An FBI characterization for Gevrey vectors on hypo-analytic structures and propagation of Gevrey singularities
AAN FBI CHARACTERIZATION FOR GEVREY VECTORS ON HYPO-ANALYTICSTRUCTURES AND PROPAGATION OF GEVREY SINGULARITIES
N. BRAUN RODRIGUES
Abstract.
In this work we prove a FBI characterization for Gevrey vectors on hypo-analytic structures,and we analyze the main differences of Gevrey regularity and hypo-analyticity concerning the FBItransform. We end with an application of this characterization on a propagation of Gevrey singularitiesresult, for solutions of the non-homogeneous system associated with the hypo-analytic structure, foranalytic structures of tube type. Introduction
In 1983 N. Hanges and F. Treves proved in [9] that on CR (embedded) manifolds, holomorphic ex-tendability for CR functions propagates along connected complex submanifolds. They actually provedtheir result on the set up of hypo-analytic structures, introduced by M.S. Baouendi, C.H. Chang, andF. Treves in [1], and they proved that hypo-analytic singularity of solutions propagates along connectedelliptic submanifolds. Propagation of holomorphic extendability is widely studied in the context of CRgeometry, for instance [12], [2] and [15]. Now for Gevrey regularity little is known concerning propaga-tion of singularities on hypo-analytic structures. In 2000 P. Caetano started the study of Gevrey vectorson hypo-analytic structures of maximum codimension in his Ph.D dissertation ([6]), and his work wascontinued in [7] and [11], but their aim was solvability questions for the associate differential complex.Our goal here is to initiate the study of regularity problems on these structures, for instance, propagationof Gevrey singularities.A very useful tool in the study of propagation of singularities is the FBI transform. Also in [1] theauthors proved that the decay of the FBI transform can be used to characterize hypo-analyticity. Theusual characterization of analytic regularity, Gevrey regularity (ultradifferential regularity) and smoothregularity of distributions on R N by the decay of the FBI transform differs from one another by the typeof their decay. Loosely speaking, a distribution u is analytic at x if | F [ χu ]( x, ξ ) | ≤ Ce − ε | ξ | , for all x in some neigborhood of x , all ξ ∈ R N , and for some positive constants C, ε , where χ is a testfunction supported in some open neighborhood of x , and F [ χu ]( x, ξ ) is the FBI transform of χu . Now u is Gevrey at x if | F [ χu ]( x, ξ ) | ≤ Ce − ε | ξ | s , for all x in some neighborhood of x , all ξ ∈ R N , and for some positive constants C, ε . So the differencebetween analytic regularity and Gevrey regularity in this context is the type of the bound. On hypo-analytic structures there is an additional difficulty that arises from its complex nature which remainsunseen when dealing with analytic regularity .For simplicity let M ⊂ C N be a smooth generic CR submanifold of codimension d , so the CR dimensionof M is n = N − d , and let p be an arbitrary point of M . Therefore there are L , . . . , L n anti-holomorphicvector fields tangent to M on a neighborhood of p , and real vector fields T , . . . , T d tangent to M ona neighborhood of p such that { L , . . . , L n , L , . . . , L n , T , . . . , T d } span the complexified tangent bundleof M on a neighborhood of p . In this set up our main theorem states that for a CR function u to be aGevrey vector for L , . . . , L n , T , . . . , T d it is necessary and sufficient that its FBI transform has the same Date : March 2020. In the context of hypo-analytic structures, by analytic regularity we mean hypo-analyticity a r X i v : . [ m a t h . C V ] J un N. BRAUN RODRIGUES bound as in the R N scenario, but only for points on the so-called real structure bundle, which is a realsubbundle of (T , M ) ⊥ (cf. section 2 . . ). Here one might notice that we are not asking any additionalregularity on the CR structure.Our initial goal was to investigate the validity of the propagation of singularities result proved in [9]for Gevrey regularity. One difficulty is that this result is deeply based on holomorphic function theory,which is not available for the Gevrey case. One of the drawbacks, when using the same techniques(the FBI approach), is that in our case we need some sort of folliation near the ”propagator”, an un-necessary assumption on [9]. On the other hand, this technique allows us to consider solutions of thenon-homogeneous system, which makes sense in the Gevrey scenario.This paper is organized as follows: In the first section we discuss what is needed from locally integrablestructures theory, and hypo-analytic structures theory. Then in section 3 we prove a FBI characterizationfor Gevery vectors, and in the last section we prove (and give some examples) a propagation of Gevreysingularities result on hypo-analytic structures of tube type.This work contains the results obtained by the author on his Ph.D. dissertation.2. Preliminaries
Locally integrable structures.
Let Ω ⊂ R N be an open set. By a locally integrable structure onΩ we mean a complex vector bundle V ⊂ C TΩ, such that [ V , V ] ⊂ V , and at every point p ∈ Ω there are Z , . . . , Z m , smooth, complex-valued functions in some open neighborhood of p in Ω, such that (cid:40) d Z ∧ · · · ∧ d Z m (cid:54) = 0;L Z j = 0 , ∀ L ∈ V , j = 1 , . . . , m. We denote by T (cid:48) ⊂ C T ∗ Ω the orthogonal bundle, with respect to the duality between forms and vectors,of the bundle V . Let p be an arbitrary point at Ω. Then there exist a local coordinate system vanishingat p on some open set U = V × W , ( x , . . . , x m , t , . . . , t n ), and smooth, real-valued functions φ , . . . , φ m ,defined on U and satisfying φ (0) = 0 and d x φ (0) = 0, such that the differentials of the functions(2.1) Z k ( x, t ) . = x k + iφ k ( x, t ) , k = 1 , . . . , m, span T (cid:48) in U . There are also linear independent, pairwise commuting, complex vector fields:M j = m (cid:88) k =1 a j,k ( x, t ) ∂∂x k , j = 1 , . . . , m, and L j = ∂∂t j − i m (cid:88) k =1 ∂φ k ∂t j ( x, t )M k , j = 1 , . . . , n, satisfying the relations L j Z k = 0 M l Z k = δ l,k L j t i = δ j,i M l t i = 0.Now let u be a distribution on U such that L j u ∈ C ∞ ( U ), for j = 1 , . . . , n , then actually u ∈C ∞ ( W ; D (cid:48) ( V )) (see proof of Proposition I.4 . K (cid:98) V , and K (cid:98) W , there exist aconstant C > q > |(cid:104) u ( x, t ) , φ ( x ) (cid:105)| ≤ C (cid:88) | α |≤ q sup x ∈ K | ∂ α φ | , ∀ φ ∈ C ∞ c ( K ) , for every t ∈ K .Now let H ⊂
Ω be a (embedded) submanifold. We say that H is maximally real if C T p Ω = V p ⊕ C T p H , ∀ p ∈ H , or equivalently, ROPAGATION OF SINGULARITIES FOR NON HOMOGENEOUS SYSTEMS 3 C T ∗ p Ω = C N ∗ p H ⊕ T (cid:48) p , ∀ p ∈ H . Hypo-analytic structures.
Let Ω ⊂ R N be an open set. A hypo-analytic structure on Ω is a pair { ( U α ) α ∈ Λ , ( Z α ) α ∈ Λ } such that • ( U α ) α ∈ Λ is an open covering for Ω; • Z α : U α −→ C m is a smooth map, for every α ∈ Λ; • d Z α, , . . . , d Z α,m are C − linear independent on U α , for every α ∈ Λ; • if α (cid:54) = β , then to each p ∈ U α ∩ U β there is a holomorphic map F such that Z α = F ◦ Z β , in aneighborhood of p in U α ∩ U β ; • if Z : U −→ C m is a smooth function such that for every p ∈ U ∩ U α there exists a holomorphicfunction F such that Z = F ◦ Z α , then ( U, Z ) = ( U β , Z β ), for some β ∈ Λ.We call each pair ( U α , Z α ) as a hypo-analytic chart. We say that a distribution u ∈ D (cid:48) (Ω) is hypo-analytic at p if for some α ∈ Λ such that p ∈ U α , there is a holomorphic function F , defined on a complexneighborhood of Z α ( p ), such that u = F ◦ Z α , in some open neighborhood of p . Given a hypo-analyticstructure on Ω we can associate a locally integrable structure V setting its orthogonal T (cid:48) as the complexbundle locally defined by the differentials d Z , . . . , d Z m . So let p ∈ Ω and (
U, Z ) a hypo-analytic chart,with p ∈ U . We can assume that there are local coordinates ( x , . . . , x m , t , . . . , t n ) in U = V × W , asdescribed in the section above, so the function Z is given by (2.1). Note that in this coordinate system,the point p is the origin. Definition 2.1.
Let s >
1. We say that a distribution u on U is a Gevrey- s vector if u is a smoothfunction on U , and for every compact set K ⊂ U there exists a constant C > ( x,t ) ∈ K | L α M β u ( x, t ) | ≤ C | α | + | β | +1 α ! s β ! s , ∀ α ∈ Z n + , β ∈ Z m + . We denote by G s ( U ; L , · · · , L n , M , · · · , M m ) the space of all Gevrey- s vectors on U . If s = 1 we saythat u is an analytic vector, and we write u ∈ C ω ( U ; L , · · · , L n , M , · · · , M m )).We have the following characterization of Gevrey vectors in terms of almost-analytic extensions: Theorem 2.2 (Theorem 1 . . Let u be a distribution on U and U = V × W (cid:98) U , where V and W are balls centered at the origin. Are equivalent: (1) u is a Gevrey- s vector on U ; (2) There exist O an open neighborhood of ( Z ( U ) , W ) on C n + m and a Gevrey function F ∈ G s ( O ) such that (cid:40) F ( Z ( x, t ) , t ) = u ( x, t ) , ∀ ( x, t ) ∈ U ; (cid:0) ∂ z + ∂ τ (cid:1) F ( z, τ ) ∼ , on ( Z ( U ) , W ) . Here f ∼ f is flat on Σ. A useful consequence of this theorem, that we shall use lateron, is the following: Corollary 2.3.
Let u be a distribution on U and U = V × W (cid:98) U , where V and W are openballs centered at the origin. Suppose that u | U ∈ G s ( U ; L , . . . L n , M , . . . , M m ) . Then there are an openneighborhood O of { Z ( x, t ) : x ∈ V , t ∈ W } on C m and a smooth function F ∈ C ∞ ( O × W ) such that (2.3) (cid:40) u ( x, t ) = F ( Z ( x, t ) , t ) , ( x, t ) ∈ V × W (cid:12)(cid:12) ∂ z F ( z, t ) (cid:12)(cid:12) ≤ C k +1 k ! s − dist( z ; W t ) k , ∀ k ∈ Z + , z ∈ O , t ∈ W , where C is a positive constant, and W t = { Z ( x, t ) : x ∈ V } . N. BRAUN RODRIGUES
This corollary is a consequence of previous theorem and the Taylor formula. Despite the difference betweenGevrey and analytic vectors beeing a power of s in their definition, they have very different properties.To illustrate this difference let us recall some well-known properties of hypo-analytic functions:Let u be a distribution on U such that L j u = 0, j = 1 , . . . , n . Then it is equivalent (see [14]):(1) u is hypo-analytic at the origin;(2) the restriction of u to a maximally real submaniold, passing through the origin, is hypo-analyticat the origin (with respect to the induced hypo-analytic structure);(3) u is an analytic vector in some open neighborhood of the origin.Let us prove that (2) ⇒ (1). So let H be a maximally real submanifold such that u | H is hypo-analytic at p . Then there exists U H an open neighborhood of p on H , and a holomorphic function F defined on O ,an open neighborhood of Z ( U H ) on C m , such that u ( p (cid:48) ) = F ( Z ( p (cid:48) )) , ∀ p (cid:48) ∈ U H . Now set ˜ u . = F ◦ Z , defined in some neighborhood of p on Ω. Since F is holomorphic we have that L ˜ u = 0, for every L ∈ V , so the same is valid for u − ˜ u , and u − ˜ u vanishes on a neighborhood of p on H . By a standard uniqueness result, based on the Baouendi-Treves approximation formula, we have that u − ˜ u vanishes on some neighborhood of p , i.e. , u is hypo-analytic at p . Note that a key ingredient ofthis argument is that the composition of a holomorphic function with the first integrals Z s are solutionsin a full neighborhood of p . So in the Gevrey scenario, where the function F would be a Gevrey functionsuch that ∂ z F is flat on Z ( U H ), we do not have this same phenomena anymore, that is, F ◦ Z is not asolution on a full neighborhood of p , just on U H , so we cannot apply the uniqueness result in this case.Conclusion: For Gevrey regularity, testing on maximally real submanifolds is not enough.2.3. The real structure bundle and the FBI transform.
An object that plays a central role in theanalysis on hypo-analytic structures is the so-called real structure bundle. It allows us to mimic some”real techniques” on this complex scenario. Let
H ⊂ C m be a maximally real submanifold ( i.e. , therestriction of the coordinate functions z , . . . , z m to H defines an hypo-analytic structure of co-rank 0).Suppose that the origin belongs to H , so H is locally the image of the map Z ( x ) = x + iφ ( x ) , where the function φ is real-valued, φ (0) = 0 and d φ (0) = 0. The real structure bundle of H is locallydefined as R T (cid:48)H = { ( Z ( x ) , t Z x ( x ) − ξ ) : x ∈ U, ξ ∈ R m } , where U is the open neighborhood of the origin where the map Z is defined. For every κ > C κ . = { ζ ∈ C m : | Im ζ | < κ | Re ζ |} . If ζ ∈ C m we write (cid:104) ζ (cid:105) . = ζ · ζ = ζ + · · · + ζ m . Using the main branch of the square root we can define (cid:104) ζ (cid:105) = [ (cid:104) ζ (cid:105) ] / , for ζ ∈ C κ , . Definition 2.4.
We shall say that the maximally real submanifold H of C m at one of its points, p , iswell positioned if there is a number κ , 0 < κ <
1, and an open neighborhood U of p in H such that ∀ z, z (cid:48) ∈ U, ζ ∈ R m , and ζ ∈ ( R T (cid:48)H | z ) ∩ ( R T (cid:48)H | z (cid:48) ) then , (2.4) (cid:40) | Im ζ | < κ | Re ζ | ;Im (cid:8) ζ · ( z − z (cid:48) ) + i (cid:104) ζ (cid:105)(cid:104) z − z (cid:48) (cid:105) (cid:9) ≥ (1 − κ ) | ζ || z − z (cid:48) | . We shall say that H is very well-positioned at p if, given any 0 < κ <
1, there is an open neighborhood U of p in H such that (2.4) is valid.After applying a biholomorphism we can always assume that a maximally real submanifold is very wellpositioned at p . Now we recall the definition of the FBI transform on maximally real submanifolds: ROPAGATION OF SINGULARITIES FOR NON HOMOGENEOUS SYSTEMS 5
Definition 2.5.
Let u ∈ E (cid:48) ( H ). So we define F [ u ]( z, ζ ) . = (cid:68) u ( z (cid:48) ) , e iζ · ( z − z (cid:48) ) −(cid:104) ζ (cid:105)(cid:104) z − z (cid:48) (cid:105) ∆( z − z (cid:48) , ζ ) (cid:69) , for z ∈ C m and ζ ∈ C , where ∆( z, ζ ) = det(Id + i ( z (cid:12) ζ ) / (cid:104) ζ (cid:105) ) , and ( z (cid:12) ζ ) = ( z i ζ j ) i,j =1 , ··· ,m .The real structure bundle is essential when it comes to estimates, as on can see in the next proposition(Proposition IX . . . and Proposition IX . . . of [14]): Proposition 2.6.
Let u ∈ E (cid:48) ( H ) . Then F [ u ]( z, ζ ) is holomorphic in ( z, ζ ) ∈ C m × C and for every K ⊂ C m compact set and every < κ < , there are constants C, R > such that | F [ u ]( z, ζ ) | ≤ Ce R | ζ | , ∀ z ∈ K, ζ ∈ C κ . Now if in addition H is well positioned at p ∈ H , then there exists an open neighborhood U of p on H such that if the support of u is contained in U , there exist an integer k > and a constant C > suchthat | F [ u ]( z, ζ ) | ≤ C (1 + | ζ | ) k , ( z, ζ ) ∈ R T (cid:48)H | U , where ( z, ζ ) ∈ R T (cid:48)H | U means that z ∈ U and ζ ∈ R T (cid:48)H | z . The FBI transform can be used to characterize holomorphic extendability, as well as other kinds ofextandabilities, as we shall see later on. To prove holomorphic extendability using the FBI transform oneneeds the following inversion formula:
Proposition 2.7 (Proposition IX . . . of [14]) . Let u ∈ E (cid:48) ( H ) . Then (2.5) u ( z ) = lim ε → + π ) m (cid:90) R m e − ε | ξ | F [ u ]( z, ξ )d ξ, where the convergence is in D (cid:48) ( H ) . So with this inversion formula one can prove the following theorem:
Theorem 2.8 (Theorem IX . . . of [14]) . Let u ∈ E (cid:48) ( H ) and p ∈ H . The following are equivalent: (1) There exists
O ⊂ C m , an open neighborhood of p , F a holomorphic function at O , such that u | O∩ Σ = F | O∩ Σ ; (2) There exists
O ⊂ C m , an open neighborhood of p , < κ (cid:48) < , and C, ε > such that | F [ u ]( z, ζ ) | ≤ Ce − ε | ζ | , ∀ ( z, ζ ) ∈ O × C κ (cid:48) (3) There exists
O ⊂ C m , an open neighborhood of p , such that F [ u ]( z, ξ ) is bounded by an integrablefunction with respect to ξ ∈ R m , uniformly in z ∈ O . The third equivalence of this theorem does not appear in the literature, but it is how actually one provethat (2) implies (1), using the inversion formula (2.5). This simple observation illustrates the advantageof having holomorphic function theory at our disposal. In this cases the FBI decay is not very importantbecause we have the control of it on a full neighborhood of p . Now if one wants to measure, for instance,smooth regularity with the FBI transform, then the estimate and where the estimate takes place are bothvery important. Theorem 2.9 (Theorem IX . . . Let u ∈ E (cid:48) ( H ) and p ∈ H . Then are equivalent: (1) u is C ∞ near p ; (2) There exists U a neighborhood of p , such that for every k ∈ Z + there is a C k > , such that | F [ u ]( z, ζ ) | ≤ C k (1 + | ζ | ) − k , ∀ ( z, ζ ) ∈ R T (cid:48)H | U . N. BRAUN RODRIGUES A FBI characterization of Gevrey vectors
Since the main result of this section, Theorem 3.6, is a local result, we will fix an arbitrary point at Ω,and for simplicity we shall call it the origin. As we saw in the previous section, there is a hypo-analyticchart (
U, Z ( x, t ) , · · · , Z m ( x, t )), with 0 ∈ U , and we can assume that the Z ’s are given by Z j ( x, t ) = x j + iφ j ( x, t ) , j = 1 , · · · , m, with ( x, t ) ∈ U , where the map φ ( x, t ) = ( φ ( x, t ) , · · · , φ m ( x, t )) is smooth, real-valued, φ (0) = 0 andd x φ (0) = 0. We can associate to it the complex vector fields { M , · · · , M m , L , · · · , L n } with the followingproperties: L j Z k = 0 M j Z k = δ j,k L j t k = δ j,k M j t k = 0.We can also assume that(3.1) | φ j ( x, t ) | ≤ C ( | x | + | t | ) , j = 1 , · · · , m, for some positive constant C , and(3.2) | φ ( x, t ) − φ ( x (cid:48) , t ) | ≤ µ | x − x (cid:48) | , with 0 < µ small as we want, for instance, less than 1 (see pg. 433 of [14]). From now on we are goingto assume that U = V × W , where V ⊂ R m and W ⊂ R n are balls centered at the origin. Under thisassumptions we can assume that for some 0 < κ < c > ∀ x, x (cid:48) ∈ V, t ∈ W, ξ ∈ R m , if ζ = t Z x ( x, t ) − ξ then , (3.3) (cid:40) | Im ζ | < κ | Re ζ | ;Im (cid:8) ζ · ( Z ( x, t ) − Z ( x (cid:48) , t )) + i (cid:104) ζ (cid:105)(cid:104) Z ( x, t ) − Z ( x (cid:48) , t ) (cid:105) (cid:9) ≥ c | ζ || Z ( x, t ) − Z ( x (cid:48) , t ) | . For every t ∈ W we define the maximally real submaniold W t as W t . = { Z ( x, t ) : x ∈ V } ⊂ C m , and we can write the real structure bundle of W t as R T (cid:48) | W t = { ( Z ( x, t ) , t Z x ( x, t ) − ξ : x ∈ V, ξ ∈ R m \ } . We can also assume that(3.4) Im (cid:26) ζ · ( Z ( x, t ) − Z ( x (cid:48) , t )) + i (cid:104) ζ (cid:105)(cid:104) Z ( x, t ) − Z ( x (cid:48) , t ) (cid:105) (cid:27) ≥ c | ζ || Z ( x, t ) − Z ( x (cid:48) , t ) | , for every x, x (cid:48) ∈ V , t ∈ W , ζ ∈ R T (cid:48) W t (cid:12)(cid:12) Z ( x,t ) ∪ R T (cid:48) W t (cid:12)(cid:12) Z ( x (cid:48) ,t ) . One consequence of (3.3) is that for every ζ ∈ ζ ∈ R T (cid:48) W t (cid:12)(cid:12) Z ( x,t ) the following is valid:(3.5) Re (cid:104) ζ (cid:105) ≥ (cid:114) − κ κ | ζ | and Im (cid:104) ζ (cid:105) ≤ | ζ | . Definition 3.1.
Let u ∈ C ∞ ( W ; E (cid:48) ( V )) and λ >
0. We define the FBI transform of u as F λ [ u ]( t ; z, ζ ) = (cid:90) V e iζ · ( z − Z ( x (cid:48) ,t )) − λ (cid:104) ζ (cid:105)(cid:104) z − Z ( x (cid:48) ,t ) (cid:105) u ( x (cid:48) , t )∆( λ ( z − Z ( x (cid:48) , t )) , ζ ))d Z ( x (cid:48) , t ) , with z ∈ C m and ζ ∈ C \ ROPAGATION OF SINGULARITIES FOR NON HOMOGENEOUS SYSTEMS 7
If we denote by (cid:101) u ( z, t ) = u ( x, t ), for z = Z ( x, t ), then we can write F λ [ u ]( t ; z, ζ ) = (cid:90) W t e iζ · ( z − z (cid:48) ) − λ (cid:104) ζ (cid:105)(cid:104) z − z (cid:48) (cid:105) (cid:101) u ( z (cid:48) , t )∆( λ ( z − z (cid:48) ) , ζ )d z (cid:48) . Note that the integral is to be understood in the dual sense. Since u has compact support in x we havethat F λ [ u ]( t ; z, ζ ) is holomorphic with respect to ( z, ζ ) ∈ C m × C \
0, and C ∞ with respect to t . Forsimplicity we write F [ u ]( t ; z, ζ ), for λ = 1. As in 2.6, we have the following bound for the FBI transform: Lemma 3.2.
Let u ∈ C ∞ ( W, E (cid:48) ( V )) . Then there exist C > and k ∈ Z + such that (3.6) | F [ u ]( t ; z, ζ ) | ≤ C (1 + | ζ | ) k , ∀ ( z, ζ ) ∈ R T (cid:48) W t . Every characterization via control of the decay/growth of the FBI transform is based on an inversionformula. The one that we will use here is not quite the same as in [14], so we will present its proof. Westart recalling the following inversion formula (usefull when dealing with holomorphic extendability, see[14]):
Proposition 3.3.
Let u ∈ C ∞ ( W ; E (cid:48) ( V )) . For every ε > set (3.7) u ε ( x, t ) . = 1(2 π ) m (cid:90) R T (cid:48) W t | Z ( x,t ) e − ε (cid:104) ζ (cid:105) F [ u ]( t ; Z ( x, t ) , ζ )d ζ. Then u ε ( x, t ) → u ( x, t ) in C ∞ ( W ; D (cid:48) ( V )) .Remark . The integral (3.7) is to be interpreted aslim δ → + (cid:90) { ζ ∈ R T (cid:48) W t | Z ( x,t ) : | ζ | >δ } e − ε (cid:104) ζ (cid:105) F [ u ]( t ; Z ( x, t ) , ζ )d ζ. We shall use this inversion formula to prove the one that we will actually use. But before doing so weneed to extend the function Z ( x, t ) with respect to the variable x to the whole R m :Let V (cid:98) V and let ψ ∈ C ∞ c ( V ) satisfying 0 ≤ ψ ≤ ψ ≡ V . Define (cid:101) Z ( x, t ) . = x + iψ ( x ) φ ( x, t ). Then (cid:101) Z defines the hypo-analytic structure in V × W , but (cid:101) Z ( x, t ) isdefined for all x ∈ R m . Also note that t (cid:101) Z x ( x, t ) − = (Id − i t ( ψφ ) x ( x, t ))(Id + t ( ψφ ) x ( x, t ) ) − . We canchoose V , W small enough so that t (cid:101) Z x ( x, t ) is invertible for all x ∈ R m and t ∈ W . From now on weshall write Z ( x, t ) instead of (cid:101) Z ( x, t ), and V and W instead of V and W . So now R T (cid:48) W t = { ( z, ζ ) ∈ C m × C : z = Z ( x, t ) , ζ = t Z x ( x, t ) − ξ for some ( x, ξ ) ∈ R m × R m } , and we can also assume that the inequality (3.2) is valid for all x ∈ R m . Note that (3.3) is still valid for( x, t ) ∈ V × W . Now we can prove the following inversion formula for the FBI transform: Theorem 3.5.
Let u ∈ C ∞ ( W ; E (cid:48) ( V )) . Then (3.8) u ( x, t ) = lim ε → + π ) m (cid:90) (cid:90) R T (cid:48) W t e iζ · ( Z ( x,t ) − z (cid:48) ) −(cid:104) ζ (cid:105)(cid:104) Z ( x,t ) − z (cid:48) (cid:105) − ε (cid:104) ζ (cid:105) F [ u ]( t ; z (cid:48) , ζ ) (cid:104) ζ (cid:105) m d z (cid:48) ∧ d ζ, where the convergence takes place in C ∞ ( W ; D (cid:48) ( V ))The proof of this theorem is very close to the one of Lemma IX.4 . . of [14]. N. BRAUN RODRIGUES
Proof.
For simplicity let us assume that u is a continuous function. Then for every ε > π ) m (cid:90) (cid:90) R T (cid:48) W t (cid:90) V e iζ · ( z (cid:48) − Z ( x (cid:48)(cid:48) ,t )) −(cid:104) ζ (cid:105)(cid:104) z (cid:48) − Z ( x (cid:48)(cid:48) ,t ) (cid:105) e iζ · ( Z ( x,t ) − z (cid:48) ) −(cid:104) ζ (cid:105)(cid:104) Z ( x,t ) − z (cid:48) (cid:105) − ε (cid:104) ζ (cid:105) ·· u ( x (cid:48)(cid:48) , t ) (cid:104) ζ (cid:105) m ∆( z (cid:48) − Z ( x (cid:48)(cid:48) , t ) , ζ )d Z ( x (cid:48)(cid:48) , t )d z (cid:48) ∧ d ζ == 1(2 π ) m (cid:90) (cid:90) (cid:90) V × R T (cid:48) W t | Z ( x (cid:48) ,t ) × R m e iζ · ( Z ( x,t ) − Z ( x (cid:48)(cid:48) ,t )) −(cid:104) ζ (cid:105)(cid:104) Z ( x (cid:48) ,t ) − Z ( x (cid:48)(cid:48) ,t ) (cid:105) −(cid:104) ζ (cid:105)(cid:104) Z ( x,t ) − Z ( x (cid:48) ,t ) (cid:105) − ε (cid:104) ζ (cid:105) ·· u ( x (cid:48)(cid:48) , t ) (cid:104) ζ (cid:105) m ∆( Z ( x (cid:48) , t ) − Z ( x (cid:48)(cid:48) , t ) , ζ )d Z ( x (cid:48)(cid:48) , t )d ζ d Z ( x (cid:48) , t )First we change the domain of the integration in the variable ζ from R T (cid:48) W t (cid:12)(cid:12) Z ( x (cid:48) ,t ) to R T (cid:48) W t (cid:12)(cid:12) Z ( x,t ) . So wecan change the order of integration and integrate in Z ( x (cid:48) , t ) first, and we shall calculate(3.9) (cid:104) ζ (cid:105) m π m (cid:90) R m e −(cid:104) ζ (cid:105) [ (cid:104) Z ( x (cid:48) ,t ) − Z ( x (cid:48)(cid:48) ,t ) (cid:105) + (cid:104) Z ( x,t ) − Z ( x (cid:48) ,t ) (cid:105) ] ∆( Z ( x (cid:48) , t ) − Z ( x (cid:48)(cid:48) , t ) , ζ )d Z ( x (cid:48) , t ) . To do so we start noticing that ω m π m (cid:90) R m e − ω (cid:104) Z ( x (cid:48) ,t ) − z (cid:105) d Z ( x (cid:48) , t ) = 1 , for every ω ∈ C , with Re ω >
0, and z ∈ C m , here note that the imaginary part of Z ( x, t ) has compactsupport, and also that ω m π m (cid:90) R m e − ω (cid:104) Z ( x (cid:48) ,t ) (cid:105) P ( Z ( x (cid:48) , t ))d Z ( x (cid:48) , t ) = 0 , for every P ( z ) polynomial such that it has degree one (exactly one) when viewed as a polynomial in eachvariable separately (in view of Fubini’s Theorem). Therefore ω m π m (cid:90) R m e − ω (cid:104) Z ( x (cid:48) ,t ) − z (cid:105) ∆( Z ( x (cid:48) , t ) − ˜ z, ζ )d Z ( x (cid:48) , t ) = ∆( z − ˜ z, ζ ) , for every z, ˜ z ∈ C m . To use this identity we must rewrite (cid:104) Z ( x (cid:48) , t ) − Z ( x (cid:48)(cid:48) , t ) (cid:105) + (cid:104) Z ( x, t ) − Z ( x (cid:48) , t ) (cid:105) . Westart noticing that (cid:28) Z ( x (cid:48) , t ) − ( Z ( x, t ) + Z ( x (cid:48)(cid:48) , t ))2 (cid:29) = (cid:28) Z ( x (cid:48) , t ) − Z ( x, t )2 + Z ( x (cid:48) , t ) − Z ( x (cid:48)(cid:48) , t )2 (cid:29) = 14 (cid:104) Z ( x (cid:48) , t ) − Z ( x, t ) (cid:105) + 14 (cid:104) Z ( x (cid:48) , t ) − Z ( x (cid:48)(cid:48) , t ) (cid:105) ++ 12 ( Z ( x (cid:48) , t ) − Z ( x, t )) · ( Z ( x (cid:48) , t ) − Z ( x (cid:48)(cid:48) , t ))= 14 (cid:104) Z ( x (cid:48) , t ) − Z ( x, t ) (cid:105) + 14 (cid:104) Z ( x (cid:48) , t ) − Z ( x (cid:48)(cid:48) , t ) (cid:105) −−
12 ( Z ( x, t ) − Z ( x (cid:48) , t )) · ( Z ( x (cid:48) , t ) − Z ( x (cid:48)(cid:48) , t )) . So we have obtained the following identity (cid:104) Z ( x (cid:48) , t ) − Z ( x, t ) (cid:105) + (cid:104) Z ( x (cid:48) , t ) − Z ( x (cid:48)(cid:48) , t ) (cid:105) = 4 (cid:28) Z ( x (cid:48) , t ) − ( Z ( x, t ) + Z ( x (cid:48)(cid:48) , t ))2 (cid:29) ++ 2( Z ( x, t ) − Z ( x (cid:48) , t )) · ( Z ( x (cid:48) , t ) − Z ( x (cid:48)(cid:48) , t )) . Also note that
ROPAGATION OF SINGULARITIES FOR NON HOMOGENEOUS SYSTEMS 9 (cid:104) Z ( x, t ) − Z ( x (cid:48)(cid:48) , t ) (cid:105) = (cid:104) Z ( x, t ) − Z ( x (cid:48) , t ) + Z ( x (cid:48) , t ) − Z ( x (cid:48)(cid:48) , t ) (cid:105) = (cid:104) Z ( x (cid:48) , t ) − Z ( x, t ) (cid:105) + (cid:104) Z ( x (cid:48) , t ) − Z ( x (cid:48)(cid:48) , t ) (cid:105) ++ 2( Z ( x, t ) − Z ( x (cid:48) , t )) · ( Z ( x (cid:48) , t ) − Z ( x (cid:48)(cid:48) , t )) . Summing up these two identities we have that (cid:104) Z ( x (cid:48) , t ) − Z ( x, t ) (cid:105) + (cid:104) Z ( x (cid:48) , t ) − Z ( x (cid:48)(cid:48) , t ) (cid:105) =(3.10) = 2 (cid:28) Z ( x (cid:48) , t ) − ( Z ( x, t ) + Z ( x (cid:48)(cid:48) , t ))2 (cid:29) ++ 12 (cid:104) Z ( x, t ) − Z ( x (cid:48)(cid:48) , t ) (cid:105) . Now we can calculate (3.9): (cid:104) ζ (cid:105) m π m (cid:90) R m e −(cid:104) ζ (cid:105) [ (cid:104) Z ( x (cid:48) ,t ) − Z ( x (cid:48)(cid:48) ,t ) (cid:105) + (cid:104) Z ( x,t ) − Z ( x (cid:48) ,t ) (cid:105) ] ∆( Z ( x (cid:48) , t ) − Z ( x (cid:48)(cid:48) , t ) , ζ )d Z ( x (cid:48) , t ) == e − (cid:104) ζ (cid:105)(cid:104) Z ( x,t ) − Z ( x (cid:48)(cid:48) ,t ) (cid:105) (cid:104) ζ (cid:105) m π m (cid:90) R m e − (cid:104) ζ (cid:105) (cid:68) Z ( x (cid:48) ,t ) − ( Z ( x,t )+ Z ( x (cid:48)(cid:48) ,t ))2 (cid:69) ·· ∆( Z ( x (cid:48) , t ) − Z ( x (cid:48)(cid:48) , t ) , ζ )d Z ( x (cid:48) , t )= e − (cid:104) ζ (cid:105)(cid:104) Z ( x,t ) − Z ( x (cid:48)(cid:48) ,t ) (cid:105) m ∆ (cid:18)(cid:18) Z ( x, t ) − Z ( x (cid:48)(cid:48) , t )2 (cid:19) , ζ (cid:19) . So let ε >
0. We have that1(2 π ) m (cid:90) (cid:90) (cid:90) V × R T (cid:48) W t | Z ( x (cid:48) ,t ) × R m e iζ · ( Z ( x,t ) − Z ( x (cid:48)(cid:48) ,t )) −(cid:104) ζ (cid:105)(cid:104) Z ( x (cid:48) ,t ) − Z ( x (cid:48)(cid:48) ,t ) (cid:105) −(cid:104) ζ (cid:105)(cid:104) Z ( x,t ) − Z ( x (cid:48) ,t ) (cid:105) − ε (cid:104) ζ (cid:105) ·· u ( x (cid:48)(cid:48) , t ) (cid:104) ζ (cid:105) m ∆( Z ( x (cid:48) , t ) − Z ( x (cid:48)(cid:48) , t ) , ζ )d Z ( x (cid:48)(cid:48) , t )d ζ d Z ( x (cid:48) , t ) == 1(4 π ) m (cid:90) R T (cid:48) W t | Z ( x (cid:48) ,t ) (cid:90) V e iζ · ( Z ( x,t ) − Z ( x (cid:48)(cid:48) ,t )) − (cid:104) ζ (cid:105)(cid:104) Z ( x,t ) − Z ( x (cid:48)(cid:48) ,t ) (cid:105) − ε (cid:104) ζ (cid:105) u ( x (cid:48)(cid:48) , t ) ·· ∆ (cid:18)(cid:18) Z ( x, t ) − Z ( x (cid:48)(cid:48) , t )2 (cid:19) , ζ (cid:19) d Z ( x (cid:48)(cid:48) , t )d ζ == 1(2 π ) m (cid:90) R T (cid:48) W t | Z ( x (cid:48) ,t ) e − ε (cid:104) ζ (cid:105) F [ u ]( t ; Z ( x, t ) , ζ )d ζ −→ ε → + u ( x, t ) . (cid:3) Now we can state the main theorem of this work:
Theorem 3.6.
Let V be a locally integrable structure on Ω ⊂ R N , and let p ∈ Ω be an arbitrary point.Consider ( V × W, x , . . . , x m , t , . . . t n ) a local coordinate system vanishing at p , as described above. Let u ∈ C ∞ ( W ; D (cid:48) ( V )) be a solution of L u = f , ... L n u = f n , where f j ∈ G s ( U ; L , · · · , L n , M , . . . , M m ) , j = 1 , . . . , n . The following are equivalent (1) There exist V ⊂ V , W ⊂ W open balls containing the origin such that u | V × W ∈ G s ( V × W ; L , · · · , L n , M , · · · , M m ) ; (2) There exists V (cid:98) V an open ball centered at the origin such that for every χ ∈ C ∞ c ( V ) , with ≤ χ ≤ and χ ≡ in some open neighborhood of the origin, there exist (cid:101) V ⊂ V , (cid:102) W ⊂ W , openballs centered at the origin, and constants C, ε > such that | F [ χu ]( t ; z, ζ ) | ≤ Ce − ε | ζ | s , ∀ t ∈ (cid:102) W , ( z, ζ ) ∈ R T (cid:48) W t (cid:12)(cid:12) (cid:101) V \ , where ( z, ζ ) ∈ R T (cid:48) W t (cid:12)(cid:12) (cid:101) V means that z = Z ( x, t ) , ζ = t Z x ( x, t ) − ξ , ξ ∈ R m \ and x ∈ (cid:101) V ; (3) For every χ ∈ C ∞ c ( V ) , with ≤ χ ≤ and χ ≡ in some open neighborhood of the origin, thereexist (cid:101) V ⊂ V , (cid:102) W ⊂ W , open balls centered at the origin, constants C, ε > such that (3.11) | F [ χu ]( t ; z, ζ ) | ≤ Ce − ε | ζ | s , ∀ t ∈ (cid:102) W , ( z, ζ ) ∈ R T (cid:48) W t (cid:12)(cid:12) (cid:101) V \ , where ( z, ζ ) ∈ R T (cid:48) W t (cid:12)(cid:12) (cid:101) V means that z = Z ( x, t ) , ζ = t Z x ( x, t ) − ξ , ξ ∈ R m \ and x ∈ (cid:101) V Before proving this theorem we shall derive a formula for the derivatives of the Gaussian:
Lemma 3.7.
Let λ > and α ∈ Z m + . Then (3.12) ∂ αx e − λ | x | = (cid:88) l +2 l = α · · · (cid:88) l m +2 l m = α m α ! l ! l ! · · · l m ! l m ! ( − λ ) l + l + ··· + l m + l m (2 x ) l · · · (2 x m ) l m e − λ | x | . Proof.
Let x ∈ R m and j = 1 , . . . , m . Consider the function f : R −→ R given by f ( t ) = e − λ { x + ··· + x j − + t + x j +1 + ··· + x m } . So f = g ◦ h ( t ), where g ( t ) = e − λt , and h ( t ) = x + · · · + x j − + t + x j +1 + · · · + x m . By Fa`a di Bruno’sformula (see for instance [5]) we have that ∂ α j x j e − λ | x | = f ( α j ) ( x j )= (cid:88) { l +2 l + ··· + α j l αj = α j } α j ! l ! · · · l α j ! g ( l + ··· + l αj ) ( h ( x j )) α j (cid:89) i =1 (cid:18) h ( i ) ( x j ) i ! (cid:19) l i = (cid:88) l +2 l = α j α j ! l ! l ! ( − λ ) l + l e − λ | x | (2 x j ) l . Since the only term in the sum above that depends on the other variables x , . . . , x j − , x j +1 , . . . , x m isthe Gaussian, we can apply this identity for each variable separately, obtaining (3.12). (cid:3) Proof of the Theorem. 1 . ⇒ . :By Corolary 2.3 we have that there exist O ⊂ C m an open neighborhood of { Z ( x, t ) : x ∈ V , t ∈ W } on C m , and F ( z, t ) ∈ C ∞ ( O × W ) such that (cid:40) F ( Z ( x, t ) , t ) = u ( x, t ) , ∀ ( x, t ) ∈ V × W ; | ∂ z F ( z, t ) | ≤ C k +1 k ! s − dist ( z, M t ) k , ∀ k > , z ∈ O , t ∈ W , where C is a positive constant, as in (2.3). Set V = V and let χ ∈ C ∞ c ( V ) be such that 0 ≤ χ ≤ χ ≡ V (cid:98) V , an open ball centered at the origin. We shall estimate F [ χu ]( t ; z, ζ ) = (cid:90) V e iζ · ( z − Z ( x,t )) −(cid:104) ζ (cid:105)(cid:104) z − Z ( x,t ) (cid:105) χ ( x ) u ( x, t )∆( z − Z ( x, t ) , ζ )d Z ( x, t ) . ROPAGATION OF SINGULARITIES FOR NON HOMOGENEOUS SYSTEMS 11
To do so we shall deform the contour of integration. So let ( z, ζ ) ∈ R T (cid:48) M t (cid:12)(cid:12) (cid:101) V be fixed, where t ∈ (cid:102) W , and (cid:101) V (cid:98) V , (cid:102) W (cid:98) W are open balls centered at the origin to be chosen latter. Let λ > V × W (cid:51) ( y, t ) (cid:55)→ Θ λ ( y, t ) . = Z ( y, t ) − iλ E V ( y ) ζ (cid:104) ζ (cid:105) , is contained in O , where E V is characteristic function of V . By Stokes theorem we obtain F [ χu ]( t ; z, ζ ) = (cid:90) V \ V e iζ · ( z − Z ( x,t )) −(cid:104) ζ (cid:105)(cid:104) z − Z ( x,t ) (cid:105) χ ( x ) u ( x, t )∆( z − Z ( x, t ) , ζ )d Z ( x, t )+ (cid:90) V e iζ · ( z − Z ( x,t )) −(cid:104) ζ (cid:105)(cid:104) z − Z ( x,t ) (cid:105) F ( Z ( x, t ) , t )∆( z − Z ( x, t ) , ζ )d Z ( x, t )= (cid:90) V \ V e iζ · ( z − Z ( x,t )) −(cid:104) ζ (cid:105)(cid:104) z − Z ( x,t ) (cid:105) χ ( x ) u ( x, t )∆( z − Z ( x, t ) , ζ )d Z ( x, t ) (cid:124) (cid:123)(cid:122) (cid:125) (1) + (cid:90) V e iζ · ( z − Θ λ ( x,t )) −(cid:104) ζ (cid:105)(cid:104) z − Θ λ ( x,t ) (cid:105) F (Θ λ ( x, t ) , t )∆( z − Θ λ ( x, t ) , ζ )d Z ( x, t ) (cid:124) (cid:123)(cid:122) (cid:125) (2) + ( − m − i (cid:90) λ (cid:90) V e iζ · ( z − Θ σ ( x,t )) −(cid:104) ζ (cid:105)(cid:104) z − Θ σ ( x,t ) (cid:105) ∂ z F (Θ σ ( x, t ) , t ) · ζ (cid:104) ζ (cid:105) d Z ( x, t ) ·· ∆( z − Θ σ ( x, t ) , ζ )d σ − (cid:124) (cid:123)(cid:122) (cid:125) (3) − (cid:90) λ (cid:90) ∂V e iζ · ( z − Θ σ ( x,t )) −(cid:104) ζ (cid:105)(cid:104) z − Θ σ ( x,t ) (cid:105) F (Θ σ ( x, t ) , t )∆( z − Θ σ ( x, t ) , ζ )d S M t d σ (cid:124) (cid:123)(cid:122) (cid:125) (4) , where d S M t is the surface measure in { Z ( x, t ) : x ∈ ∂V } . We shall estimate these four integralsseparately. Since the estimate for (1) and (4) are very similar, we will estimate them first. We startwriting (cid:101) V = B r (0), so z = Z ( x , t ) for some x ∈ B r (0). In view of (3.3) and | z − Z ( x, t ) | ≥ | x − x | , for every x , we have thatIm { ζ · ( z − Z ( x, t )) + i (cid:104) ζ (cid:105)(cid:104) z − Z ( x, t ) (cid:105) } ≥ c ( r − r ) | ζ | , for every x ∈ V \ V , where V = B r (0), and we are choosing r < r . Therefore (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) V \ V e iζ · ( z − Z ( x,t )) −(cid:104) ζ (cid:105)(cid:104) z − Z ( x,t ) (cid:105) χ ( x ) u ( x, t )∆( z − Z ( x, t ) , ζ )d Z ( x, t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Ce − c ( r − r ) | ζ | . Now the exponent of (4) can be written as iζ · ( z − Θ σ ( x, t )) − (cid:104) ζ (cid:105)(cid:104) z − Θ σ ( x, t ) (cid:105) = iζ · ( z − Z ( x, t )) − σ (cid:104) ζ (cid:105) − (cid:104) ζ (cid:105)(cid:104) z − Z ( x, t ) (cid:105) ++ σ (cid:104) ζ (cid:105) − iσ ( z − Z ( x, t )) · ζ. Now recall that in (4) we are integrating in σ from 0 to λ , so σ < λ , and using (3.3) and (3.5) we havethat Im { ζ · ( z − Θ σ ( x, t )) + i (cid:104) ζ (cid:105)(cid:104) z − Θ σ ( x, t ) (cid:105) } = Im { ζ · ( z − Z ( x, t )) + i (cid:104) ζ (cid:105)(cid:104) z − Z ( x, t ) (cid:105) } + σ Im { i (cid:104) ζ (cid:105) (1 − σ ) − z − Z ( x, t )) · ζ }≥ c | z − Z ( x, t ) | | ζ | + σ Re (cid:104) ζ (cid:105) (1 − σ ) −− σ Im { ζ · ( z − Z ( x, t )) }≥ c | z − Z ( x, t ) | | ζ | + σ (cid:114) − κ κ (1 − σ ) | ζ |−− σ | ζ || z − Z ( x, t ) |≥ | ζ || z − Z ( x, t ) | { c | z − Z ( x, t ) | − λ }≥ | ζ || z − Z ( x, t ) | [ c ( r − r ) − λ ] ≥ | ζ | ( r − r ) [ c ( r − r ) − λ ] , where we are choosing λ satisfying 2 λ < c ( r − r ). Therefore (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) λ (cid:90) ∂V e iζ · ( z − Θ σ ( x,t )) −(cid:104) ζ (cid:105)(cid:104) z − Θ σ ( x,t ) (cid:105) F (Θ σ ( x, t ) , t )∆( z − Θ σ ( x, t ) , ζ )d S M t σ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Ce − ε | ζ | , where ε = ( r − r )[ c ( r − r ) − λ ]. Before estimating (2) and (3), note that the exponent that appears ineach of them is similar to the one that we have just estimated. In (2) we have that x ∈ V , i.e. , | x | < r ,so the exponential have the following estimate: (cid:12)(cid:12)(cid:12) e iζ · ( z − Θ λ ( x,t )) −(cid:104) ζ (cid:105)(cid:104) z − Θ λ ( x,t ) (cid:105) (cid:12)(cid:12)(cid:12) ≤ e − (cid:26) c | z − Z ( x,t ) | | ζ | + λ (cid:114) − κ κ (1 − λ ) | ζ |− λ | ζ || z − Z ( x,t ) | (cid:27) ≤ e −| ζ | (cid:26) λ (cid:114) − κ κ (1 − λ )+ | z − Z ( x,t ) | [ c | z − Z ( x,t ) |− λ ] (cid:27) . When c | z − Z ( x, t ) | ≥ λ we have that (cid:12)(cid:12)(cid:12) e iζ · ( z − Θ λ ( x,t )) −(cid:104) ζ (cid:105)(cid:104) z − Θ λ ( x,t ) (cid:105) (cid:12)(cid:12)(cid:12) ≤ e −| ζ | λ (cid:114) − κ κ (1 − λ ) , and when c | z − Z ( x, t ) | ≤ λ , (cid:12)(cid:12)(cid:12) e iζ · ( z − Θ λ ( x,t )) −(cid:104) ζ (cid:105)(cid:104) z − Θ λ ( x,t ) (cid:105) (cid:12)(cid:12)(cid:12) ≤ e −| ζ | (cid:26) λ (cid:114) − κ κ (1 − λ ) − λ | z − Z ( x,t ) | (cid:27) ≤ e −| ζ | (cid:26) λ (cid:114) − κ κ (1 − λ ) − λ c (cid:27) ≤ e −| ζ | λ (cid:26)(cid:114) − κ κ (1 − λ ) − λc (cid:27) . Combining these two estimates we conclude that (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) V e iζ · ( z − Θ λ ( x,t )) −(cid:104) ζ (cid:105)(cid:104) z − Θ λ ( x,t ) (cid:105) F (Θ λ ( x, t ) , t )∆( z − Θ λ ( x, t ) , ζ )d Z ( x, t ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ Ce − λε | ζ | , where ε = (cid:113) − κ κ (1 − λ ) − λc >
0, decreasing λ if necessary. To estimate (3) we reason as before, sofor each 0 < σ ≤ λ we have that if | z − Z ( x, t ) | ≥ σ/c then (cid:12)(cid:12)(cid:12) e iζ · ( z − Θ σ ( x,t )) −(cid:104) ζ (cid:105)(cid:104) z − Θ σ ( x,t ) (cid:105) (cid:12)(cid:12)(cid:12) ≤ e −| ζ | σ (cid:114) − κ κ (1 − σ ) , and if | z − Z ( x, t ) | ≤ σ/c , ROPAGATION OF SINGULARITIES FOR NON HOMOGENEOUS SYSTEMS 13 (cid:12)(cid:12)(cid:12) e iζ · ( z − Θ σ ( x,t )) −(cid:104) ζ (cid:105)(cid:104) z − Θ σ ( x,t ) (cid:105) (cid:12)(cid:12)(cid:12) ≤ e −| ζ | (cid:26) σ (cid:114) − κ κ (1 − σ ) − σ | z − Z ( x,t ) | (cid:27) ≤ e −| ζ | (cid:26) σ (cid:114) − κ κ (1 − σ ) − σ − κ (cid:27) ≤ e −| ζ | σ (cid:26)(cid:114) − κ κ (1 − σ ) − σ − κ (cid:27) , and since (cid:113) − κ κ (1 − σ ) − σc ≥ ε , for σ < λ , we have that (cid:12)(cid:12)(cid:12) e iζ · ( z − Θ σ ( x,t )) −(cid:104) ζ (cid:105)(cid:104) z − Θ σ ( x,t ) (cid:105) (cid:12)(cid:12)(cid:12) ≤ e − σε | ζ | , for every x ∈ V . So for every k > (cid:12)(cid:12)(cid:12) ( − m − i (cid:90) λ (cid:90) V e iζ · ( z − Θ σ ( x,t )) −(cid:104) ζ (cid:105)(cid:104) z − Θ σ ( x,t ) (cid:105) ∂ z F (Θ σ ( x, t ) , t ) · ζ (cid:104) ζ (cid:105) ∆( z − Θ σ ( x, t ) , ζ )d Z ( x, t )d σ (cid:12)(cid:12)(cid:12) ≤≤ (cid:90) λ e − σε | ζ | sup ( x,t ) ∈ V × W (cid:12)(cid:12) ∂ z F (Θ σ ( x, t ) , t )∆( z − Θ σ ( x, t ) , ζ ) (cid:12)(cid:12) d σ ·· (cid:12)(cid:12)(cid:12)(cid:12) | ζ |(cid:104) ζ (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) V | d Z ( x, t ) | (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:90) λ e − σε | ζ | C k +1 k ! s − dist (Θ σ ( x, t ) , M t ) k d σ ≤ C k +1 k ! s − (cid:90) ∞ e − σε | ζ | (cid:12)(cid:12)(cid:12)(cid:12) σ | ζ |(cid:104) ζ (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) k d σ ≤ C k +1 k ! s − (cid:90) ∞ e − y (cid:18) yε | ζ | (cid:19) k ε | ζ | d y ≤ C k +1 k ! s ( ε | ζ | ) k +1 . Since the constant
C > k , and the above estimate holds for every k >
0, we havethat (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) λ (cid:90) V e iζ · ( z − Θ σ ( x,t )) −(cid:104) ζ (cid:105)(cid:104) z − Θ σ ( x,t ) (cid:105) ∂ z F (Θ σ ( x, t ) , t ) · ζ (cid:104) ζ (cid:105) ∆( z − Θ σ ( x, t ) , ζ )d Z ( x, t )d σ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Ce − ε | ζ | s , for some constants C, ε >
0. Summing up we have obtained the required estimate (3.11), with (cid:102) W = W ,and (cid:101) V = B r (0), where r > r , the radius of V . . ⇒ . :Let χ ∈ C ∞ c ( V ), and χ ∈ C ∞ c ( V ) as in . and . . Since χχ ∈ C ∞ c ( V ), and χχ ≡ | F [ χχ u ]( t ; z, ζ ) | ≤ Ce − ε | ζ | s , t ∈ (cid:102) W , ( z, ζ ) ∈ R T (cid:48) W t (cid:12)(cid:12) (cid:101) V . Now note that χ − χχ ≡ V (cid:98) V . Write V = B ρ (0). So if x (cid:48) ∈ B ρ (0), x ∈ V \ V , t ∈ W , and ζ ∈ R T (cid:48) W t (cid:12)(cid:12) x , we have thatIm { ζ · ( Z ( x, t ) − Z ( x (cid:48) , t )) + i (cid:104) ζ (cid:105)(cid:104) Z ( x, t ) − Z ( x (cid:48) , t ) (cid:105) } ≥ c | Z ( x, t ) − Z ( x (cid:48) , t ) | | ζ |≥ c ρ | ζ | . Therefore if we set V = B ρ (0) ∩ (cid:101) V we have that | F [( χ − χχ ) u ]( t ; z, ζ ) | ≤ Ce − ε (cid:48) | ζ | , t ∈ W, ( z, ζ ) ∈ R T (cid:48) W t (cid:12)(cid:12) V . Combining these two decays we obtain | F [ χu ]( t ; z, ζ ) | ≤ Ce − (cid:101) ε | ζ | s , t ∈ (cid:102) W , ( z, ζ ) ∈ R T (cid:48) W t (cid:12)(cid:12) V . . ⇒ . :Let (cid:101) V ⊂ V and (cid:102) W be open balls centered at the origin, χ ∈ C ∞ c ( V ) with 0 ≤ χ ≤
1, and χ ≡ C, ˜ ε > | F [ χu ]( t ; z, ζ ) | ≤ Ce − ˜ ε | ζ | s , for every z = Z ( x, t ) and ζ = t Z x ( x, t ) − ξ , where x ∈ (cid:101) V , t ∈ (cid:102) W and ξ ∈ R m \
0. Note that we canchoose supp χ as small as we want, keeping in mind that (cid:101) V depends on χ . Since we already have thatL j u ∈ G s ( U ; L , . . . , L n , M , . . . , M m ), we only have to prove that there exist V ⊂ V and W ⊂ W ,open balls centered at the origin, such that, writing U = V × W , u | U ∈ G s ( U ; M , . . . , M m ), sincethe complex vector fields { L , . . . , L n , M , . . . , M m } are pair-wise commuting. We write V = B r (0) and W = B δ (0). By (3.8) we have that χ ( x ) u ( x, t ) = lim ε → + π ) m (cid:90) (cid:90) R T (cid:48) W t e iζ · ( Z ( x,t ) − z (cid:48) ) −(cid:104) ζ (cid:105)(cid:104) Z ( x,t ) − z (cid:48) (cid:105) − ε (cid:104) ζ (cid:105) F [ χu ]( t ; z (cid:48) , ζ ) (cid:104) ζ (cid:105) m d z (cid:48) ∧ d ζ. We shall split this integral in three regions: Q t . = { ( z (cid:48) , ζ ) : z = Z ( x (cid:48) , t ) , ζ = t Z x ( x (cid:48) , t ) − ξ, for some | x (cid:48) | < ˜ r and ξ ∈ R m } Q t . = { ( z (cid:48) , ζ ) : z = Z ( x (cid:48) , t ) , ζ = t Z x ( x (cid:48) , t ) − ξ, for some ˜ r ≤ | x (cid:48) | < r and ξ ∈ R m } Q t . = { ( z (cid:48) , ζ ) : z = Z ( x (cid:48) , t ) , ζ = t Z x ( x (cid:48) , t ) − ξ, for some r ≤ | x (cid:48) | and ξ ∈ R m } , where ˜ r and r are the radii of (cid:101) V and V . For ε > j = 1 , ,
3, we setI εj ( x, t ) . = 1(2 π ) m (cid:90) (cid:90) Q jt e iζ · ( Z ( x,t ) − z (cid:48) ) −(cid:104) ζ (cid:105)(cid:104) Z ( x,t ) − z (cid:48) (cid:105) − ε (cid:104) ζ (cid:105) F [ χu ]( t ; z (cid:48) , ζ ) (cid:104) ζ (cid:105) m d z (cid:48) ∧ d ζ, so we can write χ ( x ) u ( x, t ) = lim ε → + I ε ( x, t ) + I ε ( x, t ) + I ε ( x, t )To prove . it is enough to prove the following: there exists a sequence { ε j } j ∈ Z + with ε j → ε j and I ε j converge to analytic vectors for M , . . . , M m , and that I ε converges to a Gevrey vec-tor for M , . . . , M m . To do so we shall prove that there exist G ε ( z, t ), G ε ( z, t ), G ( z, t ) and G ( z, t ),holomorphic functions in some open neighborhood of the origin such that I ε ( x, t ) = G ε ( Z ( x, t ) , t ),I ε ( x, t ) = G ε ( Z ( x, t ) , t ), and G ε j ( z, t ) −→ G ( z, t ) and G ε j ( z, t ) −→ G ( z, t ) uniformly in z , for somesequence { ε j } j ∈ Z + satisfying ε j →
0, and we shall also prove that there exists a positive constant C suchthat | M α I ε ( x, t ) | ≤ C | α | +1 α ! s , ∀ α ∈ Z m + , for all ( x, t ) ∈ U and ε > ε ( x, t ):Let ( z (cid:48) , ζ ) ∈ Q t . Since z (cid:48) = Z ( x (cid:48) , t ), with x (cid:48) ∈ V , we can use (3.3) and (3.2) to obtain ROPAGATION OF SINGULARITIES FOR NON HOMOGENEOUS SYSTEMS 15 Im { ζ · ( Z (0 , t ) − Z ( x (cid:48) , t )) + i (cid:104) ζ (cid:105)(cid:104) Z (0 , t ) − Z ( x (cid:48) , t ) (cid:105) } ≥ c | ζ || Z (0 , t ) − Z ( x (cid:48) , t ) | ≥ c | ζ | (1 − µ ) | x (cid:48) | ≥ c (1 − µ )˜ r | ζ | , in other words sup ( z (cid:48) ,ζ ) ∈ Q t Im { ζ · ( Z (0 , t ) − z (cid:48) ) + i (cid:104) ζ (cid:105)(cid:104) Z (0 , t ) − z (cid:48) (cid:105) }| ζ | ≥ c (1 − µ )˜ r, and this is valid for every t ∈ W . So there are O ⊂ C m an open neighborhood of the origin and W (cid:98) W an open neighborhood of the origin, such thatsup ( z (cid:48) ,ζ ) ∈ Q t Im { ζ · ( z − z (cid:48) ) + i (cid:104) ζ (cid:105)(cid:104) z − z (cid:48) (cid:105) }| ζ | ≥ c (1 − µ )˜ r , ∀ z ∈ O , t ∈ W . Now using (3.6) we obtain(3.13) (cid:12)(cid:12)(cid:12) e iζ · ( z − z (cid:48) ) −(cid:104) ζ (cid:105)(cid:104) z − z (cid:48) (cid:105) F [ χu ]( t ; z (cid:48) , ζ ) (cid:104) ζ (cid:105) m (cid:12)(cid:12)(cid:12) ≤ C (1 + | ζ | ) k + m e − c (1 − µ r | ζ | , for some k ≥ z ∈ O , ( z (cid:48) , ζ ) ∈ Q t , and t ∈ W . Now setG ε ( z, t ) . = 1(2 π ) m (cid:90) (cid:90) Q t e iζ · ( z − z (cid:48) ) −(cid:104) ζ (cid:105)(cid:104) z − z (cid:48) (cid:105) − ε (cid:104) ζ (cid:105) F [ χu ]( t ; z (cid:48) , ζ ) (cid:104) ζ (cid:105) m d z (cid:48) ∧ d ζ, and G ( z, t ) . = 1(2 π ) m (cid:90) (cid:90) Q t e iζ · ( z − z (cid:48) ) −(cid:104) ζ (cid:105)(cid:104) z − z (cid:48) (cid:105) F [ χu ]( t ; z (cid:48) , ζ ) (cid:104) ζ (cid:105) m d z (cid:48) ∧ d ζ, for ε > z ∈ O , and t ∈ W . Let V (cid:98) V and W (cid:98) W such that { Z ( x, t ) : ( x, t ) ∈ V × W } ⊂ O ,so G ε ( Z ( x, t ) , t ) = I ε ( x, t ) for every ( x, t ) ∈ V × W . Define I ( x, t ) . = G ( Z ( x, t ) , t ), for ( x, t ) ∈ V × W .In view of (3.13) we have that G ε ( z, t ) and G ( z, t ) are holomorphic with respect to z , and G ε ( z, t ) −→ G ( z, t ) uniformly on O × W .I ε ( x, t ):We can deform the domain of integration with respect to the variable ζ , moving the contour of theintegration from R T (cid:48) W t (cid:12)(cid:12) Z ( x (cid:48) ,t ) to R m , obtainingI ε ( x, t ) = 1(2 π ) m (cid:90) (cid:90) Q t e iζ · ( Z ( x,t ) − z (cid:48) ) −(cid:104) ζ (cid:105)(cid:104) Z ( x,t ) − z (cid:48) (cid:105) − ε (cid:104) ζ (cid:105) F [ χu ]( t ; z (cid:48) , ζ ) (cid:104) ζ (cid:105) m d z (cid:48) ∧ d ζ = 1(2 π ) m (cid:90) R m (cid:90) r ≤| x (cid:48) | e iξ · ( Z ( x,t ) − Z ( x (cid:48) ,t )) −| ξ |(cid:104) Z ( x,t ) − Z ( x (cid:48) ,t ) (cid:105) − ε | ξ | F [ χu ]( t ; Z ( x (cid:48) , t ) , ξ ) | ξ | m d Z ( x (cid:48) , t )d ξ = 1(2 π ) m (cid:90) R m (cid:90) r ≤| x (cid:48) | (cid:68) u ( x (cid:48)(cid:48) , t ) , χ ( x (cid:48)(cid:48) ) e iξ · ( Z ( x,t ) − Z ( x (cid:48)(cid:48) ,t ))) −| ξ | (cid:2) (cid:104) Z ( x,t ) − Z ( x (cid:48) ,t ) (cid:105) + (cid:104) Z ( x (cid:48) ,t ) − Z ( x (cid:48)(cid:48) ,t ) (cid:105) (cid:3) ·· e − ε | ξ | | ξ | m ∆( Z ( x (cid:48) , t ) − Z ( x (cid:48)(cid:48) , t ) , ξ ) det Z x ( x (cid:48)(cid:48) , t ) (cid:69) d Z ( x (cid:48) , t )d ξ. Now for every ε > ε ( z, t ) . = 1(2 π ) m (cid:90) R m (cid:90) r ≤| x (cid:48) | (cid:68) u ( x (cid:48)(cid:48) , t ) , χ ( x (cid:48)(cid:48) ) | ξ | m ∆( Z ( x (cid:48) , t ) − Z ( x (cid:48)(cid:48) , t ) , ξ ) · (3.14) · e iξ · ( z − Z ( x (cid:48)(cid:48) ,t ))) −| ξ | (cid:2) (cid:104) z − Z ( x (cid:48) ,t ) (cid:105) + (cid:104) Z ( x (cid:48) ,t ) − Z ( x (cid:48)(cid:48) ,t ) (cid:105) (cid:3) − ε | ξ | det Z x ( x (cid:48)(cid:48) , t ) (cid:69) d Z ( x (cid:48) , t )d ξ for z ∈ C m , and t ∈ W . As usual, we begin estimating the exponential, but first for z = Z (0 , t ): (cid:12)(cid:12)(cid:12)(cid:12) e iξ · ( Z (0 ,t ) − Z ( x (cid:48)(cid:48) ,t ) −| ξ | (cid:2) (cid:104) Z (0 ,t ) − Z ( x (cid:48) ,t ) (cid:105) + (cid:104) Z ( x (cid:48) ,t ) − Z ( x (cid:48)(cid:48) ,t ) (cid:105) (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) ≤ e | ξ || φ (0 ,t ) − φ ( x (cid:48)(cid:48) ,t ) |−| ξ | (cid:2) | x (cid:48) | −| φ (0 ,t ) − φ ( x (cid:48) ,t ) | (cid:3) ·· e −| ξ | (cid:2) | x (cid:48) − x (cid:48)(cid:48) | −| φ ( x (cid:48) ,t ) − φ ( x (cid:48)(cid:48) ,t ) | (cid:3) ≤ e | ξ | µ | x (cid:48)(cid:48) |−| ξ | [ | x (cid:48) | − µ | x (cid:48) | + | x (cid:48) − x (cid:48)(cid:48) | − µ | x (cid:48) − x (cid:48)(cid:48) | ] ≤ e −| ξ | (cid:2) (1 − µ ) | x (cid:48) − x (cid:48)(cid:48) | +(1 − µ ) | x (cid:48) | − µ | x (cid:48)(cid:48) | (cid:3) , where x (cid:48)(cid:48) ∈ supp χ , and r ≤ | x (cid:48) | . Note that the previous argument (for I ε ) does not depend on the ”size”of supp χ , therefore we can shrink it as we want to. So we can assume that | x (cid:48)(cid:48) | is small enough so (cid:12)(cid:12)(cid:12)(cid:12) e iξ · ( Z (0 ,t ) − Z ( x (cid:48)(cid:48) ,t ) −| ξ | (cid:2) (cid:104) Z (0 ,t ) − Z ( x (cid:48) ,t ) (cid:105) + (cid:104) Z ( x (cid:48) ,t ) − Z ( x (cid:48)(cid:48) ,t ) (cid:105) (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) ≤ e −| ξ | (1 − µ ) | x (cid:48) | . Now, for z ∈ C m we have that (cid:12)(cid:12)(cid:12)(cid:12) e iξ · ( z − Z ( x (cid:48)(cid:48) ,t ) −| ξ | (cid:2) (cid:104) z − Z ( x (cid:48) ,t ) (cid:105) + (cid:104) Z ( x (cid:48) ,t ) − Z ( x (cid:48)(cid:48) ,t ) (cid:105) (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) == (cid:12)(cid:12)(cid:12)(cid:12) e iξ · ( Z (0 ,t ) − Z ( x (cid:48)(cid:48) ,t ) −| ξ | (cid:2) (cid:104) Z (0 ,t ) − Z ( x (cid:48) ,t ) (cid:105) + (cid:104) Z ( x (cid:48) ,t ) − Z ( x (cid:48)(cid:48) ,t ) (cid:105) (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) ·· (cid:12)(cid:12)(cid:12)(cid:12) e iξ · ( z − Z (0 ,t )) −| ξ | (cid:2) (cid:104) z − Z (0 ,t ) (cid:105) +2 i ( z − Z (0 ,t )) · ( Z (0 ,t ) − Z ( x (cid:48) ,t )) (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) ≤ e −| ξ | (1 − µ ) | x (cid:48) | e | ξ || z − Z (0 ,t ) | (cid:2) | z − Z (0 ,t ) | +2 | Z (0 ,t ) − Z ( x (cid:48) ,t ) | (cid:3) ≤ e −| ξ | (1 − µ ) | x (cid:48) | e | ξ || z − Z (0 ,t ) | (cid:2) | z − Z (0 ,t ) | +2(1+ µ ) | x (cid:48) | (cid:3) . By continuity we can choose ρ > | z − Z (0 , t ) | < ρ , then(1 − µ )2 | x (cid:48) | − | z − Z (0 , t ) | (cid:2) | z − Z (0 , t ) | + 2(1 + µ ) | x (cid:48) | (cid:3) ≥ , ∀| x (cid:48) | ≥ r . We can shrink, if necessary, W , such that sup t ∈ W | Z (0 , t ) | < ρ . So if we define O ⊂ C m as O . = (cid:26) z ∈ C m : sup t ∈ W | z − Z (0 , t ) | < ρ (cid:27) , then for every z ∈ O , t ∈ W , and r ≤ | x (cid:48) | , we have that (cid:12)(cid:12)(cid:12)(cid:12) e iξ · ( z − Z ( x (cid:48)(cid:48) ,t ) −| ξ | (cid:2) (cid:104) z − Z ( x (cid:48) ,t ) (cid:105) + (cid:104) Z ( x (cid:48) ,t ) − Z ( x (cid:48)(cid:48) ,t ) (cid:105) (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) ≤ e −| ξ | (1 − µ | x (cid:48) | . Since supp χ and W are compact sets, there exist k ∈ Z + and C >
0, such that (cid:12)(cid:12)(cid:12)(cid:68) u ( x (cid:48)(cid:48) , t ) , χ ( x (cid:48)(cid:48) ) | ξ | m ∆( Z ( x (cid:48) , t ) − Z ( x (cid:48)(cid:48) , t ) , ξ ) det Z x ( x (cid:48)(cid:48) , t ) ·· e iξ · ( z − Z ( x (cid:48)(cid:48) ,t ))) −| ξ | (cid:2) (cid:104) z − Z ( x (cid:48) ,t ) (cid:105) + (cid:104) Z ( x (cid:48) ,t ) − Z ( x (cid:48)(cid:48) ,t ) (cid:105) (cid:3) − ε | ξ | (cid:69)(cid:12)(cid:12)(cid:12) ≤≤ C (cid:88) | α |≤ k sup x (cid:48)(cid:48) ∈ supp χ (cid:12)(cid:12)(cid:12) ∂ αx (cid:48)(cid:48) (cid:110) χ ( x (cid:48)(cid:48) ) | ξ | m ∆( Z ( x (cid:48) , t ) − Z ( x (cid:48)(cid:48) , t ) , ξ ) det Z x ( x (cid:48)(cid:48) , t ) ·· e iξ · ( z − Z ( x (cid:48)(cid:48) ,t ))) −| ξ | (cid:2) (cid:104) z − Z ( x (cid:48) ,t ) (cid:105) + (cid:104) Z ( x (cid:48) ,t ) − Z ( x (cid:48)(cid:48) ,t ) (cid:105) (cid:3) − ε | ξ | (cid:111)(cid:12)(cid:12)(cid:12) ≤ C | ξ | k + m e −| ξ | (1 − µ | x (cid:48) | , for every z ∈ O , and t ∈ W , where the constant C > χ , and k . Therefore theintegrand in (3.14) is dominated by ROPAGATION OF SINGULARITIES FOR NON HOMOGENEOUS SYSTEMS 17 (3.15) C | ξ | k + m e −| ξ | (1 − µ | x (cid:48) | e −| ξ | (1 − µ r . Now since the integral of e −| ξ | (1 − µ | x (cid:48) | , with respect to x (cid:48) , is bounded by a constant times | ξ | − m , wehave that (3.15) is an integrable function with respect to ( x (cid:48) , ξ ) in R m × R m . Therefore by Montel’sTheorem, we have that there exists a sequence { ε j } j ∈ Z + , with ε j →
0, such that G ε j ( z, t ) −→ G ( z, t )uniformly in O × W , and G ( z, t ) is holomorphic with respect to z , and it is given byG ( z, t ) . = 1(2 π ) m (cid:90) R m (cid:90) r ≤| x (cid:48) | (cid:68) u ( x (cid:48)(cid:48) , t ) , χ ( x (cid:48)(cid:48) ) | ξ | m ∆( Z ( x (cid:48) , t ) − Z ( x (cid:48)(cid:48) , t ) , ξ ) ·· e iξ · ( z − Z ( x (cid:48)(cid:48) ,t ))) −| ξ | (cid:2) (cid:104) z − Z ( x (cid:48) ,t ) (cid:105) + (cid:104) Z ( x (cid:48) ,t ) − Z ( x (cid:48)(cid:48) ,t ) (cid:105) (cid:3) det Z x ( x (cid:48)(cid:48) , t ) (cid:69) d x (cid:48) d ξ. So if we take V ⊂ V and W ⊂ W neighborhoods of the origin, such that { Z ( x, t ) : x ∈ V , t ∈ W } ⊂ O , we have that I ε j ( x, t ) −→ G ( Z ( x, t ) , t ), for every ( x, t ) ∈ V × W .I ε ( x, t ):Let ( x, t ) ∈ B r (0) × B δ (0) and α ∈ Z m + . ThenM α I ε ( x, t ) = 1(2 π ) m (cid:90) (cid:90) Q t M α (cid:110) e iζ · ( Z ( x,t ) − z (cid:48) ) −(cid:104) ζ (cid:105)(cid:104) Z ( x,t ) − z (cid:48) (cid:105) (cid:111) e − ε (cid:104) ζ (cid:105) F [ χu ]( t ; z (cid:48) , ζ ) (cid:104) ζ (cid:105) m d ζ d z (cid:48) . Since the vectors fields { M , . . . , M m } are pairwise commuting and M j Z k ( x, t ) = δ j,k , we can use formula(3.12) to calculate M α (cid:110) e iζ · ( Z ( x,t ) − z (cid:48) ) −(cid:104) ζ (cid:105)(cid:104) Z ( x,t ) − z (cid:48) (cid:105) (cid:111) , obtainingM α I ε ( x, t ) = 1(2 π ) m (cid:88) β ≤ α (cid:18) αβ (cid:19) (cid:90) (cid:90) Q t M α − β e iζ · ( Z ( x,t ) − z (cid:48) ) M β e −(cid:104) ζ (cid:105)(cid:104) Z ( x,t ) − z (cid:48) (cid:105) ·· e − ε (cid:104) ζ (cid:105) F [ χu ]( t ; z (cid:48) , ζ ) (cid:104) ζ (cid:105) m d ζ d z (cid:48) = 1(2 π ) m (cid:88) β ≤ α (cid:18) αβ (cid:19) (cid:88) l +2 l = β · · · (cid:88) l m +2 l m = β m β ! l ! l ! · · · l m ! l m ! ·· (cid:90) (cid:90) Q t e iζ · ( Z ( x,t ) − z (cid:48) ) −(cid:104) ζ (cid:105)(cid:104) Z ( x,t ) − z (cid:48) (cid:105) − ε (cid:104) ζ (cid:105) F [ χu ]( t ; z (cid:48) , ζ ) (cid:104) ζ (cid:105) m ·· ( −(cid:104) ζ (cid:105) ) l + l + ··· + l m + l m ( iζ ) α − β (2( Z ( x, t ) − z (cid:48) )) l · · · (2( Z m ( x, t ) − z (cid:48) m )) l m d ζ d z (cid:48) . Therefore by (3.11) there exists ˜ ε > | M α I ε ( x, t ) | ≤ π ) m (cid:88) β ≤ α (cid:18) αβ (cid:19) (cid:88) l +2 l = β · · · (cid:88) l m +2 l m = β m β ! l ! l ! · · · l m ! l m ! (cid:90) (cid:90) Q t e − (1 − κ ) | ζ || Z ( x,t ) − z (cid:48) | ·· | ζ | | α − β | + l + l + ··· + l m + l m + m | F [ χu ]( t ; z (cid:48) , ζ ) | | d ζ d z (cid:48) |≤ C | α | +11 (cid:88) β ≤ α (cid:18) αβ (cid:19) (cid:88) l +2 l = β · · · (cid:88) l m +2 l m = β m β ! l ! l ! · · · l m ! l m ! (cid:90) (cid:90) Q t e − ˜ ε | ζ | s ·· | ζ | | α − β | + l + l + ··· + l m + l m + m | d ζ d z (cid:48) |≤ C | α | +12 (cid:88) β ≤ α (cid:18) αβ (cid:19) (cid:88) l +2 l = β · · · (cid:88) l m +2 l m = β m β ! l ! l ! · · · l m ! l m ! (cid:90) ∞ e − ˜ ερ s ·· ρ | α − β | + l + l + ··· + l m + l m + m + m − d ρ ≤ C | α | +13 (cid:88) β ≤ α (cid:18) αβ (cid:19) (cid:88) l +2 l = β · · · (cid:88) l m +2 l m = β m β ! l ! l ! · · · l m ! l m ! α ! s β ! s ( l + l )! s · · · ( l m + l m )! s = C | α | +13 (cid:88) β ≤ α (cid:18) αβ (cid:19) (cid:88) l +2 l = β · · · (cid:88) l m +2 l m = β m β ! l !(2 l )! · · · l m !(2 l m )! α ! s β ! s ( l + l )! s (2 l )! l ! · · ·· · · ( l m + l m )! s (2 l m )! l m ! ≤ C | α | +14 (cid:88) β ≤ α (cid:18) αβ (cid:19) (cid:88) l +2 l = β · · · (cid:88) l m +2 l m = β m β ! l !(2 l )! · · · l m !(2 l m )! α ! s β ! s ( l + l )! s l ! · · ·· · · ( l m + l m )! s l m ! ≤ C | α | +15 (cid:88) β ≤ α (cid:18) αβ (cid:19) (cid:88) l +2 l = β · · · (cid:88) l m +2 l m = β m β ! l !(2 l )! · · · l m !(2 l m )! α ! s β ! s ( l + 2 l )! s · · ·· · · ( l m + 2 l m )! s = C | α | +15 α ! s (cid:88) β ≤ α (cid:18) αβ (cid:19) (cid:88) l +2 l = β · · · (cid:88) l m +2 l m = β m β ! l !(2 l )! · · · l m !(2 l m )! ≤ C | α | +15 α ! s (cid:88) β ≤ α (cid:18) αβ (cid:19) (cid:88) l + l = β · · · (cid:88) l m + l m = β m β ! l ! l ! · · · l m ! l m ! ≤ C | α | +16 α ! s , where the constant C does not depend on ε . The constant C can be taken as 3 C , in view of Lemma4 . . of [5]. (cid:3) Remark . Note that for the implications . ⇒ . ⇒ . we can take (cid:102) W = W . Also by a closerinspection on the proof of . ⇒ . we can take V as V , so that if χ ∈ C ∞ c ( V ) such that χ ≡ V anopen ball centered at the origin, then the inequality (3.11) is valid for every open ball (cid:101) V (cid:98) V , centeredat the origin. 4. Propagation of singularities
In 1983 N. Hanges and F. Treves ([9]) proved that hypo-analytic regularity propagates along ellipticsubmanifolds, and in their proof they actually showed that the decay of the FBI transform propagates.But since then all the propagation of singularities results, concerning systems of complex vector fields,were obtained in the setting of CR geometry, for instance holomorphic extendabillity of CR functions,propagation along CR orbits, sector extendability, (see [15], [12] and [3]) and so on. We did not find inthe literature any other result concerning propagation of Gevrey singularities in this set up.
ROPAGATION OF SINGULARITIES FOR NON HOMOGENEOUS SYSTEMS 19
We shall consider only analytic tube structures, i.e ., locally the hypo-analytic structure is given by Z ( x, t ) = x + iφ ( t ), defined on U = V × W , and φ ( t ) is analytic. One of the reasons we are only dealingwith tube structures is that the real structure bundle, R T (cid:48) W t , is trivial for every t , i.e. , it is equal to Z ( U ) × R m . Now we will recall a simple comparison result for the FBI transform for solutions (seeproposition IX.5 . Proposition 4.1.
There are open balls V (cid:98) V (cid:98) V in R m and W (cid:98) W in R n , all centered at theorigin, and constants r, κ, R > such that, if χ ∈ C ∞ c ( V ) is equal to in V , then, to every solution u in U = V × W , there is a constant C > such that | F [ χu ]( t ; z, ζ ) − F [ χu ]( t (cid:48) ; z, ζ ) | ≤ Ce −| ζ | /R , in the region t, t (cid:48) ∈ W , z ∈ C m , | z | < r, ζ ∈ C κ . This proposition can be used to show that hypo-analyticity propagates along connected fibers (recall thata fiber is locally a level set of the map Z ( x, t )). Let us just indicate how it is done. Suppose that u ishypo-analytic at the origin and let t ∈ W be such that Z (0 , t ) = 0. To show that u is hypo-analyticat (0 , t ) it is enough to show that u | H t is hypo-analytic at (0 , t ), where H t = { ( x, t ) : x ∈ V } , butthis is equivalent to | F [ χu ]( t ; z, ζ ) | ≤ Ce − ε | ζ | , for some C, ε >
0, and z in some open neighborhood of the origin and ζ ∈ C κ , for some 0 < κ <
1. Butsince u is hypo-analytic at the origin, we have that | F [ χu ](0; z, ζ ) | ≤ Ce − ε | ζ | , for some C, ε >
0, and z in some open neighborhood of the origin and ζ ∈ C κ , for some 0 < κ < t in view of Proposition 4.1. One can follow the end of the proofof the Theorem 5.2 to globalize this argument to connected fibers. So why we can not use this sameargument for Gevrey vectors? First, we do not have the property that ensures the desired regularity byonly looking to restrictions on maximally real submanifolds. And the second reason is that for Gevreyregularity, to use a FBI transform argument, ( z, ζ ) must belong to the real structure bundle R T (cid:48) W t , thatdepends on t . So to avoid this dependence we are restringing ourselves to tube structures. To deal with”restringing to maximally real submanifolds is not enough” problem we need some sort of foliation nearthe ”propagators”, and for that it is important for the structure to be analytic.5. Propagation of Gevrey regularity for solutions of the non homogeneous system
In this section we will define the sets that will propagate the Gevrey regularity, the ”propagators”,and then exhibit the proof of the second main theorem of this work. Let Σ ⊂ Ω be a connected subset ofΩ, satisfying the following properties:(1) For every p ∈ Σ there is (
U, Z ), a hypo-analytic chart, with p ∈ U , such that Σ ∩ U ⊂ Z − (0);(2) In the same situation as above, for every q ∈ Σ ∩ U , and (cid:101) U (cid:98) U , an open neighborhood of p ,there is (cid:101) U (cid:98) U , an open neighborhood of q , such that the connected component of the fiber Z − ( Z ( q (cid:48) )) that contains q (cid:48) intersects (cid:101) U , for every q (cid:48) ∈ (cid:101) U ;(3) the map Σ (cid:51) p (cid:55)→ sup { r > B r ( p ) ⊂ U } is continuous.Condition (3) is not exactly a condition, because we can always shrink the open set U for each p , thereforewe can choose U to be a ball with radius varying continuously on p . Condition (2) implies that for every q (cid:48) ∈ (cid:101) U there is a curve γ q (cid:48) : [0 , −→ U satisfying • γ q (cid:48) (0) = q (cid:48) ; • Z ( γ q (cid:48) ( σ )) = Z ( q (cid:48) ), for every 0 ≤ σ ≤ • γ q (cid:48) (1) ∈ (cid:101) U ;Since the structure is analytic, the level sets of Z ( x, t ) are subanalytic sets, therefore the curves { γ q (cid:48) } q (cid:48) ∈ (cid:101) U have bounded length, see for instance section 8 of [10] or pg. 39 of [13] (in the appendix wrote by B.Teissier). Let p ∈ Σ, and let (
U, Z ) be the hypo-analytic chart described above. Take local coordinatesin ( U , x , . . . , x m , t , . . . , t n ), such that in this coordinates p = 0, U = V × W , and the real structurebundle on V , R T (cid:48) W t (cid:12)(cid:12) V , is well positioned for every t ∈ W , i.e. , there exists c > { ξ · ( Z ( x, t ) − Z ( y, t )) + i | ξ |(cid:104) Z ( x, t ) − Z ( y, t ) (cid:105) } ≥ c | ξ || Z ( x, t ) − Z ( y, t ) | , for every x, y ∈ V , t ∈ W , and ξ ∈ R m . Lemma 5.1.
Let ρ > be such that B ρ (0) (cid:98) V , and let f ∈ C ∞ ( W ; E (cid:48) ( K \ B ρ (0))) , where B ρ (0) ⊂ K (cid:98) V , is a compact set. Then | F [ f ]( t ; Z ( x, t ) , ξ ) | ≤ Ce − ε | ξ | , ∀ x ∈ B ρ/ (0) , t ∈ W, ξ ∈ R m . Proof.
Let x ∈ B ρ/ (0) and y ∈ K \ B ρ (0), thenIm { ξ · ( Z ( x, t ) − Z ( y, t )) + i | ξ |(cid:104) Z ( x, t ) − Z ( y, t ) (cid:105) } ≥ c | ξ || Z ( x, t ) − Z ( y, t ) | ≥ c | ξ || x − y | ≥ c | ξ | ρ , for every t ∈ W and ξ ∈ R m . Therefore | F [ f ]( t ; Z ( x, t ) , ξ ) | = (cid:12)(cid:12)(cid:12)(cid:68) f ( y, t ) , e iξ · ( Z ( x,t ) − Z ( y,t )) −| ξ |(cid:104) Z ( x,t ) − Z ( y,t ) (cid:105) ∆( Z ( x, t ) − Z ( y, t ) , ξ ) ·· det Z x ( y, t ) (cid:69)(cid:12)(cid:12)(cid:12) ≤ C (cid:88) | α |≤ λ sup y ∈ K \ B ρ/ (0) (cid:12)(cid:12)(cid:12) ∂ αy (cid:110) e iξ · ( Z ( x,t ) − Z ( y,t )) −| ξ |(cid:104) Z ( x,t ) − Z ( y,t ) (cid:105) ·· ∆( Z ( x, t ) − Z ( y, t ) , ξ ) det Z x ( y, t ) (cid:111)(cid:12)(cid:12)(cid:12) ≤ C | ξ | λ e − c | ξ | ρ ≤ Ce − c ρ | ξ | , for every x ∈ B ρ/ (0), t ∈ W , and ξ ∈ R m . (cid:3) Theorem 5.2.
Let Ω ⊂ R n + m be an open set endowed with an analytic hypo-analytic structure of tubetype. Let Σ ⊂ Ω be a connected submanifold as described above. If u ∈ D (cid:48) (Ω) is such that L u ∈ G s (Ω) ,then singsupp s u ∩ Σ = ∅ or Σ ⊂ singsupp s u .Proof. Let p ∈ Σ, and suppose that p / ∈ singsupp s u . Let ( U, Z ) be the hypo-analytic chart de-scribed before. Consider in U the local coordinates ( x , . . . , x m , t , . . . , t n ), and the complex vectorfields { M , . . . , M m , L , . . . , L n } , as in the previous chapter. In this coordinates system p = 0, and wewrite U = V × W , where V ⊂ R n and W ⊂ R m are both open neighborhoods of the origin. We also havethat Z k ( x, t ) = x k + iφ k ( t ) , k = 1 , . . . , m, and L j Z k = 0 M l Z k = δ l,k L j t i = δ j,i M l t i = 0. ROPAGATION OF SINGULARITIES FOR NON HOMOGENEOUS SYSTEMS 21
We can also assume that the real structure bundle R T (cid:48) W t is well positioned, for every t ∈ W . Now let ρ > B ρ (0) ⊂ V , and χ ∈ C ∞ c ( V ) be such that χ ≡ B ρ (0). Since L u ∈ G s (Ω), we havethat L j u ∈ G s ( U ; L , . . . , L n , M , . . . , M m ), then u ∈ C ∞ ( W ; D (cid:48) ( V )). By Theorem 3.6 and Remark 3.8we have that(5.1) (cid:12)(cid:12) F [ χ L j u ]( t ; Z ( x, t ) , ξ ) (cid:12)(cid:12) ≤ Ce − ε | ξ | s , ∀ x ∈ B ρ (cid:48) (0) , ∀ t ∈ W, ∀ ξ ∈ R m , j = 1 , . . . , n, for some C, ε >
0, where ρ/ ≤ ρ (cid:48) < ρ . We are assuming that u | U ∈ G s ( U ; L , . . . , L n , M , . . . , M m ),for some open neighborhood of the origin, U = V × W . Then by Theorem 3.6 and Remark 3.8 thereexist V (cid:98) V , an open neighborhood of the origin, and positive constants C, ε , such that(5.2) | F [ χu ]( t ; Z ( x, t ) , ξ ) | ≤ Ce − ε | ξ | s , ∀ ( x, t ) ∈ V × W , ∀ ξ ∈ R m . By condition (2), for every ( x , t ) ∈ Σ ∩ (cid:0) B ρ/ (0) × W (cid:1) there exists (cid:101) V × (cid:102) W ⊂ B ρ/ × W , an openneighborhood of ( x , t ), such that, for every ( x (cid:48) , t (cid:48) ) ∈ (cid:101) V × (cid:102) W there is a curve γ ( x (cid:48) ,t (cid:48) ) : [0 , −→ U ,satisfying: • γ ( x (cid:48) ,t (cid:48) ) (0) = ( x (cid:48) , t (cid:48) ); • Z ( γ ( x (cid:48) ,t (cid:48) ) ( σ )) = Z ( x (cid:48) , t (cid:48) ), for every 0 ≤ σ ≤ • γ ( x (cid:48) ,t (cid:48) ) (1) ∈ V × W ; • There exists C > (cid:90) (cid:107) γ (cid:48) ( x (cid:48) ,t (cid:48) ) ( σ ) (cid:107) d σ ≤ C , for every ( x (cid:48) , t (cid:48) ) ∈ (cid:101) V × (cid:102) W .Now let ( x (cid:48) , t (cid:48) ) ∈ (cid:101) V × (cid:102) W be fixed. We write γ ( x (cid:48) ,t (cid:48) ) ( σ ) = ( γ (1)( x (cid:48) ,t (cid:48) ) ( σ ) , γ (2)( x (cid:48) ,t (cid:48) ) ( σ )). By Stokes theorem wehave that F [ χu ]( t (cid:48) ; Z ( x (cid:48) , t (cid:48) ) , ξ ) − F [ χu ]( γ (2)( x (cid:48) ,t (cid:48) ) (1) , Z ( γ ( x (cid:48) ,t (cid:48) ) (1)) , ξ ) == (cid:90) ∂∂σ F [ χu ]( γ (2)( x (cid:48) ,t (cid:48) ) ( σ ); Z ( x (cid:48) , t (cid:48) ) , ξ )d σ = (cid:90) n (cid:88) j =1 F [L j ( χu )]( γ (2)( x (cid:48) ,t (cid:48) ) ( σ ); Z ( x (cid:48) , t (cid:48) ) , ξ ) dd σ γ (2)( x (cid:48) ,t (cid:48) ) j ( σ )d σ = (cid:90) n (cid:88) j =1 F [ u L j χ ]( γ (2)( x (cid:48) ,t (cid:48) ) ( σ ); Z ( x (cid:48) , t (cid:48) ) , ξ ) dd σ γ (2)( x (cid:48) ,t (cid:48) ) j ( σ )d σ ++ (cid:90) n (cid:88) j =1 F [ χ L j u ]( γ (2)( x (cid:48) ,t (cid:48) ) ( σ ); Z ( x (cid:48) , t (cid:48) ) , ξ ) dd σ γ (2)( x (cid:48) ,t (cid:48) ) j ( σ )d σ. Now we analyze these two terms separately. First we note that L j χ vanishes on B ρ (0), for j = 1 , . . . , n therefore, by the previous lemma, we have that there exist C, ε > | F [ u L j χ ]( t ; Z ( x, t ) , ξ ) | ≤ Ce − ε | ξ | , ∀ x ∈ B ρ/ (0) , t ∈ W, ξ ∈ R m , j = 1 , . . . , n. Therefore | F [ u L j χ ]( γ (2)( x (cid:48) ,t (cid:48) ) ( σ ); Z ( x (cid:48) , t (cid:48) ) , ξ ) | ≤ Ce − ε | ξ | , ≤ σ ≤ , ∀ ξ ∈ R m , j = 1 , . . . , n. In view of (5.1) we also have that | F [ χ L j u ]( γ (2)( x (cid:48) ,t (cid:48) ) ( σ ); Z ( x (cid:48) , t (cid:48) ) , ξ ) | ≤ Ce − ε | ξ | s , ≤ σ ≤ , ∀ ξ ∈ R m , j = 1 , . . . , n. Summing up we have obtained | F [ χu ]( t (cid:48) ; Z ( x (cid:48) , t (cid:48) ) , ξ ) | ≤ | F [ χu ]( γ (2)( x (cid:48) ,t (cid:48) ) (1) , Z ( γ ( x (cid:48) ,t (cid:48) ) (1)) , ξ ) | + Ce − ε | ξ | s ≤ Ce − ε | ξ | s for every ( x (cid:48) , t (cid:48) ) ∈ (cid:101) V × (cid:102) W , since γ ( x (cid:48) ,t (cid:48) ) (1) ∈ V × W , where ε = min { ε , ε } and ε = min { ε , ε } .So we conclude that for p ∈ Σ there exists a neighborhood U (cid:98) U , such that if p / ∈ singsupp s u , thenΣ ∩ U ⊂ (cid:123) singsupp s u . Moreover, since the map Σ (cid:51) p (cid:55)→ sup { r > B r ( p ) ⊂ U } is continuous, thesame can be assumed for the map Σ (cid:51) p (cid:55)→ sup { r > B r ( p ) ⊂ U} . To indicate the dependence of p in U , we shall write U = U ( p ). Now we claim that Σ ∩ singsupp s u is an open set. So take { p k } k ∈ Z + a sequence on Σ ∩ (cid:123) singsupp s u , such that p k → p ∈ Σ. Now there exists δ > U ( p k ), as described above, contains a ball, centered at p k , of radius at least δ , for every k . So thereexists k > p ∈ U ( p k ). Since p k / ∈ singsupp s u we have that Σ ∩ U ( p k ) ⊂ (cid:123) singsupp s u , i.e , p / ∈ singsupp s u . Clearly Σ ∩ singsupp s u is closed. Therefore Σ ∩ singsupp s u = ∅ or Σ ⊂ singsupp s u . (cid:3) Examples
Consider in R a real valued, real-analytic function φ satisfying φ (0) = 0, and consider in R thestructure V defined by the complex vector fieldsL = ∂∂t − i φ ( t , t ) ∂φ∂t ( t , t ) ∂∂x , L = ∂∂t − i φ ( t , t ) ∂φ∂t ( t , t ) ∂∂x . The first integral for these complex vector fields is given by Z ( x, t , t ) = x + iφ ( t , t ) , and the characteristic set T is equal toT = { ( x, t , t , ξ, η , η ) ∈ R × (cid:0) R \ (cid:1) : φ ( t , t ) = 0 or ∇ φ ( t , t ) = 0 } . Now suppose that the collection Σ α = φ − ( α ) forms a folliation of a neighborhood of Σ on R byconnected, smooth curves. So we can apply our Theorem 5.2 for this structure V and for { x } × Σ , forevery x ∈ R . Note that in this example it is important that { x } × Σ is contained in the base projectionof T , otherwise the structure would be elliptic on { x } × Σ , and we would not need to use our theoremin this case. Now we give some simple examples of such functions φ :(1) φ ( t , t ) = ( t − + ( t − − φ ( t , t ) = t − t ;(3) φ ( t , t ) = ( t + 1)( t + 1) − Acknowledgements
I wish to express my gratitude to Prof. Paulo D. Cordaro for his careful guidance during my Ph.D.,and I also wish to thank the reserach group at the University of S˜ao Paulo (S˜ao Paulo and S˜ao Carlos),and at the Federal University of S˜ao Carlos for the helpful seminars and conversations, and in especialLuis F. Ragognette for his careful reading of the preprint. Finally I wish to thank CNPq for the financialsupport.
ROPAGATION OF SINGULARITIES FOR NON HOMOGENEOUS SYSTEMS 23
References [1] M.S. Baouendi, C.H. Chang and F. Treves.
Microlocal hypo-analyticity and extension of CR functions . Journal ofDifferential Geometry, 18(13), 331–391, 1983.[2] M.S. Baouendi, F. Treves.
About the holomorphic extension of CR functions on real hypersurfaces in complex space .Duke Mathematical Journal, 51(1), 77–107, 1984.[3] L. Baracco, G. Zampieri.
Propagation of CR extendibility along complex tangent directions . Complex Variables, Theoryand Application: An International Journal, 50(12),967–975, 2005.[4] S. Berhanu, P. D. Cordaro and J. Hounie.
An introduction to involutive structures . New Mathematical Monographs,6. Cambridge University Press, Cambridge, 2008.[5] E. Bierstone, P. Milman.
Resolution of singuarities in Denjoy-Carleman classes . Selecta Mathematica, 10, 1–28, 2004.[6] P.A.S. Caetano
Classes de Gevrey em estruturas hipo-analiticas
Ph.D dissertation, University of S˜ao Paulo, 2000.[7] P. Caetano, P.D. Cordaro.
Gevrey solvability and Gevrey regularity in differential complexes associated to locallyintegrable structures . Transactions of the American Mathematical Society, 363(1), 185–201, 2011.[8] P.D. Cordaro, F. Treves.
Homology and cohomology in hypo-analytic structures of the hypersurface type . The Journalof Geometric Analysis, 1(1), 39–70, 1991.[9] N. Hanges, F. Treves.
Propagation of holomorphic extendability of CR functions . Mathematische Annalen, 263(2),157–177, 1983[10] R.M. Hardt.
Some analytic bounds for subanalytic sets . Differential geometric control theory (R. Brockett, R. Millmanand H. Sussmann, editors), Progress in Mathematics 26, Birkhuser, 259–267, 1982.[11] L.F. Ragognette.
Ultradifferential operators in the study of Gevrey solvability and regularity . MathematischeNachrichten, 292(2), 409–427, 2019.[12] J-M. Trepreau.
Sur la propagation des singularit´es dans les vari´et´es CR . Bulletin de la Soci´et´e Math´ematique deFrance, 118(4), 403–450, 1990.[13] F. Treves.
On the local solvability and the local integrability of systems of vector fields . Acta mathematica, 151, 1–48,1983.[14] F. Treves.
Hypo-Analytic Structures, vol. 40 of Princeton Mathematical Series . Princiton University Press, Princeton,NJ, 1992[15] A. Tumanov.