of order 0 as (cid:15) → ST ), and theyconverge strongly to the identity when (cid:15) tends to 0, both as operators on L ( M )and H (or any Sobolev space, see Corollary 2.4 and the proof of Corollary 2.3).We deduce then from [ GGK00 , Theorem IV.5.5] thatTr H L ρ ( z − P ) − m = lim (cid:15) → Tr H ( I (cid:15) L ρ ( z − P ) − m I (cid:15) ) . We consider now u a generalized eigenvector of I (cid:15) L ρ ( z − P ) − m I (cid:15) , associatedto an eigenvalue λ (cid:54) = 0. Since I (cid:15) maps H continuously in E ,R for some R > u ∈ H , then u ∈ E ,R . In particular, u ∈ L . Reciprocally,if u ∈ L , then u ∈ H . Since I (cid:15) L δ ( z − P ) − m I (cid:15) has the same eigenvectors andeigenvalues on L and H , according to Lidskii’s theorem, its trace is the same onboth spaces, so that we haveTr H L ρ ( z − P ) − m = lim (cid:15) → Tr L ( I (cid:15) L ρ ( z − P ) − m I (cid:15) ) . In order to compute the limit when (cid:15) tends to 0 of the trace of I (cid:15) L δ ( z − P ) − m I (cid:15) ,we will use the notion of flat trace. We refer to [ DZ16 ] and references thereinfor basic properties of this object and insightful discussion of its properties in thecontext of Anosov flows.Let us show that the flat trace of L ρ ( z − P ) − m is well-defined. Recall thatthe flat trace of an operator A : C ∞ ( M ) → D (cid:48) ( M ) is defined if the intersectionbetween its wave front set W F (cid:48) ( A ) (defined in [ DZ16 , C.1]) and the conormal tothe diagonal ∆( T ∗ M ) = { ( x, ξ ; x, ξ ) | ( x, ξ ) ∈ T ∗ M } ⊆ T ∗ M × T ∗ M is empty. Then, according to [ DZ16 , Proposition 3.3], the wave front set of ( z − P ) − is contained in ∆( T ∗ M ) ∪ Ω − ∪ E ∗ s × E ∗ u , (3.28) 62 3. RUELLE–POLLICOTT RESONANCES AND GEVREY ANOSOV FLOWS where Ω − = { ( x, ξ ; φ t ( x ) , T d φ t ( x ) − ξ )) | t ≥ , ξ ( X ( x )) = 0 } . (we have adjusted the signs because they study the transfer operator instead ofthe Koopman operator). Then, by [ DZ19 , Proposition E.40], we see that the wavefront set of ( z − P ) − m is also contained in (3.28). Then, from [ H¨or03a , Theorem8.2.4], we know that the wave front set of L δ ( z − P ) − m is contained in F ρ ∪ Ω ρ − ∪ E ∗ s × E ∗ u , where Ω ρ − is defined by replacing the condition t ≥ t ≥ ρ in the definition ofΩ − and F ρ = { ( x, ξ ; φ ρ ( x ) , T d φ ρ ( x ) − ξ )) | t ≥ , ( x, ξ ) ∈ T ∗ M } In particular, provided that ρ > φ t , we see that the wave front set of L ρ ( z − P ) − m does not intersect∆ (cid:0) T ∗ M (cid:1) (the wave front set of an operator does not intersect the zero section), sothat the flat trace of L ρ ( z − P ) − m is well-defined.In order to compute this flat trace, we just notice that the argument usedin [ DZ16 , Section 4] to compute the flat trace of L ρ ( z − P ) − also applies to L ρ ( z − P ) − m . Let us just mention that this argument is based on Guillemin traceformula [ Gui77 ]: for t > DZ16 , AppendixB] for details) Tr (cid:91) L t = (cid:88) γ T γ exp (cid:82) γ V | det 1 − P γ | δ ( t − T γ ) . Consequently, recalling (3.27), we find without surprise thatTr (cid:91) (cid:16) L ρ ( z − P ) − m (cid:17) = 1( m − (cid:88) γ T γ e (cid:82) γ V (cid:12)(cid:12) det (cid:0) I − P γ (cid:1)(cid:12)(cid:12) (cid:0) T γ − ρ (cid:1) m − e − z ( T γ − ρ ) . Hence, we only need to prove thatlim (cid:15) → Tr | L (cid:16) I (cid:15) L ρ ( z − P ) − m I (cid:15) (cid:17) = Tr (cid:91) (cid:16) L ρ ( z − P ) − m (cid:17) . Since for (cid:15) > I (cid:15) L ρ ( z − P ) − m I (cid:15) has a C ∞ kernel, its trace acting on L ( M ) coincides with its flat trace. Consequently, we want to prove thatlim (cid:15) → Tr (cid:91) (cid:16) I (cid:15) L ρ ( z − P ) − m I (cid:15) (cid:17) = Tr (cid:91) (cid:16) L ρ ( z − P ) − m (cid:17) . (3.29)According to [ H¨or03b , Definition 8.2.2 and Theorem 8.2.4], in order to prove(3.29), we only need to prove that the Schwartz kernel of I (cid:15) L ρ ( z − P ) − m I (cid:15) convergesweakly to the kernel of L ρ ( z − P ) − m (when (cid:15) tends to 0) and that the wavefront set condition needed to define the flat trace is uniformly satisfied by the I (cid:15) L ρ ( z − P ) − m I (cid:15) ’s.In order to check the wave front set condition, we just use the fact that the I (cid:15) ’s form a family of pseudors uniformly of order 0, so that the Schwartz kernel of I (cid:15) L ρ ( z − P ) − m I (cid:15) is the image of the kernel of L ρ ( z − P ) − m by a pseudor J (cid:15) = I (cid:15) ⊗ t I (cid:15) uniformly of order 0 when (cid:15) tends to 0. To get the weak convergence, we just need torecall from the proof of Corollary 2.1 that I (cid:15) converges pointwise to the identity onany Sobolev space on M (see also Corollary 2.4), so that J (cid:15) has the same propertyon M × M . (cid:3) With Lemma 3.5, we are ready to relate the dynamical determinant ζ X,V witha regularized determinant. See [ GGK00 , Chapter XI] for the general theory ofregularized determinants. .2. TRACES AND I-LAGRANGIAN SPACES 163 Lemma . Let z be a complex number with large and positive real part and m be the smallest integer strictly larger than ns . Let Q z be the polynomial of order atmost m − Q z ( λ ) = − m − (cid:88) (cid:96) =0 (cid:32)(cid:88) γ T γ e (cid:82) γ V e − zT γ T (cid:96) − γ (cid:12)(cid:12) det (cid:0) I − P γ (cid:1)(cid:12)(cid:12) (cid:33) ( z − λ ) (cid:96) (cid:96) ! . Then for every λ ∈ C we have ζ X,V ( λ ) = det m (cid:16) I + ( λ − z ) ( z − P ) − (cid:17) exp ( Q z ( λ )) , where det m denotes the regularized determinant of order m . Proof. By analytic continuation principle, we only need to prove this resultfor λ close to z . For such a λ the regularized determinant is defined bydet m (cid:16) I + ( λ − z ) ( z − P ) − (cid:17) = exp − (cid:88) (cid:96) ≥ m ( z − λ ) (cid:96) (cid:96) Tr (cid:16) ( z − P ) − (cid:96) (cid:17) = exp − (cid:88) (cid:96) ≥ m ( z − λ ) (cid:96) (cid:96) ! (cid:88) γ T γ e (cid:82) γ V (cid:12)(cid:12) det (cid:0) I − P γ (cid:1)(cid:12)(cid:12) T (cid:96) − γ e − zT γ = exp − (cid:88) γ T γ T γ e (cid:82) γ V e − zT γ (cid:12)(cid:12) det (cid:0) I − P γ (cid:1)(cid:12)(cid:12) (cid:88) (cid:96) ≥ m (cid:0) ( z − λ ) T γ (cid:1) (cid:96) (cid:96) ! = exp (cid:32) − (cid:88) γ T γ T γ e (cid:82) γ V e − zT γ (cid:12)(cid:12) det (cid:0) I − P γ (cid:1)(cid:12)(cid:12) (cid:32) e ( z − λ ) T γ − m − (cid:88) (cid:96) =0 (cid:0) ( z − λ ) T γ (cid:1) (cid:96) (cid:96) ! (cid:33)(cid:33) = ζ X,V ( λ ) e − Q z ( λ ) . The applications of Fubini’s Theorem are justified when Re z (cid:29) | z − λ | smallenough by Margulis’ bound. (cid:3) We are now ready to prove Theorem 8. Proof of Theorem 8. Let m be as in Lemma 3.6. Recall the Weierstrassprimary factor (the second expression is valid when | λ | < E ( λ, m − 1) = (1 − λ ) exp (cid:32) m − (cid:88) (cid:96) =1 (cid:96) λ (cid:96) (cid:33) = exp (cid:32) − + ∞ (cid:88) (cid:96) = m (cid:96) λ (cid:96) (cid:33) . It follows from Lidskii’s Trace Theorem that when Re z (cid:29) λ ∈ C we havedet m (cid:16) I − ( z − λ )( z − P ) − (cid:17) = + ∞ (cid:89) k =0 E (cid:18) λ − zλ k − z , m − (cid:19) , (3.30)where ( λ k ) k ∈ N denotes the sequence of Ruelle resonances of P . We want to usethis expression with Lemma 3.6 in order to prove Theorem 8, but let us make anobservation first. If λ is a complex number such that (cid:12)(cid:12)(cid:12)(cid:12) λ | λ | − (cid:12)(cid:12)(cid:12)(cid:12) ≤ λ ≥ | λ | / 2. Hence, using the expression (3.3) and dominatedconvergence, we see that when | λ | tends to + ∞ while satisfying (3.31), the function 64 3. RUELLE–POLLICOTT RESONANCES AND GEVREY ANOSOV FLOWS ζ X,V ( λ ) tends to 1. In particular, ζ X,V ( λ ) remains bounded when λ satisfies (3.31).Hence, we may assume in the following that (cid:12)(cid:12)(cid:12)(cid:12) λ | λ | − (cid:12)(cid:12)(cid:12)(cid:12) ≥ , (3.32)and that | λ | is large of course. When | λ | is large enough, we may apply Lemma 3.6with z = | λ | . Then, notice that Q | λ | ( λ ) tends to 0 when | λ | tends to + ∞ and thuswe may ignore the factor exp ( Q z ( λ )) from Lemma 3.6. The other factor is given by(3.30) (with z = | λ | ).Notice that if | λ k | ≥ | λ | then | λ − | λ || / | λ k − | λ || ≤ / (cid:12)(cid:12)(cid:12)(cid:12) E (cid:18) λ − | λ | λ k − | λ | , m − (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) λ − | λ | λ k − | λ | (cid:12)(cid:12)(cid:12)(cid:12) m ≤ m +1 (cid:18) | λ || λ k | − | λ | (cid:19) m ≤ (cid:18) (cid:19) m (cid:12)(cid:12)(cid:12)(cid:12) λλ k (cid:12)(cid:12)(cid:12)(cid:12) m . (3.33)On the other hand if | λ k | < | λ | , we have, since we assume (3.32), (cid:12)(cid:12)(cid:12)(cid:12) λ − | λ | λ k − | λ | (cid:12)(cid:12)(cid:12)(cid:12) ≥ 12 1 (cid:12)(cid:12)(cid:12) λ k | λ | − (cid:12)(cid:12)(cid:12) ≥ . Hence, we havelog (cid:12)(cid:12)(cid:12)(cid:12) E (cid:18) λ − | λ | λ k − | λ | , m − (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ log (cid:12)(cid:12)(cid:12)(cid:12) − λ − | λ | λ k − | λ | (cid:12)(cid:12)(cid:12)(cid:12) + m − (cid:88) (cid:96) =1 (cid:96) (cid:12)(cid:12)(cid:12)(cid:12) λ − | λ | λ k − | λ | (cid:12)(cid:12)(cid:12)(cid:12) (cid:96) ≤ (cid:12)(cid:12)(cid:12)(cid:12) λ − | λ | λ k − | λ | (cid:12)(cid:12)(cid:12)(cid:12) + m − (cid:88) (cid:96) =1 (cid:12)(cid:12)(cid:12)(cid:12) λ − | λ | λ k − | λ | (cid:12)(cid:12)(cid:12)(cid:12) (cid:96) ≤ (cid:32) m − (cid:88) (cid:96) =0 m − − (cid:96) (cid:33) (cid:12)(cid:12)(cid:12)(cid:12) λ − | λ | λ k − | λ | (cid:12)(cid:12)(cid:12)(cid:12) m − ≤ m (cid:12)(cid:12)(cid:12)(cid:12) λ − | λ | λ k − | λ | (cid:12)(cid:12)(cid:12)(cid:12) m − . (3.34)Then, introduce a constant C such that Re λ k ≤ C for all k ∈ N (such a constantexists because P is the generator of a strongly continuous semi-group) and noticethat for | λ | large enough, we have (cid:12)(cid:12)(cid:12)(cid:12) λ − | λ | λ k − | λ | (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) λ k − C | λ | − C | λ | (cid:12)(cid:12)(cid:12) ≤ − C | λ | ≤ . And thus, (3.34) becomes:log (cid:12)(cid:12)(cid:12)(cid:12) E (cid:18) λ − | λ | λ k − | λ | , m − (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ m . (3.35)Now, gathering (3.33) and (3.35), that are valid respectively when | λ k | ≥ | λ | and | λ k | < | λ | , we find thatlog (cid:12)(cid:12)(cid:12) det m (cid:16) I − ( | λ | − λ )( | λ | − P ) − (cid:17)(cid:12)(cid:12)(cid:12) ≤ × (cid:18) (cid:19) m | λ | m (cid:88) | λ k |≥ | λ | | λ k | − m + 48 m { k ∈ N : | λ k | < | λ |} . (3.36)Then, from the counting bound in Theorem 10, we see that { k ∈ N : | λ k | < | λ |} = O ( | λ | ns ) , (3.37) .3. PERTURBATIVE RESULTS 165 and that, (cid:88) | λ k |≥ | λ | | λ k | − m = O (cid:16) | λ | ns − m (cid:17) . (3.38)We end the proof of Theorem 8 by plugging (3.37) and (3.38) in (3.36). (cid:3) This section isdedicated to the proof of Theorem 9. Recall that we are considering perturbationsof P of the form P (cid:15) := P + (cid:15) ∆ , where (cid:15) ≥ G s semi-classical pseudor, self-adjoint, negative and classicallyelliptic of order m > 1. The proof of Theorem 9 will be split into two parts. We willfirst prove that the spectrum of P (cid:15) on H is discrete and coincides with σ L ( P (cid:15) ),and then prove the convergence results working directly on H . In this section, wewill need to work with other adapted Lagrangians than Λ that we introduced in § (cid:48) , keeping the notation Λ forthe Lagrangian that we introduced to study P . We start with a technical lemma. Lemma . Let τ be small enough and Λ (cid:48) be a ( τ , s ) -adapted Lagrangian.Then, for h small enough, for every N > , (cid:15) > , λ ∈ C and k ∈ N , there is aconstant C such that for every u ∈ H km Λ (cid:48) , we have (cid:107) u (cid:107) H km Λ (cid:48) ≤ C (cid:18)(cid:13)(cid:13)(cid:13) ( P (cid:15) − λ ) k u (cid:13)(cid:13)(cid:13) H (cid:48) + (cid:107) u (cid:107) H − N Λ (cid:48) (cid:19) . Proof. We use the assumption that τ and h are small enough to be able toapply Proposition 2.10 and write B Λ (cid:48) T Λ (cid:48) P S Λ (cid:48) B Λ (cid:48) = B Λ (cid:48) σ P B Λ (cid:48) + R and B Λ (cid:48) T Λ (cid:48) σ ∆ S Λ (cid:48) B Λ (cid:48) = B Λ (cid:48) σ ∆ B Λ (cid:48) + R , where σ P and σ ∆ are symbols on Λ (cid:48) (of order 1 and m respectively), and R and R are negligible operators. Then, provided that τ is small enough, it follows from theellipticity of ∆ that there is a constant C > α ∈ Λ (cid:48) , we haveRe σ ∆ ( α ) ≤ − C (cid:104)| α |(cid:105) m + C. (3.39)Then, applying Proposition 2.12, we see that B Λ (cid:48) (cid:104)| α |(cid:105) mk B Λ (cid:48) T (cid:48) Λ ( P (cid:15) − λ ) k S Λ (cid:48) B Λ (cid:48) = B Λ (cid:48) σ (cid:15),k,λ B Λ (cid:48) + R , (3.40)where R is negligible and σ (cid:15),k,λ is a symbol of order 2 mk . Moreover, it followsfrom (3.39) that for some C > α ∈ Λ (cid:48) we haveRe σ (cid:15),k,λ ( α ) ≤ − C (cid:104)| α |(cid:105) mk + C (cid:104)| α |(cid:105) − N + mk , (3.41)where the constant C > (cid:15), Λ (cid:48) , N, λ and k . By Cauchy–Schwarz inequality, we find that (the scalar product is in L (cid:0) Λ (cid:48) (cid:1) ) (cid:12)(cid:12)(cid:12) Re (cid:68) (cid:104)| α |(cid:105) mk T Λ (cid:48) ( P (cid:15) − λ ) k u, T Λ (cid:48) u (cid:69)(cid:12)(cid:12)(cid:12) ≤ (cid:107) u (cid:107) H mk Λ (cid:48) (cid:13)(cid:13)(cid:13) ( P (cid:15) − λ ) k u (cid:13)(cid:13)(cid:13) H (cid:48) . (3.42)Then, using (3.40) and (3.41), we find that (for some new constant C > 0, we alsoapply Cauchy–Schwarz inequality)Re (cid:68) (cid:104)| α |(cid:105) mk T Λ (cid:48) ( P (cid:15) − λ ) k u, T Λ (cid:48) u (cid:69) = Re (cid:10) σ (cid:15),k,λ T Λ (cid:48) u, T Λ (cid:48) u (cid:11) + Re (cid:10) R T Λ (cid:48) u, T Λ (cid:48) u (cid:11) ≤ − C (cid:107) u (cid:107) H mk Λ (cid:48) + C (cid:107) u (cid:107) H − N Λ (cid:48) (cid:107) u (cid:107) H mk Λ (cid:48) . 66 3. RUELLE–POLLICOTT RESONANCES AND GEVREY ANOSOV FLOWS Hence, with (3.42), we find that1 C (cid:107) u (cid:107) H mk Λ (cid:48) ≤ (cid:107) u (cid:107) H mk Λ (cid:48) (cid:13)(cid:13)(cid:13) ( P (cid:15) − λ ) k u (cid:13)(cid:13)(cid:13) H (cid:48) + C (cid:107) u (cid:107) H − N Λ (cid:48) (cid:107) u (cid:107) H mk Λ (cid:48) , and the result follows. (cid:3) We prove now that P (cid:15) has still discrete spectrum after the Lagrangian pertur-bation of T ∗ M . Lemma . Assume that τ is small enough and let Λ (cid:48) be a ( τ , s ) -adaptedLagrangian. Then, for h small enough and (cid:15) > , the operator P (cid:15) acting on H (cid:48) with domain D ( P (cid:15) ) = (cid:110) u ∈ H (cid:48) : P (cid:15) u ∈ H (cid:48) (cid:111) = H m Λ (cid:48) has a discrete spectrum σ H (cid:48) ( P (cid:15) ) made of eigenvalues of finite multiplicity. Proof. By a parametrix construction (using Proposition 2.12), we prove that D ( P (cid:15) ) = H m Λ (cid:48) . As in the proof of Proposition 3.3, we use the negativity of thereal part of the symbol of ∆ to prove that the resolvent set of P (cid:15) acting on H (cid:48) isnon-empty. Finally, we apply Lemma 3.7 with λ = 0 , k = 1 and N = 0 to see thatthe resolvent of P (cid:15) sends H (cid:48) continuously into H m Λ (cid:48) , and is consequently compactas an endomorphism on H (cid:48) (recall Proposition 2.13). (cid:3) We prove now that the spectrum of an elliptic operator is unchanged undersmall Lagrangian deformations. The proof of Lemma 3.9 is an adaptation of theproof of [ GZ19a , Lemma 7.8]. Lemma . Assume that τ is small enough and let Λ (cid:48) be a ( τ , s ) -adaptedLagrangian. Then, for h small enough and every (cid:15) > , we have σ H (cid:48) ( P (cid:15) ) = σ L ( P (cid:15) ) .The (generalized) eigenvectors also coincide (in particular, they belong to both L ( M ) and H (cid:48) ). Proof. We only need to prove that, for (cid:15) > 0, the generalized eigenvectorsof P (cid:15) on H (cid:48) and L ( M ) coincides. Let us prove for instance that the generalizedeigenvectors of P (cid:15) that belong to H (cid:48) also belong to L ( M ) (the other way issimilar). Let G (cid:48) be the symbol that defines Λ (cid:48) , as in Definition 2.2. Choose a C ∞ function χ : R → [0 , 1] such that χ ( x ) = 0 for x ≤ χ ( x ) = 1 for x ≥ 2. Thendefine for r ∈ R the symbol G (cid:48) r ( α ) = χ ( r (cid:104)| α |(cid:105) ) G (cid:48) ( α ) . Notice that for r > G − r (cid:48) = G (cid:48) and that G (cid:48) = 0. For r ∈ R , let Λ (cid:48) r denotes the Lagrangian defined by G (cid:48) r , using (2.4). Notice that, since Λ (cid:48) is ( τ , s )-adapted, then for some c > (cid:48) r are uniformly ( cτ , s )-adapted.Hence, if τ and h are small enough, we can apply Lemma 3.7 uniformly to theLagrangians Λ (cid:48) r . Let u ∈ H (cid:48) be such that there are λ ∈ C , k ∈ N ∗ and (cid:15) > P (cid:15) − λ ) k u = 0. We notice that, for every r > 0, the Lagrangians Λ (cid:48) r and Λ (cid:48) coincide outside of a compact set. Hence, so does T Λ (cid:48) u and T Λ (cid:48) r u . Since T Λ (cid:48) r u iscontinuous, it is bounded on any compact set and consequently, for every r > 0, wehave u ∈ H (cid:48) r . Hence, we may apply Lemma 3.7 to find a constant C > r > (cid:107) u (cid:107) H (cid:48) r ≤ C (cid:107) u (cid:107) H − (cid:48) r . .3. PERTURBATIVE RESULTS 167 Here, the constant C does not depend on r > 0, since the Lagrangians Λ (cid:48) r satisfyuniformly the hypotheses of Lemma 3.7. Then, for large M > 0, we have, for r > (cid:107) u (cid:107) H (cid:48) r ≤ C (cid:16)(cid:13)(cid:13)(cid:13) {(cid:104)| α |(cid:105)≤ M } (cid:104)| α |(cid:105) − T Λ (cid:48) r u (cid:13)(cid:13)(cid:13) L + (cid:13)(cid:13)(cid:13) {(cid:104)| α |(cid:105) >M } (cid:104)| α |(cid:105) − T Λ (cid:48) r u (cid:13)(cid:13)(cid:13) L (cid:17) ≤ C (cid:18)(cid:13)(cid:13)(cid:13) {(cid:104)| α |(cid:105)≤ M } (cid:104)| α |(cid:105) − T Λ (cid:48) r u (cid:13)(cid:13)(cid:13) L + M − (cid:107) u (cid:107) H (cid:48) r (cid:19) . Thus, if M ≥ C , we find that (cid:107) u (cid:107) H (cid:48) r ≤ C (cid:13)(cid:13)(cid:13) {(cid:104)| α |(cid:105)≤ M } (cid:104)| α |(cid:105) − T Λ (cid:48) r u (cid:13)(cid:13)(cid:13) L ≤ C (cid:48) , where the constant C (cid:48) > u but not on r (we only use the factthat T u is continuous on a conical neighbourhood of T ∗ M and hence bounded oncompact sets). Since (cid:107) u (cid:107) H (cid:48) r = (cid:13)(cid:13)(cid:13) T Λ (cid:48) r u (cid:13)(cid:13)(cid:13) L , we may apply Fatou’s Lemma (after a change of variable to write the norm asthe square root of an integral over T ∗ M for instance) to find that (notice thatΛ (cid:48) = T ∗ M ) (cid:107) u (cid:107) L ( M ) = (cid:107) u (cid:107) H (cid:48) ≤ C (cid:48) < + ∞ . Hence, u belongs to L ( M ), and the proof of the lemma is over. (cid:3) By choosing for Λ (cid:48) the adapted Lagrangian defined by the symbol G (cid:48) ( α ) = − τ (cid:104)| α |(cid:105) s with τ (cid:28) 1, we deduce from Lemma 3.9 and Lemma 2.14 the followingresult, that was already known for less general G s pseudors (see for instance [ BG72 ,Th´eor`eme 4.3], we could adapt our proof to deal with any elliptic G s pseudor in thesense of Definition 1.6). Corollary . There is R > such that, for every (cid:15) > , the L eigenvectorsof P (cid:15) belongs to E s,R ( M ) . Now that we proved that the spectrum of P (cid:15) is invariant under Lagrangiandeformation (in the sense of Lemma 3.9), we may go back to the case Λ (cid:48) = Λ (theLagrangian deformation introduced in § P = P ). We assume inaddition that τ and h are small enough so that the machinery from the first chapterapplies to both P and ∆. First, we prove that the hypo-ellipticity still holds forsmall (cid:15) ≥ Lemma . Assume that h and τ are small enough. Then, for every k ∈ R ,if z is a large enough positive real number, then there is a constant C > such that,for every (cid:15) ∈ [0 , the number z belongs to the resolvent set of P (cid:15) acting on H k Λ and (cid:13)(cid:13)(cid:13) ( z − P (cid:15) ) − (cid:13)(cid:13)(cid:13) H k Λ →H k Λ ≤ C. (3.43) Moreover, for every (cid:15) ∈ ]0 , , the resolvent ( z − P (cid:15) ) − is bounded from H k Λ to H m + k Λ and (cid:13)(cid:13)(cid:13) ( z − P (cid:15) ) − (cid:13)(cid:13)(cid:13) H k Λ →H k + m Λ ≤ C(cid:15) . (3.44) Proof. We may write (using Proposition 2.10) B Λ T Λ P S Λ B Λ = B Λ p Λ B Λ + R and B Λ T Λ ∆ S Λ B Λ = B Λ σ ∆ B Λ + R , where p Λ and σ ∆ are symbols of order respectively 1 and m . The operators R and R are negligible. Moreover, Λ has been constructed so that for some C > 68 3. RUELLE–POLLICOTT RESONANCES AND GEVREY ANOSOV FLOWS α ∈ Λ we have Re p Λ ( α ) ≤ C (see the proof of Proposition 3.3). Furthermore, byellipticity of ∆, up to making C larger, we also have, for every α ∈ Λ,Re σ ∆ ( α ) ≤ − C (cid:104)| α |(cid:105) m + C. Then, we apply Proposition 2.10 to write B Λ (cid:104)| α |(cid:105) k T Λ P S Λ B Λ = B Λ (cid:104)| α |(cid:105) k p Λ B Λ + R (cid:48) and B Λ (cid:104)| α |(cid:105) k T Λ ∆ S Λ B Λ = B Λ (cid:104)| α |(cid:105) k σ ∆ B Λ + R (cid:48) , where R (cid:48) and R (cid:48) are negligible. The symbols (cid:104)| α |(cid:105) k p Λ and (cid:104)| α |(cid:105) k σ ∆ are givenat leading order by p Λ (cid:104)| α |(cid:105) k and σ ∆ (cid:104)| α |(cid:105) k . Hence, for some large C > α ∈ Λ we haveRe (cid:16) (cid:104)| α |(cid:105) k p Λ (cid:17) ( α ) ≤ C (cid:104)| α |(cid:105) k and Re (cid:16) (cid:104)| α |(cid:105) k σ ∆ (cid:17) ( α ) ≤ C. Consequently, for u ∈ H m + k Λ we have that (up to making C larger and with (cid:15) ∈ [0 , (cid:104) P (cid:15) u, u (cid:105) H k Λ = Re (cid:68) (cid:104)| α |(cid:105) k T Λ P (cid:15) u, T Λ u (cid:69) = Re (cid:68)(cid:16) (cid:104)| α |(cid:105) k p Λ + (cid:15) (cid:104)| α |(cid:105) k σ ∆ (cid:17) T Λ u, T Λ u (cid:69) + Re (cid:10)(cid:0) R (cid:48) + (cid:15)R (cid:48) (cid:1) T Λ u, T Λ u (cid:11) ≤ C (cid:68) (cid:104)| α |(cid:105) k T Λ u, T Λ u (cid:69) + C (cid:107) u (cid:107) H k Λ ≤ C (cid:107) u (cid:107) H k Λ . As in the proof of Proposition 3.3, it follows that if z is a real number such that z > C , then z belongs to the resolvent set of P (cid:15) and (cid:13)(cid:13)(cid:13) ( z − P (cid:15) ) − (cid:13)(cid:13)(cid:13) H k Λ →H k Λ ≤ z − C . In particular, (3.43) holds for such a z . We turn now to the proof of (3.44). To doso, we proceed as above, replacing the factor (cid:104)| α |(cid:105) k by (cid:104)| α |(cid:105) k + m , then as above wehave Re (cid:16) (cid:104)| α |(cid:105) k + m p Λ (cid:17) ( α ) ≤ C (cid:104)| α |(cid:105) k + m and Re (cid:16) (cid:104)| α |(cid:105) k + m σ ∆ (cid:17) ( α ) ≤ − C (cid:104)| α |(cid:105) k +2 m + C (cid:104)| α |(cid:105) k + m . It follows that (for a larger C > 0, the scalar product is in L (Λ))Re (cid:68) (cid:104)| α |(cid:105) k + m T Λ P (cid:15) u, T Λ u (cid:69) ≤ C (cid:107) u (cid:107) H k Λ (cid:107) u (cid:107) H m + k Λ − C (cid:15) (cid:107) u (cid:107) H m + k Λ . And, as in the proof of Lemma 3.3, we find that (cid:15) (cid:107) u (cid:107) H m + k Λ ≤ C (cid:16) (cid:107) P (cid:15) u (cid:107) H k Λ + (cid:107) u (cid:107) H k Λ (cid:17) , and (3.44) follows using (3.43). (cid:3) We want now to use Lemma 3.10 to deduce other resolvent bounds that areneeded for the proof of Theorem 9. The idea behind the proof of Lemma 3.11 belowis that for k , k ∈ R and θ ∈ [0 , 1] then the complex interpolation space [ H k Λ , H k Λ ] θ is H k θ Λ (with equivalent norms), where k θ = (1 − θ ) k + θk . The proof of this factis elementary and we will not provide it, since the interpolation bound that we needfor the proof of Lemma 3.11 follows directly from H¨older’s inequality. .3. PERTURBATIVE RESULTS 169 Lemma . Assume that h and τ are small enough. Let z be a large enoughpositive real number. Then there is a constant C > such that for every (cid:15) ∈ [0 , we have (recall that δ = 1 /s ) (cid:13)(cid:13)(cid:13) ( z − P ) − − ( z − P (cid:15) ) − (cid:13)(cid:13)(cid:13) H →H ≤ C(cid:15) δm , (3.45) and (cid:13)(cid:13)(cid:13) ( z − P (cid:15) ) − (cid:13)(cid:13)(cid:13) H →H δ Λ ≤ C. (3.46) Proof. Write( z − P ) − − ( z − P (cid:15) ) − = − (cid:15) ( z − P (cid:15) ) − ∆ ( z − P ) − . (3.47)From Proposition 2.4 and Lemma 3.3, we know that ∆ is bounded from H δ Λ to H δ − m Λ and that ( z − P ) − is bounded from H to H δ Λ . Hence, we see with (3.44)that, for some C > (cid:15) ∈ [0 , (cid:13)(cid:13)(cid:13) ( z − P (cid:15) ) − ∆ ( z − P ) − (cid:13)(cid:13)(cid:13) H →H δ Λ ≤ C(cid:15) , and (3.46) follows by (3.47).Notice that, since ∆ ( z − P ) − is bounded from H to H δ − m Λ , the estimate(3.45) will follow from (3.47) if we are able to prove that, for some C > (cid:15) ∈ [0 , (cid:13)(cid:13)(cid:13) ( z − P (cid:15) ) − (cid:13)(cid:13)(cid:13) H δ − m Λ →H ≤ C(cid:15) δm − . (3.48)To see that (3.48) holds, just notice that if u ∈ H δ − m Λ then by H¨older’s inequalitywe have (cid:13)(cid:13)(cid:13) ( z − P (cid:15) ) − u (cid:13)(cid:13)(cid:13) H ≤ (cid:13)(cid:13)(cid:13) ( z − P (cid:15) ) − u (cid:13)(cid:13)(cid:13) − δm H δ Λ (cid:13)(cid:13)(cid:13) ( z − P (cid:15) ) − u (cid:13)(cid:13)(cid:13) δm H δ − m Λ and apply Lemma 3.10. (cid:3) We are now ready to prove Theorem 9. Proof of Theorem 9. The proof relies on the results from [ Ban04 ]. Firstrecall that, λ ∈ σ ( P (cid:15) ) ∪ {∞} if and only if 1 / ( z − λ ) belongs to the spectrum of σ (( z − P (cid:15) ) − ). Hence, we have d z,H ( σ ( P ) ∪ {∞} , σ ( P (cid:15) ) ∪ {∞} ) = d H (cid:16) σ (cid:16) ( z − P ) − (cid:17) , σ (cid:16) ( z − P (cid:15) ) − (cid:17)(cid:17) , (3.49)where d H denotes the usual Hausdorff distance on compact subsets of C . Thenchoose p > n/δ = ns . It follows from Lemma 3.11 and Proposition 2.13 that( z − P (cid:15) ) − (seen as an endomorphism of H ) is uniformly in the Schatten class S p for (cid:15) ∈ [0 , Ban04 , Theorem 5.2] and (3.45) that,for some C > (cid:15) ∈ [0 , d H (cid:16) σ (cid:16) ( z − P (cid:15) ) − (cid:17) , σ (cid:16) ( z − P ) − (cid:17)(cid:17) ≤ Cf p (cid:32) (cid:15) − δm C (cid:33) − . Here, f p is the inverse of the function g p : R + (cid:51) x (cid:55)→ x exp (cid:0) a p x p + b p (cid:1) ∈ R + forsome a p , b p > 0. We see that f p ( x ) ∼ x → + ∞ (ln x ) p . Hence, for some new constant C > 0, we have that d H (cid:16) σ (cid:16) ( z − P (cid:15) ) − (cid:17) , σ (cid:16) ( z − P ) − (cid:17)(cid:17) ≤ C | ln (cid:15) | − p , and the result follows from (3.49). (cid:3) 70 3. RUELLE–POLLICOTT RESONANCES AND GEVREY ANOSOV FLOWS We want now to apply our machinery to study howthe Ruelle spectrum vary under a deterministic perturbation of the dynamics. Todo so, we consider a perturbation (cid:15) (cid:55)→ X (cid:15) of our vector field X = X . Here, the perturbation is defined for (cid:15) in a neighbourhoodof 0 and is assumed to be (at least) C ∞ from this neighbourhood of zero to a spaceof G s sections of the tangent bundle of M (see Remark 1.4). We can also considera perturbation (cid:15) (cid:55)→ V (cid:15) with the same features and then form for (cid:15) near zero theoperator P (cid:15) := X (cid:15) + V (cid:15) . (3.50)This kind of perturbation of the operator P is very different from the one weconsidered in the last section.Notice that, when (cid:15) is close enough to 0, then the vector field X (cid:15) generates anAnosov flow, so that the Ruelle spectrum of P (cid:15) is well-defined. It is then natural towonder how this spectrum varies with (cid:15) . The situation is pretty well-understood infinite differentiability and in the C ∞ case (see [ BL07, BL13, Bon18 ]).Let us consider the most simple example: in the C ∞ case, if P has a simpleresonances λ , then it will extend to a C ∞ family of simple resonances (cid:15) (cid:55)→ λ (cid:15) (provided that the perturbation is C ∞ ). However, even if the perturbation (cid:15) (cid:55)→ X (cid:15) is real-analytic in the C ∞ category, then we do not know that the family (cid:15) (cid:55)→ λ (cid:15) isreal-analytic (in fact, it is reasonable to expect that it is not). We will see belowthat if the perturbation (cid:15) (cid:55)→ X (cid:15) is real-analytic in the real-analytic category, thenthe family (cid:15) (cid:55)→ λ (cid:15) is real-analytic (this is an immediate consequence of Theorem11). Our first result is the following (the operator P (cid:15) is the one defined by (3.50)),and the Lagrangian Λ is the one that we introduced in § Proposition . Assume that δ > / (i.e. s < ). Let (cid:96) ∈ R + \ N . Assumethat h and τ are small enough. Then for (cid:15) small enough, the spectrum of P (cid:15) actingon H is the Ruelle spectrum of P (cid:15) . Moreover, if k = ( (cid:96) + 1) δ − (cid:96) and r is a largeenough positive real number, then the the map (cid:15) (cid:55)→ ( r − P (cid:15) ) − ∈ L (cid:16) H , H k Λ (cid:17) is C (cid:96) on a neighbourhood of . As in the case of stochastic perturbations, we may prove a global bound onthe way the spectrum of P (cid:15) tends to the spectrum of P . Indeed, using the samearguments as in § Corollary . Assume that δ > / (i.e. s < ). Then, for every p > ns and r ∈ R + large enough, there is a constant C > such that for every (cid:15) closeenough to we have d r,H ( σ ( P ) ∪ {∞} , σ ( P (cid:15) ) ∪ {∞} ) ≤ C | ln | (cid:15) || − p . Finally, in the real-analytic case we are able to improve Proposition 3.4 in thefollowing way. Theorem . Assume that s = 1 and that the perturbations (cid:15) (cid:55)→ X (cid:15) and (cid:15) (cid:55)→ V (cid:15) are real-analytic. Assume that h and τ are small enough. Then, for r ∈ R + largeenough, the map (cid:15) (cid:55)→ ( r − P (cid:15) ) − ∈ L (cid:16) H , H (cid:17) is real-analytic on a neighbourhood of zero. .3. PERTURBATIVE RESULTS 171 Theorem 11 allows to apply Kato theory on analytic perturbations of operators[ Kat66 ] and to deduce in particular Theorem 2.In order to prove Proposition 3.4, we recall that Proposition 2.5 allows us towrite for (cid:15) near 0: T Λ P (cid:15) S Λ = p Λ ,(cid:15) π Λ + R (cid:15) , (3.51)where p Λ ,(cid:15) is the restriction to Λ of an almost analytic extension of the principalsymbol of P (cid:15) and R (cid:15) is a bounded operator from L (Λ) to L − (Λ) (the bound isuniform when (cid:15) remains in a neighbourhood of 0).When constructing Λ, we took care to ensure that the real part of p Λ , isbounded from above. Unfortunately, it does not need to be true anymore when (cid:15) (cid:54) = 0. This issue will beget some technical difficulties: it is not clear anymore that P (cid:15) is the generator of a strongly continuous semi-group on H for instance. This iswhy we need the additional assumption δ > / G s semi-classical pseudor ∆. We denote by m the order of ∆ and assume for conveniencethat 1 < m ≤ δ . Then for ν ≥ (cid:15) near 0, we form the operator P (cid:15),ν = P (cid:15) + ν ∆ . According to Lemma 3.9, when ν > 0, the operator P (cid:15),ν has discrete spectrum on H , and this spectrum converges to the Ruelle spectrum of P when ν tends to 0(see the proof of Theorem 9). Notice that we can also apply Proposition 2.5 to ∆ inorder to write T Λ ∆ S Λ = σ ∆ π Λ + R ∆ , (3.52)where σ ∆ is a symbol of order m and the operator R ∆ is bounded between thespaces L (Λ) and L − m + (Λ). We need now to prove the following key lemma. Lemma . Assume that τ is small enough. There are r > and a constant C > , such that for every real number r ≥ r , every α ∈ Λ , every ν ≥ and (cid:15) near , we have (cid:12)(cid:12) r − p Λ ,(cid:15) ( α ) − νσ ∆ ( α ) (cid:12)(cid:12) ≥ C (cid:16) r + max (cid:16) (cid:104)| α |(cid:105) δ , ν (cid:104)| α |(cid:105) m (cid:17)(cid:17) . Proof. The definition of the Lagrangian Λ ensures that there are constants C, C > α ∈ Λ we haveRe σ ∆ ≤ − C (cid:104)| α |(cid:105) m + C, | Im σ ∆ ( α ) | ≤ C (cid:104)| α |(cid:105) m , and (cid:12)(cid:12) Re p Λ ,(cid:15) ( α ) (cid:12)(cid:12) ≤ C (cid:104)| α |(cid:105) . Moreover, the constant C > τ to besmall. In addition, the ellipticity conditions that are satisfied by p Λ , are preservedunder small perturbations, so that for (cid:15) small enough and up to making C largerwe have Re p Λ ,(cid:15) ( α ) ≤ − C (cid:104)| α |(cid:105) δ + C or (cid:12)(cid:12) Im p Λ ,(cid:15) (cid:12)(cid:12) ≥ C (cid:104)| α |(cid:105) − C. (3.53)We start by writing that (cid:12)(cid:12) r − p Λ ,(cid:15) − νσ ∆ (cid:12)(cid:12) ≥ √ (cid:0)(cid:12)(cid:12) r − Re p Λ ,(cid:15) − ν Re σ ∆ (cid:12)(cid:12) + (cid:12)(cid:12) Im p Λ ,(cid:15) + ν Im σ δ (cid:12)(cid:12)(cid:1) . (3.54) 72 3. RUELLE–POLLICOTT RESONANCES AND GEVREY ANOSOV FLOWS Now, if the first alternative holds in (3.53), we just write that (for ν ≤ r ≥ C ) r − Re p Λ ,(cid:15) ( α ) − ν Re σ ∆ ( α ) ≥ r + 1 C (cid:104)| α |(cid:105) δ − C + νC (cid:104)| α |(cid:105) m − C ≥ min (cid:18) , C (cid:19) (cid:16) r + max (cid:16) (cid:104)| α |(cid:105) δ , ν (cid:104)| α |(cid:105) m (cid:17)(cid:17) . Hence, we may focus on the second case in (3.53). In that case, we have (cid:12)(cid:12) Im p Λ ,(cid:15) ( α ) + ν Im σ ∆ ( α ) (cid:12)(cid:12) ≥ (cid:12)(cid:12) Im p Λ ,(cid:15) ( α ) (cid:12)(cid:12) − ν | Im σ ∆ ( α ) |≥ C (cid:104)| α |(cid:105) − C − νC (cid:104)| α |(cid:105) m . On the other hand, we still have the general bound (cid:12)(cid:12) r − Re p Λ ,(cid:15) ( α ) − ν Re σ ∆ ( α ) (cid:12)(cid:12) ≥ r − C (cid:104)| α |(cid:105) + νC (cid:104)| α |(cid:105) m − νC, and consequently (3.54) gives that (cid:12)(cid:12) r − p Λ ,(cid:15) ( α ) − νσ ∆ ( α ) (cid:12)(cid:12) ≥ √ (cid:18) r + (cid:18) C − C (cid:19) (cid:104)| α |(cid:105) + ν (cid:18) C − C (cid:19) (cid:104)| α |(cid:105) m − C (1 + ν ) (cid:19) . Then, by taking τ small enough, we ensure that C > C , and the result follows (weget rid of the term C (1 + ν ) by taking r large enough). (cid:3) Now, we use Lemma 3.12 to get a new construction for the resolvent (cid:0) r − P (cid:15),ν (cid:1) − (the positivity argument a priori does not work when ν = 0 and (cid:15) (cid:54) = 0). Lemma . Assume that h and τ are small enough. Then there is r suchthat, for (cid:15) and ν ≥ small enough, if r ≥ r then r belongs to the resolvent set of P (cid:15),ν acting on H (with its natural domain). Moreover, for every k ∈ R , if r > is large enough then the resolvent ( r − P (cid:15),ν ) − is bounded from H k Λ to H k + δ Λ withuniform bound in (cid:15), ν . Proof. Let r ∈ R + be large enough. The formulae (3.51) and (3.52) suggestto consider the following approximate inverse for r − P (cid:15),ν : A r,(cid:15),ν := S Λ r − p Λ ,(cid:15) − νσ ∆ T Λ . According to Lemma 3.12, if r is large enough and (cid:15) and ν are close enough to 0,the operator T r,(cid:15),ν is well-defined and is bounded from H to H δ Λ and from H − δ Λ to H with uniform bounds. Then, we compute A r,(cid:15),ν (cid:0) r − P (cid:15),ν (cid:1) = S Λ r − p Λ ,(cid:15) − νσ ∆ T Λ ( r − P (cid:15),ν ) S Λ T Λ = I + S Λ r − p Λ ,(cid:15) − νσ ∆ ( R (cid:15) + νR ∆ ) T Λ . In order to control the remainder term, notice using Lemma 3.12 that the multipli-cation by ( r − p Λ ,(cid:15) − νσ ∆ ) − is bounded from L − (Λ) to L (Λ) with normsup α ∈ Λ (cid:104)| α |(cid:105) r − p Λ ,(cid:15) ( α ) − νσ ∆ ( α ) ≤ C sup t ∈ R + tr + t δ ≤ C (2 δ − − δ δ r δ − . .3. PERTURBATIVE RESULTS 173 Similarly, if 0 < ν ≤ 1, then the multiplication by ( r − p Λ ,(cid:15) − νσ ∆ ) − is boundedfrom L − m + (Λ) to L (Λ) with normsup α ∈ Λ (cid:104)| α |(cid:105) m − r − p Λ ,(cid:15) ( α ) − νσ ∆ ( α ) ≤ Cν sup t ∈ R + tr + t mm − ≤ C m (2 m − m +12 m r − m ν . Hence, we see that the operator S Λ r − p Λ ,(cid:15) − νσ ∆ ( R (cid:15) + νR ∆ ) T Λ is bounded from H to itself with norm an O ( r − m + r δ − ) when r → + ∞ (uniformly in (cid:15) and ν near 0). Consequently, if r is large enough, we may invert theoperator I + S Λ r − p Λ ,(cid:15) − νσ ∆ ( R (cid:15) + νR ∆ ) T Λ by mean of Neumann series and (cid:18) I + S Λ r − p Λ ,(cid:15) − νσ ∆ ( R (cid:15) + νR ∆ ) T Λ (cid:19) − A r,(cid:15),ν is a left inverse for r − P (cid:15),ν . We construct similarly a right inverse for r − P (cid:15),ν .Indeed, in (3.51) we may replace p Λ ,(cid:15) π Λ by π Λ p Λ ,(cid:15) (with a different remainder ofcourse), and similarly in (3.52). Finally, if r is large enough, it belongs to theresolvent set of P (cid:15),ν for (cid:15) and ν near 0, and the resolvent writes (cid:0) r − P (cid:15),ν (cid:1) − = A r,(cid:15),ν (cid:16) I + (cid:101) R (cid:15),ν (cid:17) − , (3.55)where the operator (cid:101) R (cid:15),ν is bounded on H with norm less than . Proceeding asabove, we see that, for k ∈ R and r > (cid:101) R (cid:15),ν is alsobounded on H k Λ with norm less than and it follows from (3.55) and Lemma 3.12that (cid:0) r − P (cid:15),ν (cid:1) − is bounded from H k Λ to H k + δ Λ . (cid:3) Proof of Proposition 3.4. First of all, from Lemma 3.13, we know thatthe resolvent (cid:0) r − P (cid:15),ν (cid:1) − is compact and hence P (cid:15),ν has discrete spectrum on H .This fact holds in particular for P (cid:15) = P (cid:15), . In order to see that the spectrum of P (cid:15) acting on H coincides with its Ruelle spectrum, notice that (cid:0) r − P (cid:15),ν (cid:1) − = ( r − P (cid:15) ) − + ν ( r − P (cid:15) ) − ∆ (cid:0) r − P (cid:15),ν (cid:1) − . Then, using Proposition 2.4 and Lemma 3.13 (recall that the order of ∆ is lessthan 2 δ ), we see that (cid:0) r − P (cid:15),ν (cid:1) − converges to ( r − P (cid:15) ) − in operator norm when ν tends to 0. It follows that the spectrum of P (cid:15),ν converges to the spectrum of P (cid:15) ,but thanks to Lemma 3.9 and [ DZ15 , Theorem 1], the spectrum of P (cid:15),ν convergesto the Ruelle spectrum of P (cid:15) . Hence, the spectrum of P (cid:15) acting on H is the Ruellespectrum of P (cid:15) .We prove now the regularity of the map (cid:15) (cid:55)→ ( r − P (cid:15) ) − . We start with the case (cid:96) ∈ ]0 , 1[ by writing (cid:0) r − P (cid:15) (cid:48) (cid:1) − = ( r − P (cid:15) ) − + ( r − P (cid:15) ) − (cid:0) P (cid:15) (cid:48) − P (cid:15) (cid:1) (cid:0) r − P (cid:15) (cid:48) (cid:1) − . 74 3. RUELLE–POLLICOTT RESONANCES AND GEVREY ANOSOV FLOWS Then we notice that, for some constant C > 0, we have (cid:13)(cid:13)(cid:13)(cid:0) r − P (cid:15) (cid:48) (cid:1) − − ( r − P (cid:15) ) − (cid:13)(cid:13)(cid:13) H →H δ − ≤ (cid:13)(cid:13)(cid:13)(cid:0) r − P (cid:15) (cid:48) (cid:1) − (cid:13)(cid:13)(cid:13) H δ − →H δ − (cid:13)(cid:13) P (cid:48) (cid:15) − P (cid:15) (cid:13)(cid:13) H δ Λ →H δ − (cid:13)(cid:13)(cid:13)(cid:0) r − P (cid:15) (cid:48) (cid:1) − (cid:13)(cid:13)(cid:13) H →H δ Λ ≤ C (cid:12)(cid:12) (cid:15) − (cid:15) (cid:48) (cid:12)(cid:12) . (3.56)Here, we applied Proposition 2.4 and Lemma 3.13. It follows also from Lemma 3.13that, up to making C larger, we have (cid:13)(cid:13)(cid:13)(cid:0) r − P (cid:15) (cid:48) (cid:1) − − ( r − P (cid:15) ) − (cid:13)(cid:13)(cid:13) H →H δ Λ ≤ C. (3.57)Applying H¨older’s inequality as in the proof of Lemma 3.11, we deduce from (3.56)and (3.57) that, for (cid:15), (cid:15) (cid:48) near 0, we have (cid:13)(cid:13)(cid:13)(cid:0) r − P (cid:15) (cid:48) (cid:1) − − ( r − P (cid:15) ) − (cid:13)(cid:13)(cid:13) H →H (1 − (cid:96) ) δ + (cid:96) (2 δ − ≤ C (cid:12)(cid:12) (cid:15) − (cid:15) (cid:48) (cid:12)(cid:12) (cid:96) . The result is then proved when (cid:96) ∈ ]0 , (cid:96) ∈ ]1 , 2[ (the general result will follow by induction).Letting ˙ P (cid:15) denotes the derivative of (cid:15) (cid:55)→ P (cid:15) , we write, for (cid:15), (cid:15) (cid:48) near 0, (cid:0) r − P (cid:15) (cid:48) (cid:1) − − ( r − P (cid:15) ) − − (cid:0) (cid:15) (cid:48) − (cid:15) (cid:1) ( r − P (cid:15) ) − ˙ P (cid:15) ( r − P (cid:15) ) − = ( r − P (cid:15) ) − (cid:16) P (cid:15) (cid:48) − P (cid:15) − ( (cid:15) (cid:48) − (cid:15) ) ˙ P (cid:15) (cid:17) ( r − P (cid:15) ) − + ( r − P (cid:15) ) − (cid:0) P (cid:15) (cid:48) − P (cid:15) (cid:1) (cid:16)(cid:0) r − P (cid:15) (cid:48) (cid:1) − − ( r − P (cid:15) ) − (cid:17) . (3.58)From Proposition 2.4 and Taylor’s formula, we see that (provided τ is small enough) (cid:13)(cid:13)(cid:13) P (cid:15) (cid:48) − P (cid:15) − ( (cid:15) (cid:48) − (cid:15) ) ˙ P (cid:15) (cid:13)(cid:13)(cid:13) H δ Λ →H δ − ≤ C (cid:12)(cid:12) (cid:15) − (cid:15) (cid:48) (cid:12)(cid:12) , (3.59)and (cid:13)(cid:13) P (cid:15) (cid:48) − P (cid:15) (cid:13)(cid:13) H k +1 − δ Λ →H k − δ Λ ≤ C (cid:12)(cid:12) (cid:15) − (cid:15) (cid:48) (cid:12)(cid:12) . (3.60)Then, we find, applying the previous case (with (cid:96) − (cid:96) ), that (cid:13)(cid:13)(cid:13) ( r − P (cid:15) ) − − (cid:0) r − P (cid:15) (cid:48) (cid:1) − (cid:13)(cid:13)(cid:13) H →H k − δ Λ ≤ C (cid:12)(cid:12) (cid:15) (cid:48) − (cid:15) (cid:12)(cid:12) (cid:96) − . (3.61)Putting (3.59), (3.60), (3.61) and Lemma 3.13 in (3.58), we find that, for a newconstant C > (cid:15), (cid:15) (cid:48) near 0, we have (cid:13)(cid:13)(cid:13)(cid:0) r − P (cid:15) (cid:48) (cid:1) − − ( r − P (cid:15) ) − − (cid:0) (cid:15) (cid:48) − (cid:15) (cid:1) ( r − P (cid:15) ) − ˙ P (cid:15) ( r − P (cid:15) ) − (cid:13)(cid:13)(cid:13) H →H k Λ ≤ C (cid:12)(cid:12) (cid:15) (cid:48) − (cid:15) (cid:12)(cid:12) (cid:96) . It follows that the map (cid:15) (cid:55)→ ( r − P (cid:15) ) − ∈ L (cid:16) H , H k Λ (cid:17) is differentiable with deriva-tive (cid:15) (cid:55)→ ( r − P (cid:15) ) − ˙ P (cid:15) ( r − P (cid:15) ) − . (3.62)Moreover, this derivative is ( (cid:96) − (cid:15), (cid:15) (cid:48) near 0, we may write (cid:0) (cid:15) − (cid:15) (cid:48) (cid:1) (cid:16) ( r − P (cid:15) ) − ˙ P (cid:15) ( r − P (cid:15) ) − − (cid:0) r − P (cid:15) (cid:48) (cid:1) − ˙ P (cid:15) (cid:48) (cid:0) r − P (cid:15) (cid:48) (cid:1) − (cid:17) = (cid:16) ( r − P (cid:15) ) − − (cid:0) r − P (cid:15) (cid:48) (cid:1) − − (cid:0) (cid:15) − (cid:15) (cid:48) (cid:1) (cid:0) r − P (cid:15) (cid:48) (cid:1) − ˙ P (cid:15) (cid:48) (cid:0) r − P (cid:15) (cid:48) (cid:1) − (cid:17) + (cid:16)(cid:0) r − P (cid:15) (cid:48) (cid:1) − − ( r − P (cid:15) ) − − (cid:0) (cid:15) (cid:48) − (cid:15) (cid:1) ( r − P (cid:15) ) − ˙ P (cid:15) ( r − P (cid:15) ) − (cid:17) . .3. PERTURBATIVE RESULTS 175 To get the result in the case (cid:96) > 2, we proceed by induction. By applying thecase (cid:96) − 1, we find as above that the map (cid:15) (cid:55)→ ( r − P (cid:15) ) − ∈ L (cid:16) H , H k Λ (cid:17) is (cid:98) (cid:96) (cid:99) − (cid:98) l (cid:99) − (cid:88) (cid:96) + ··· + (cid:96) r = (cid:98) (cid:96) (cid:99)− a (cid:96) ,...,(cid:96) r ( r − P (cid:15) ) − d (cid:96) d (cid:15) (cid:96) ( P (cid:15) ) . . . ( r − P (cid:15) ) − d (cid:96) r d (cid:15) (cid:96) r ( P (cid:15) ) ( r − P (cid:15) ) − , where the a (cid:96) ,...,(cid:96) r ’s are integral coefficients. Now, reasoning as in the case (cid:96) ∈ ]1 , C (cid:96) −(cid:98) (cid:96) (cid:99)− with the expected derivative, endingthe proof of the proposition. (cid:3) Proof of Theorem 11. One could prove Theorem 11 by showing that (cid:15) (cid:55)→ ( r − P (cid:15) ) − is the sum of its Taylor series at 0 on a neighbourhood of 0. However,we will rather rely on Cauchy’s formula.From the analyticity assumption on (cid:15) (cid:55)→ P (cid:15) , we know that we may extendthis map to a holomorphic map on a neighbourhood of 0. It follows then fromProposition 2.4 that (cid:15) (cid:55)→ P (cid:15) defines a holomorphic family of operators from H to H on a neighbourhood of 0 (in particular, it satisfies Cauchy’s formula). Noticehowever that when (cid:15) is complex then X (cid:15) is a first order differential operator that maynot necessarily be interpreted as a vector field on M (this is a section of T M ⊗ C ).Working as in the proof of Proposition 3.4, we see that if r ∈ R + is large enoughthe, for (cid:15) in a complex neighbourhood of 0, the operator ( r − P (cid:15) ) − is well-definedand sends H into H with uniform bound. The fact that X (cid:15) is not necessarily avector field does not play any role here, since the proof of Proposition 3.4 in thecase s = 1 only relies on the fact that P (cid:15) is a small perturbation of P (this is truesince δ = 1 so that the ellipticity condition on the real part of p Λ ,(cid:15) is stable by smallperturbations). Consequently, we may define for (cid:15) near 0 the operator Q (cid:15) = 12 iπ (cid:90) γ ( r − P w ) − w − (cid:15) d w, where γ is a small circle around 0 in C . By interverting series and integral, we seethat Q (cid:15) is holomorphic in (cid:15) near 0. Moreover, for (cid:15) near 0, we have Q (cid:15) ( r − P (cid:15) ) = 12 iπ (cid:90) γ ( r − P w ) − w − (cid:15) (cid:18) iπ (cid:90) γ ( r − P z ) z − (cid:15) d z (cid:19) d w = 12 iπ (cid:90) γ w − (cid:15) (cid:18) iπ (cid:90) γ z − (cid:15) d z (cid:19) d w + 12 iπ (cid:90) γ ( r − P w ) − w − (cid:15) (cid:18) iπ (cid:90) γ ( P w − P z ) z − (cid:15) d z (cid:19) d w = I + 12 iπ (cid:90) γ ( r − P w ) − w − (cid:15) ( P w − P (cid:15) ) d w = I. Thus, Q (cid:15) = ( r − P (cid:15) ) − and the result follows. (cid:3) We deduce now Theorem 2 from Theorem 11. Proof of Theorem 11. The uniqueness of the SRB measure for (cid:15) near 0follows from [ BL07 , Lemma 5.1] and the fact that Ruelle resonances are intrinsicallydefined.From [ BL07 , Lemma 5.1], we also know that for (cid:15) near 0, the point 0 is a simpleeigenvalues for X (cid:15) acting on H , and that the associated normalized left eigenvector µ (cid:15) is the SRB measure for the Anosov flow generated by X (cid:15) . 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