Exponential decay for products of Fourier integral operators
aa r X i v : . [ m a t h . A P ] J u l EXPONENTIAL DECAY FOR PRODUCTS OF FOURIER INTEGRALOPERATORS
NALINI ANANTHARAMAN
Abstract.
This text contains an alternative presentation, and in certain cases an im-provement, of the “hyperbolic dispersive estimate” proved in [1, 3], where it was used tomake progress towards the quantum unique ergodicity conjecture. The main statement isa sufficient condition to have exponential decay of the norm of a product of sub-unitaryFourier integral operators. The improved version presented here is needed in the twopapers [5] and [6]. Introduction
On a Hilbert space H , consider the product ˆ P n ˆ P n − · · · ˆ P of a large number of operators ˆ P j , with k ˆ P j k = 1 . Think, for instance, of the case where each operator ˆ P j is an orthogonalprojector, or a product of an orthogonal projector and a unitary operator. What kind ofgeometric considerations can be helpful to prove that the norm k ˆ P n ˆ P n − · · · ˆ P k is strictlyless than ? or better, that it decays exponentially fast with n ? In Section 2, we will de-scribe a situation in which H = L ( R d ) , and the operators ˆ P j are Fourier integral operatorsassociated to a sequence of canonical transformations κ j . We will give a “hyperbolicity”condition, on the sequence of transformations κ j and on the symbols of the operators ˆ P j ,under which we can prove exponential decay of the norm k ˆ P n ˆ P n − · · · ˆ P k .This technique was introduced in [1, 3], and is used in [1, 3, 4, 18, 19, 6] to prove resultsrelated to the quantum unique ergodicity conjecture. In [1, 3], the proofs are written ona riemannian manifold of negative curvature, for ˆ P n = e iτ ~ △ ˆ χ n , where the operators ˆ χ n belong to a finite family of pseudodifferential operators, supported inside compact sets ofsmall diameters, and where △ is the laplacian and τ > is fixed. The exponential decayis then used to prove a lower bound on the “entropy” of eigenfunctions, answering by thenegative the long-standing question : can a sequence of eigenfunctions concentrate on aclosed geodesic, as the eigenvalue goes to infinity ? An expository paper can be found in[15], see also the forthcoming paper [2]. We give here an alternative presentation, basedon the use of local adapted symplectic coordinates, which leads in certain cases to animprovement, needed in the two papers [5] and [6].Let us also mention the work of Nonnenmacher-Zworski [16, 17], Christianson [8, 9, 10],Datchev [11], and Burq-Guillarmou-Hassell [7], who showed how to use these techniquesin scattering situations, to prove the existence of a gap below the real axis in the resolventspectrum, and to get local smoothing estimates with loss, as well as Strichartz estimates. In this context, the idea of proving exponential decay for Fourier integral operators wasalso present, although in an implicit form, in Doi’s work [12].The technique is presented in the first four sections, and the applications needed in [5, 6]are stated in section 5. 2.
A hyperbolic dispersion estimate
In this section, R d × ( R d ) ∗ is endowed with the canonical symplectic form ω o = P dj =1 dx j ∧ dξ j , where dx j denotes the projection on the j -th vector of the canonical basis in R d , and dξ j is the projection on the j -th vector of the dual basis in ( R d ) ∗ . The space R d will alsobe endowed with its usual scalar product, denoted h ., . i , and we use it to systematicallyidentify R d with ( R d ) ∗ .We consider a sequence of smooth ( C ∞ ) canonical transformations κ n : R d × R d −→ R d × R d , preserving ω o ( n ∈ N ). We will only be interested in the restriction of κ to afixed relatively compact neighbourhood Ω of , and it is actually sufficient for us to assumethat the product κ n ◦ κ n − ◦ · · · ◦ κ is well defined, for all n , on Ω . The Darboux-Lietheorem ensures that every lagrangian foliation can be mapped, by a symplectic changeof coordinates, to the foliation of R d × R d by the “horizontal” leaves L ξ = { ( x, ξ ) ∈ R d × R d , ξ = ξ } . For our purposes (section 5), there is no loss of generality if we make thesimplifying assumption that each symplectic transformation κ n preserves this horizontalfoliation. It means that κ n is of the form ( x, ξ ) ( x ′ , ξ ′ = p n ( ξ )) where p n : R d −→ R d is a smooth function. In more sophisticated words, κ n has a generating function of theform S n ( x, x ′ , θ ) = h p n ( θ ) , x ′ i − h θ, x i + α n ( θ ) (where x, x ′ , θ ∈ R d , and α n : R d −→ R d is asmooth function). We have the equivalence (cid:2) ( x ′ , ξ ′ ) = κ n ( x, ξ ) (cid:3) ⇐⇒ (cid:2) ξ = − ∂ x S n ( x, x ′ , θ ) , ξ ′ = ∂ x ′ S n ( x, x ′ , θ ) , ∂ θ S n ( x, x ′ , θ ) = 0 (cid:3) . The product κ n ◦ . . . ◦ κ ◦ κ also preserves the horizontal foliation, and it admits thegenerating function h p n ◦ . . . ◦ p ( θ ) , x ′ i − h θ, x i + α ( θ ) + α ( p ( θ )) + . . . + α n ( p n − ◦ . . . ◦ p ( θ ))= h p n ◦ . . . ◦ p ( θ ) , x ′ i − h θ, x i + A n ( θ ) , where the equality defines A n ( θ ) .If p is a map R d −→ R d , we will denote ∇ p the matrix ( ∂p i ∂θ j ) ij , which represents itsdifferential in the canonical basis. Assumptions (H) : We shall be interested in the following operators, acting on L ( R d ) : ˆ P n f ( x ′ ) = 1(2 π ~ ) d Z x ∈ R d ,θ ∈ R d e iSn ( x,x ′ ,θ ) ~ a ( n ) ( x, x ′ , θ, ~ ) f ( x ) dxdθ, where ~ > is a parameter destined to go to . We will assume the following :(H1) The functions p n are smooth diffeomorphisms, and all the derivatives of p n , of p − n and of α n are bounded uniformly in n .(H2) For a given ~ > , the function ( x, x ′ , θ ) a ( n ) ( x, x ′ , θ, ~ ) is of class C ∞ ; RODUCTS OF FOURIER INTEGRAL OPERATORS 3 (H3) The function a (1) ( x, x ′ , θ, ~ ) is supported in Ω with respect to the variable x ;(H4) With respect to the variables ( x ′ , θ ) , the functions a ( n ) ( x, x ′ , θ, ~ ) have a compactsupport x ′ ∈ Ω , θ ∈ Ω , independent of n and ~ ;(H5) When ~ −→ , each a ( n ) ( x, x ′ , θ, ~ ) has an asymptotic expansion a ( n ) ( x, x ′ , θ, ~ ) ∼ (det ∇ p n ( θ )) / ∞ X k =0 ~ k a ( n ) k ( x, x ′ , θ ) , valid up to any order and in all the C ℓ norms on compact sets. Besides, theseasymptotic expansions are uniform with respect to n ;(H6) If ( x ′ , θ ′ ) = κ n ( x, θ ) , we have | a ( n )0 ( x, x ′ , θ ) | ≤ . This condition ensures that k ˆ P n k L −→ L ≤ O ( ~ ) .The operators ˆ P n are (semiclassical) Fourier integral operators associated with the trans-formations κ n (section §6).2.1. Propagation of a single plane wave.
The following theorem is essentially provedin [1]. We denote e ξ , ~ the function e ξ , ~ ( x ) = e i h ξ ,x i ~ . Theorem 2.1.
Fix ξ ∈ R d . Denote ξ n = p n ◦ . . . ◦ p ( ξ ) .In addition to the assumptions (H) above, assume that lim sup k −→ + ∞ k log k∇ ( p n + k ◦ p n + k − ◦ . . . ◦ p n +1 )( ξ n ) k ≤ , uniformly in n .Fix K > arbitrary, and an integer M ∈ N . Then we have, for n = K| log ~ | , and ˜ ǫ > arbitrary, ˆ P n ◦ . . . ◦ ˆ P ◦ ˆ P e ξ , ~ ( x ) = e i An ( ξ ~ e ξ n , ~ ( x )(det ∇ p n ◦ . . . ◦ p ( ξ )) / " M − X k =0 ~ k b ( n ) k ( x, ξ n ) + O ( ~ M (1 − ˜ ǫ ) ) . The functions b ( n ) k , defined on R d × R d , are smooth, supported in Ω × Ω , and b ( n )0 ( x n , ξ n ) = n Y j =1 a ( j )0 ( x j , x j +1 , ξ j ) , where we denote ξ n = p n ◦ . . . ◦ p ( ξ ) , x n = x and the other terms are defined by therelations ( x j , ξ j ) = κ j ◦ . . . ◦ κ ( x , ξ ) .The functions b ( n ) k , for k > , have the same support as b ( n )0 . We have | b ( n )0 ( x n , ξ n ) | ≤ ,and besides, we have bounds k d jx b ( n ) k k ∞ ≤ C ( k, j, ǫ ) n j +3 k e ǫ ( j +2 k ) n valid for arbitrary ǫ > , where the prefactor C ( k, j, ǫ ) does not depend on n . N. ANANTHARAMAN If n is fixed, and if we write ˆ P n ◦ . . . ◦ ˆ P ◦ ˆ P e ξ , ~ ( x ) explicitly as an integral over ( R d ) n , this theorem is a straightforward application of the stationary phase method. If n is allowed to go to infinity as ~ −→ , our result amounts, in some sense, to applyingthe method of stationary phase on a space whose dimension goes to ∞ , and this is knownto be very delicate. The theorem was first proved this way, in an unpublished version(available on request or on my webpage) of the paper [1]. A nicer proof, written with thecollaboration of Stéphane Nonnenmacher, is available in [1], and has also appeared underdifferent forms in [3, 16]. In these papers, the proofs are written on a riemannian manifold,for ˆ P n = e iτ ~ △ ˆ χ n , where the operators ˆ χ n belong to a finite family of pseudodifferentialoperators, whose symbols are supported inside compact sets of small diameters, and where △ is the laplacian and τ > is fixed. In local coordinates, and on a manifold of constantnegative sectional curvature, the calculations done in [1, 3] amount to the simpler statementpresented here (see section 5).In all the papers cited above, the dynamical systems under study satisfy a uniformhyperbolicity (or Anosov) property, ensuring an exponential decay(2.1) sup ξ ∈ Ω k∇ ( p n ◦ . . . ◦ p ◦ p )( ξ ) k ≤ Ce − λn , with uniform constants C, λ > . This is why, following [16], we call our result a hyperbolicdispersion estimate .2.2. Estimating the norm of ˆ P n ◦ . . . ◦ ˆ P ◦ ˆ P . We use the ~ -Fourier transform F ~ u ( ξ ) = 1(2 π ~ ) d/ Z R d u ( x ) e − i h ξ,x i ~ dx, the inversion formula u ( x ) = 1(2 π ~ ) d/ Z R d F ~ u ( ξ ) e i h ξ,x i ~ dξ, and the Plancherel formula k u k L ( R d ) = kF ~ u k L ( R d ) . We denote by e Ω an open, relatively compact subset of R d , that contains the closure Ω .Using the Fourier inversion formula, Theorem 2.1 implies, in a straightforward manner,the following Theorem 2.2.
In addition to the assumptions (H) above, assume that lim sup k −→ + ∞ k log k∇ ( p n + k ◦ p n + k − ◦ . . . ◦ p n +1 )( ξ n ) k ≤ , uniformly in n and ξ ∈ e Ω (with ξ n = p n ◦ . . . ◦ p ( ξ ) ).Fix K > arbitrary. Then, for n = K| log ~ | , k ˆ P n ◦ . . . ◦ ˆ P ◦ ˆ P k L −→ L ≤ | e Ω | / (2 π ~ ) d/ sup ξ ∈ e Ω | det ∇ p n ◦ . . . ◦ p ( ξ ) | / (1 + O ( ~ n e ǫn )) , where | e Ω | denotes the volume of e Ω (and ǫ > is arbitrary). RODUCTS OF FOURIER INTEGRAL OPERATORS 5
Of course, since k ˆ P j k L −→ L ≤ O ( ~ ) , we always have the trivial bound k ˆ P n ◦ . . . ◦ ˆ P ◦ ˆ P k L −→ L ≤ O ( ~ | log ~ | ) . Since we are working in the limit ~ −→ , our estimatecan only have an interest if we have an upper bound of the form(2.2) sup ξ ∈ e Ω | det ∇ p n ◦ . . . ◦ p ( ξ ) | / ≤ Ce − λn , λ > , and if K is large enough. Note that (2.2) is weaker than the condition (2.1).We now state a refinement of Theorem 2.2. We consider the same family ˆ P i , satisfyingassumptions (H). The multiplicative constants in our estimate have no importance, and inwhat follows we will often omit them. Theorem 2.3.
Assume as above that lim sup k −→ + ∞ k log k∇ ( p n + k ◦ p n + k − ◦ . . . ◦ p n +1 )( ξ n ) k ≤ , uniformly in n and ξ ∈ e Ω .Let r ≤ d , and assume that the coisotropic foliation by the leaves { ξ r +1 = c r +1 , . . . , ξ d = c d } is invariant by each canonical transformation κ n . In other words, the map p n is of theform p n (( ξ , . . . , ξ r ) , ( ξ r +1 , . . . , ξ d )) = ( m n ( ξ , . . . , ξ d ) , ˜ p n ( ξ r +1 , . . . , ξ d )) , where m n : R d −→ R r and ˜ p n : R d − r −→ R d − r . Fix K > arbitrary. Then there exists ~ K > such that, for n = K| log ~ | , and for ~ < ~ K , k ˆ P n ◦ . . . ◦ ˆ P ◦ ˆ P k L −→ L ≤ π ~ ) ( r − ǫ ) / sup ξ ∈ e Ω | (det ∇ p n ◦ . . . ◦ p ( ξ )) | / inf ξ ∈ e Ω | (det ∇ ˜ p n ◦ . . . ◦ ˜ p ( ξ )) | / (1 + O ( n ~ e ǫn )) for ǫ > arbitrary.In addition, if we make the stronger assumption that k∇ ( p n + k ◦ p n + k − ◦ . . . ◦ p n +1 )( ξ n ) k is bounded above, uniformly in n, k and for ξ ∈ ˜Ω , we have k ˆ P n ◦ . . . ◦ ˆ P ◦ ˆ P k L −→ L ≤ π ~ ) r/ sup ξ ∈ e Ω | (det ∇ p n ◦ . . . ◦ p ( ξ )) | / inf ξ ∈ e Ω | (det ∇ ˜ p n ◦ . . . ◦ ˜ p ( ξ )) | / (1 + O ( n ~ )) . Theorem 2.3 is an improvement of Theorem 2.2 in the case where we have π ~ ) d/ sup ξ ∈ Ω | (det ∇ p n ◦ . . . ◦ p ( ξ )) / | ≫ but π ~ ) r/ sup ξ ∈ Ω | (det ∇ p n ◦ . . . ◦ p ( ξ )) | / inf ξ ∈ Ω | (det ∇ ˜ p n ◦ . . . ◦ ˜ p ( ξ )) | / ≪ . As a trivial example, when each κ n is the identity, Theorem 2.2 gives a non-optimal bound,whereas we can take r = 0 in Theorem 2.3, and recover the (almost) optimal bound k ˆ P n ◦ . . . ◦ ˆ P ◦ ˆ P k L −→ L ≤ O ( ~ | log ~ | ) . A less trivial example will be given in section5.
N. ANANTHARAMAN Proof of Theorem 2.1
The ideas below are contained in [1, 3]; however, our notations here are quite different,and we recall (without giving all details) the main steps. In all this section, M is a fixedinteger, and all the calculations are done modulo remainders of order ~ M (with explicitcontrol of the constants).In all that follows, it is useful to keep in mind the following : if ( x ′ , ξ ′ ) = κ n ◦ . . . ◦ κ ◦ κ ( x, ξ ) , we have ξ ′ = p n ◦ . . . ◦ p ◦ p ( ξ ) , and x = ∇ ( p n ◦ . . . ◦ p ◦ p ) ⊺ x ′ + ∇ A n ( ξ ) .3.1. One step of the iteration.
Let us first fix ξ ∈ R d , and look at the action of theoperator ˆ P n on a function of the form b ξ ( x ) = e i h ξ,x i ~ b ( x ) where b ( x ) = M − X k =0 ~ k b k ( x ) , and where the functions b k are of class C ∞ .We introduce the following notation : ( T ξn a )( x ′ ) = a ( n )0 ( x, x ′ , ξ ) a ( x ) where x is the point such that ( x ′ , p n ( ξ )) = κ n ( x, ξ ) (in other words, x = ∇ p n ( ξ ) ⊺ x ′ + ∇ α n ( ξ ) ). In the case a ( n )0 ≡ , we note that the operator U ξn : a (det ∇ p n ( ξ )) / T ξn a isunitary on L ( R d ) . If we assume (as above) that | a ( n )0 ( x, x ′ , ξ ) | ≤ , it defines a boundedoperator on L ( R d ) , of norm ≤ .A standard application of the stationary phase method yields : Proposition 3.1. ˆ P n b ξ ( x ′ ) = e i αn ( ξ )+ h pn ( ξ ) ,x ′i ~ (det ∇ p n ( ξ )) / " M − X k =0 ~ k b ′ k ( x ′ ) + ~ M R M ( x ′ ) , where : • b ′ ( x ′ ) = ( T ξn b )( x ′ ) ; • b ′ k ( x ′ ) = P ≤ l ≤ k − D k − l ) n b l ( x ′ ) + ( T ξn b k )( x ′ ) , where the operator D k − l ) n is a differ-ential operator of order k − l ) (whose expression also depends on ξ , although itdoes not appear in our notations). Its coefficients can be expressed in terms of thederivatives of order ≤ k − l ) of a ( n ) l , and of order ≤ k − l ) + 3 of p n , p − n and α n , at the point ( x, x ′ , ξ ) , where ( x ′ , p n ( ξ )) = κ n ( x, ξ ) ). • There exists an integer N d (depending only on the dimension d ), and a positive realnumber C such that k R M k L ( R d ) ≤ C M − X k =0 k b k k C M − k )+ Nd . RODUCTS OF FOURIER INTEGRAL OPERATORS 7
The constant C can be expressed in terms of a fixed finite number of derivatives of thefunctions a ( n ) l ( l ≤ M − ), p n , p − n and α n at the point ( x, x ′ , ξ ) . Under our assumptions(H1) and (H5), C is uniformly bounded for all n . Also note that, under (H4), the functions b ′ k are always supported inside the relatively compact set Ω .3.2. After many iterations.
We can now describe the action of the product ˆ P n ◦ . . . ◦ ˆ P ◦ ˆ P on e ξ , ~ . We will give an approximate expression of ˆ P n ◦ . . . ◦ ˆ P ◦ ˆ P e ξ , ~ ( x ) , in theform e i An ( ξ ~ e ξ n , ~ ( x )(det ∇ p n ◦ . . . ◦ p ( ξ )) / " M − X k =0 ~ k b ( n ) k ( x ) , as announced in the theorem. This expression will approximate ˆ P n ◦ . . . ◦ ˆ P ◦ ˆ P e ξ , ~ up toan error of order ~ M (1 − ˜ ǫ ) for any ˜ ǫ > . The function b ( n ) k ( x ) depends, of course, on ξ , andin the final statement of the theorem we indicated this dependence by writing b ( n ) k ( x, ξ n ) (with ξ n = p n ◦ . . . ◦ p ( ξ ) ).The method consists in iterating the method described in Section 3.1, controlling care-fully how the remainders grow with n in the L norm. We recall that k ˆ P n k L ( R d ) ≤ O ( ~ ) ,uniformly in n .Suppose that, after n iterations, we have proved that ˆ P n ◦ . . . ◦ ˆ P ◦ ˆ P e ξ , ~ ( x ) = e i An ( ξ ~ e ξ n , ~ ( x )(det ∇ p n ◦ . . . ◦ p ( ξ )) / " M − X k =0 ~ k b ( n ) k ( x ) + ~ M R ( n ) M ( x ) . The calculations done in Section 3.1 allows to describe the action of ˆ P n +1 on e ξ n , ~ hP M − k =0 ~ k b ( n ) k i :(3.1) ˆ P n +1 e ξ n , ~ " M − X k =0 ~ k b ( n ) k ( x )= e i αn +1( ξn )+ h pn +1( ξn ) ,x i ~ (det ∇ p n +1 ( ξ n )) / " M − X k =0 ~ k b ( n +1) k ( x ) + ~ M R ( n +1) M ( x ) . Note that n Y ℓ =1 (det ∇ p ℓ ( ξ ℓ − )) / = (det ∇ p n ◦ . . . ◦ p ( ξ )) / , and A n ( ξ ) = α ( ξ ) + α ( ξ ) + · · · + α n ( ξ n − ) , so that ˆ P n +1 ˆ P n ◦ . . . ◦ ˆ P ◦ ˆ P e ξ , ~ ( x ) = e i An +1( ξ ~ e ξ n +1 , ~ ( x )(det ∇ p n +1 ◦ . . . ◦ p ( ξ )) / " M − X k =0 ~ k b ( n +1) k ( x ) + ~ M R ( n +1) M ( x ) , N. ANANTHARAMAN with the relation R ( n +1) M = e i An ( ξ ~ (det ∇ p n ◦ . . . ◦ p ( ξ )) / R ( n +1) M + ˆ P n +1 R ( n ) M .We need to control how each term in these expansions will grow with n , and in particular,to control the remainder terms. We form an array B ( n ) that contains all the functions b ( n ) k ,and a certain number of higher order differentials : B ( n ) j,k = d j b ( n ) k , with ≤ k ≤ M − and ≤ j ≤ M − k ) + N d . The index k indicates the power of ~ , and the index j indicates the number of differentials. Note that d j b ( n ) k is a (symmetric)covariant tensor field of order j on R d . If σ is a covariant tensor field of order j on R d , wedefine k σ k ∞ = sup x ∈ R d | σ x | , where | σ x | is the norm of the j -linear form σ x . By assumption(H4), the forms d j b ( n ) k all vanish outside the compact set Ω .There is a linear relation between B ( n ) and B ( n +1) , that we can make a little moreexplicit. We extend the definition of the operators T ξn (previously defined on functions) tocovariant tensor fields, by letting (if σ is of order j ) ( T ξn σ ) x ′ ( v , . . . , v j ) = a ( n )0 ( x, x ′ , ξ ) σ x ( ∇ p n ( ξ ) ⊺ v , . . . , ∇ p n ( ξ ) ⊺ v j ) , where x = ∇ p n ( ξ ) ⊺ x ′ . Taking successive derivatives of the relation b ( n +1) k = X ≤ l ≤ k − D k − l ) n +1 b ( n ) l + T ξn +1 b ( n ) k , which appears in Proposition 3.1, we obtain a linear relation of the form : B ( n +1) = K n +1 B ( n ) + L n +1 B ( n ) + T ξn +1 B ( n ) , where T ξn +1 acts “diagonally”, meaning that [ T ξn +1 B ( n ) ] j,k = T ξn +1 (cid:16) B ( n ) j,k (cid:17) . The only informa-tion we need about the other terms is that [ K n +1 B ( n ) ] j,k depends only on the components B ( n ) j ′ ,l , for l ≤ k − and j ′ ≤ k − l ) + j ; and [ L n +1 B ( n ) ] j,k depends only on the components B ( n ) j ′ ,k , with j ′ ≤ j − . Besides, we have max j,k k [ K n +1 B ( n ) ] j,k k ∞ ≤ C max j,k k B ( n ) j,k k ∞ , where C does not depend on n by our assumptions (H4) (and the same holds with K n +1 replaced by L n +1 ).By induction, we see that B ( n ) can be expressed as B ( n ) = X A ℓ ∈{ T ξℓ ,K ℓ ,L ℓ } A n ◦ A n − ◦ · · · ◦ A B (0) . In a product of the form A n ◦ A n − ◦ · · · ◦ A (where A ℓ ∈ { T ξℓ , K ℓ , L ℓ } for all ℓ = 1 , . . . , n ),we see that there can be at most M indices ℓ for which A ℓ = K ℓ , and M + N d indices k such that A ℓ = L ℓ (otherwise the product A n ◦ A n − ◦ · · · ◦ A vanishes). Even more RODUCTS OF FOURIER INTEGRAL OPERATORS 9 precisely, when we write(3.2) B ( n ) j,k = X A ℓ ∈{ T ξℓ ,K ℓ ,L ℓ } A n ◦ A n − ◦ · · · ◦ A B (0) j,k , in the right-hand side there can be at most k indices ℓ with A ℓ = K ℓ , and k + j indices ℓ with A ℓ = L ℓ . Hence, the sum has at most k + j C k + jn ∼ C ( k, j ) n k + j terms.We now use our assumption that lim sup k −→ + ∞ k log k∇ ( p n + k ◦ p n + k − ◦ . . . ◦ p n +1 )( ξ n ) k ≤ , uniformly in n . Combined with (3.2), this implies that, for any ǫ > , we have k B ( n ) j,k k ∞ ≤ C ( k, j, ǫ ) n k + j e ǫn ( j +2 k ) . These estimates (combined with Proposition 3.1) imply that k R ( n +1) M k L ( R d ) ≤ C M − X k =0 2( M − k )+ N d X j =0 k B ( n ) j,k k ∞ ≤ C ( M, j, ǫ ) n M + N d e ǫn (2 M + N d ) . Remember the induction relation R ( n +1) M = e i An ( ξ ~ (det ∇ p n ◦ . . . ◦ p ( ξ )) / R ( n +1) M + ˆ P n +1 R ( n ) M . We have k ˆ P n +1 R ( n ) M k L ( R d ) ≤ (1 + O ( ~ )) kR ( n ) M k L ( R d ) . If we restrict our attention to n ≤K| log ~ | (where K is fixed), this induction relation implies that kR ( n ) M k L ( R d ) = O ( ~ − ˜ ǫM ) for any ˜ ǫ > . 4. Proof of Theorems 2.2 and 2.3
Theorem 2.2.
The proof of Theorem 2.2 is now very easy. Let u ∈ L ( R d ) . Weknow that u ( x ) = 1(2 π ~ ) d/ Z R d F ~ u ( ξ ) e i h ξ,x i ~ dξ. Let e Ω be an open set containing the closure of Ω . We decompose u = u + u , where u ( x ) = 1(2 π ~ ) d/ Z e Ω F ~ u ( ξ ) e i h ξ,x i ~ dξ and u ( x ) = 1(2 π ~ ) d/ Z R d \ e Ω F ~ u ( ξ ) e i h ξ,x i ~ dξ. Since ˆ P ∗ ˆ P is a pseudodifferential operator, whose complete symbol is supported in Ω × Ω ,we have k ˆ P u k L ( R d ) = O ( ~ ∞ ) k u k L ( R d ) . Concerning u , we apply Theorem 2.1 for each ξ ∈ e Ω . We take n = K| log ~ | and choose M accordingly, large enough so that O ( ~ M (1 − ˜ ǫ ) ) ≪ sup ξ ∈ e Ω | det ∇ p n ◦ . . . ◦ p ( ξ ) | / . From Theorem 2.1, we know that k ˆ P n ◦ . . . ◦ ˆ P ◦ ˆ P e ξ, ~ k L ( R d ) ≤ | det ∇ p n ◦ . . . ◦ p ( ξ ) | / (1 + O ( ~ n e ǫn )) (for ǫ > arbitrary). By a direct application of the triangular inequality, it follows that k ˆ P n ◦ . . . ◦ ˆ P ◦ ˆ P u k L ( R d ) ≤ π ~ ) d/ sup ξ ∈ e Ω | det ∇ p n ◦ . . . ◦ p ( ξ ) | / (1+ O ( ~ n e ǫn )) kF ~ u k L ( e Ω ) ≤ π ~ ) d/ sup ξ ∈ e Ω | det ∇ p n ◦ . . . ◦ p ( ξ ) | / (1 + O ( ~ n e ǫn )) | e Ω | / kF ~ u k L ( R d ) and our result follows.4.2. Theorem 2.3.
The Cotlar-Stein lemma.
Lemma 4.1.
Let
E, F be two Hilbert spaces. Let ( A α ) ∈ L ( E, F ) be a countable family ofbounded linear operators from E to F . Assume that for some R > we have sup α X β k A ∗ α A β k ≤ R and sup α X β k A α A ∗ β k ≤ R Then A = P α A α converges strongly and A is a bounded operator with k A k ≤ R . The Cotlar-Stein lemma is often used to bound in a precise manner the norm of pseu-dodifferential operators.4.2.2. Remember that we assume everywhere that n = K| log ~ | , with K fixed. In orderto bound the norm of ˆ P n ◦ . . . ◦ ˆ P (modulo ~ N for arbitrary N ), the results of the previoussections show that it is enough to bound the norm of the operator A defined by(4.1) Af ( x ′ )= 1(2 π ~ ) d Z ξ ∈ e Ω ,x ∈ R d (det ∇ p n ◦ . . . ◦ p ( ξ )) / " M − X k =0 ~ k b ( n ) k ( x ′ , ξ n ) e i ~ ( h ξ n ,x ′ i + A n ( ξ ) −h ξ,x i ) f ( x ) dxdξ = 1(2 π ~ ) d/ Z ξ ∈ e Ω ,x ∈ R d (det ∇ p n ◦ . . . ◦ p ( ξ )) / " M − X k =0 ~ k b ( n ) k ( x ′ , ξ n ) e i ~ ( h ξ n ,x ′ i + A n ( ξ )) F ~ f ( ξ ) dξ for a suitable choice of M , large. We denote everywhere ξ n = p n ◦ . . . ◦ p ( ξ ) . RODUCTS OF FOURIER INTEGRAL OPERATORS 11
We decompose R d = R r × R d − r , and write any ξ ∈ R d as ξ = ( ξ ( r ) , ˜ ξ ) where ξ ( r ) ∈ R r and ˜ ξ ∈ R d − r . Under our current assumptions, ξ n decomposes as ξ n = ( ξ n ( r ) , ˜ ξ n ) , where ˜ ξ n = ˜ p n ◦ . . . ◦ ˜ p ( ˜ ξ ) . We now introduce a (real-valued) smooth compactly supported χ on R d − r , such that ≤ χ ≤ , and having the property that X ℓ ∈ Z d − r χ ( ˜ ξ − ℓ ) = 1 for all ˜ ξ ∈ R d − r . For ~ > , ℓ ∈ Z d − r and ˜ ξ ∈ R d − r , we denote χ ~ ,ℓ ( ˜ ξ ) = χ (cid:16) ˜ ξ π ~ − ℓ (cid:17) . Usingthe same notation as in (4.1), we define(4.2) A ℓ f ( x ′ )= 1(2 π ~ ) d Z (det ∇ p n ◦ . . . ◦ p ( ξ )) / " M − X k =0 ~ k b ( n ) k ( x ′ , ξ n ) χ ~ ,ℓ ( ˜ ξ n ) e i ~ ( h ξ n ,x ′ i + A n ( ξ ) −h ξ,x i ) f ( x ) dxdξ = 1(2 π ~ ) d/ Z ξ ∈ e Ω (det ∇ p n ◦ . . . ◦ p ( ξ )) / " M − X k =0 ~ k b ( n ) k ( x ′ , ξ n ) χ ~ ,ℓ ( ˜ ξ n ) e i ~ ( h ξ n ,x ′ i + A n ( ξ )) F ~ f ( ξ ) dξ. It is clear that A = P ℓ ∈ Z d − r A ℓ . A crucial remark is that the function ξ χ ~ ,ℓ ( ˜ ξ n ) , definedon Ω , is supported in a set of volume ≤ (2 π ~ ) d − r ξ ∈ e Ω2 | (det ∇ ˜ p n ◦ ... ◦ ˜ p ( ξ )) | . We are going to apply the Cotlar-Stein lemma to this decomposition. Let us writeexplicitly the expression for the adjoint :(4.3) A ∗ ℓ f ( x )= 1(2 π ~ ) d Z (det ∇ p n ◦ . . . ◦ p ( ξ )) / " M − X k =0 ~ k b ( n ) k ( x ′ , ξ n ) χ ~ ,ℓ ( ˜ ξ n ) e − i ~ ( h ξ n ,x ′ i + A n ( ξ ) −h ξ,x i ) f ( x ′ ) dx ′ dξ. We shall evaluate the norm of A ∗ m A ℓ and A ℓ A ∗ m , for all m, ℓ ∈ Z d − r .4.2.3. Norm of A ∗ m A ℓ . We evaluate the norm of A ∗ m A ℓ acting on L ( R d ) by studying thescalar product h A ℓ f, A m f i for f ∈ L ( R d ) . Using expression (4.2) and bilinearity of thescalar product, we will bound the scalar product h A ℓ f, A m f i by studying separately eachbracket(4.4) χ ~ ,ℓ ( ˜ ξ n ) χ ~ ,m ( ˜ ξ ′ n ) *" M − X k =0 ~ k b ( n ) k ( x ′ , ξ n ) e i ~ h ξ n ,x ′ i , " M − X k =0 ~ k b ( n ) k ( x ′ , ξ ′ n ) e i ~ h ξ ′ n ,x ′ i + L x ′ . Using the notation of §4.2.2, we decompose the complex phase h ξ n , x ′ i − h ξ ′ n , x ′ i into h ξ n ( r ) , x ′ ( r ) i − h ξ ′ n ( r ) , x ′ ( r ) i + h ˜ ξ n , ˜ x ′ i − h ˜ ξ ′ n , ˜ x ′ i . In the integral defining the scalar product(4.4), we perform an integration by parts with respect to ˜ x ′ ∈ R d − r : we integrate N timesthe function e i ~ h ˜ ξ n , ˜ x ′ i−h ˜ ξ ′ n , ˜ x ′ i and differentiate the functions b ( n ) k ( x ′ , ξ n ) . Using the estimatesof Theorem 2.1, we obtain Proposition 4.2. (4.5) χ ~ ,ℓ ( ˜ ξ n ) χ ~ ,m ( ˜ ξ ′ n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)*" M − X k =0 ~ k b ( n ) k ( x ′ , ξ n ) e i ~ h ξ n ,x ′ i , " M − X k =0 ~ k b ( n ) k ( x ′ , ξ ′ n ) e i ~ h ξ ′ n ,x ′ i +(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( ǫ ) e ǫNn k m − ℓ k + 1) N for ǫ > arbitrary. The integer N will be chosen soon. We now use the bilinearity of the scalar product,and the fact that k χ ~ ,ℓ ( ˜ ξ n ) F ~ f ( ξ ) k L ( e Ω ) ≤ (2 π ~ ) ( d − r ) / ξ ∈ e Ω | (det ∇ ˜ p n ◦ . . . ◦ ˜ p ( ξ )) | / kF ~ f ( ξ ) k L ( R d ) . Combined with expression (4.2), this yields that(4.6) k A ∗ m A ℓ k ≤ C ( ǫ ) e ǫNn k m − ℓ k + 1) N π ~ ) r sup ξ ∈ e Ω | (det ∇ p n ◦ . . . ◦ p ( ξ )) | inf ξ ∈ e Ω | (det ∇ ˜ p n ◦ . . . ◦ ˜ p ( ξ )) | . Looking at the statement of the Cotlar-Stein lemma, we see that we must choose N largeenough such that P ℓ ∈ Z d − r k ℓ k +1) N/ < + ∞ . Remark 4.3.
If we make the assumption that k∇ ( p n + k ◦ p n + k − ◦ . . . ◦ p n +1 )( ξ n ) k is boundedabove, uniformly in n, k and ξ ∈ e Ω , we see that we can take ǫ = 0 in all the statementsmade above.4.2.4. Norm of A ℓ A ∗ m . This step is actually shorter than the previous one. We now haveto evaluate the scalar product h A ∗ ℓ f, A ∗ m f i for f ∈ L ( R d ) , and we use the expression (4.3)of the adjoint. We do not need integration by parts, as we see directly that h A ∗ ℓ f, A ∗ m f i vanishes as soon as k m − ℓ k is too large (in fact, the supports of χ ~ ,ℓ and χ ~ ,m are disjointif k m − ℓ k > C , where C is fixed and depends only on the support of χ ). In what followswe consider the case k m − ℓ k ≤ C . We see that A ∗ ℓ f is the F ~ -transform of F ℓ : ξ π ~ ) d/ Z (det ∇ p n ◦ . . . ◦ p ( ξ )) / " M − X k =0 ~ k b ( n ) k ( x ′ , ξ n ) χ ~ ,ℓ ( ˜ ξ n ) e − i ~ ( h ξ n ,x ′ i + A n ( ξ )) f ( x ′ ) dx ′ . We recall that each b ( n ) k ( x ′ , ξ n ) is supported in { x ′ ∈ Ω } , and we bound k F ℓ k L ( R d ) ≤ π ~ ) d/ sup ξ ∈ e Ω | (det ∇ p n ◦ . . . ◦ p ( ξ )) | / (2 π ~ ) ( d − r ) / ξ ∈ e Ω | (det ∇ ˜ p n ◦ . . . ◦ ˜ p ( ξ )) | / k f k L (Ω ) , and k f k L (Ω ) ≤ | Ω | / k f k L ( R d ) . We again obtain the bound(4.7) k A ℓ A ∗ m k ≤ π ~ ) r sup ξ ∈ e Ω | (det ∇ p n ◦ . . . ◦ p ( ξ )) | inf ξ ∈ e Ω | (det ∇ ˜ p n ◦ . . . ◦ ˜ p ( ξ )) | , RODUCTS OF FOURIER INTEGRAL OPERATORS 13 and k A ℓ A ∗ m k = 0 if k ℓ − m k > C . Estimates (4.6) and (4.7), combined with the Cotlar-Stein lemma, yield Theorem 2.3. The last statement of the theorem comes from Remark4.3. 5. Examples
We now give examples of application of Theorems 2.2 and 2.3. These results are neededin [6] and [5].Let Y be a d -dimensional C ∞ manifold. The cotangent bundle T ∗ Y is endowed withits canonical symplectic form, denoted ω . Let H : T ∗ Y −→ R be a smooth function(hamiltonian), and let Φ tH : T ∗ Y −→ T ∗ Y be the corresponding hamiltonian flow. Weassume for simplicity that (Φ tH ) is complete. We fix a time step τ > , arbitrary. Beforespecifying the operators ˆ P n to which we will apply the previous results, we have to makeseveral assumptions concerning the underlying geometric situation.We assume that we have a smooth foliation F of T ∗ Y by lagrangian leaves (in the sequelwe shall simply speak about a “lagrangian foliation”), and will denote F ( n ) = Φ nτH ( F ) . Let O ⊂ T ∗ Y be an open, relatively compact subset of T ∗ Y ; we assume that we have a finiteopen covering of O , O ⊂ O ∪ O ∪ . . . ∪ O K , with the following properties :for any n > and any sequence ( α , α , . . . , α n − ) ∈ { , . . . , K } n such that O α ∩ Φ − τH ( O α ) ∩ . . . ∩ Φ − ( n − τH ( O α n − ) = ∅ , we can find for all k ≤ n − a smoothsymplectic coordinate chart Ψ k : ( O α k , ω ) −→ ( R d , ω o ) which maps O α k to aball in R d , and the foliation F ( k ) ⌉ O αk to the horizontal foliation of that ball; thecollection of coordinate charts (Ψ k ) may depend on the sequence ( α , α , . . . , α n − ) ,however, it can be chosen so that all the derivatives of Ψ k and Ψ − k are bounded,independently of n , ( α , α , . . . , α n − ) , and k .We can now give some details about the operators ˆ P k to which we shall apply the mainresults. We fix a family ˆ χ , . . . , ˆ χ K of ~ -pseudodifferential operators (see the appendix),such that the full symbol of ˆ χ k is compactly supported inside O k . We also assume that itsprincipal symbol χ k (which is a smooth function on T ∗ Y ) satisfies k χ k k C ≤ .Let ˆ H be a self-adjoint ~ -pseudodifferential operator with principal symbol H .Fix, finally, a sequence ( α , α , . . . , α n − ) ∈ { , . . . , K } n . We shall use Theorems 2.2and 2.3 to estimate the norm of the product Q n − k =0 ˆ χ α k +1 e − iτ ~ ˆ H ~ ˆ χ α k . The operator ˆ P k willbe ˆ χ α k +1 e − iτ ~ ˆ H ~ ˆ χ α k , read in an adapted coordinate system :Once the sequence ( α , α , . . . , α n − ) is fixed, we consider the family of coordinates Ψ k described in our assumptions. We fix a collection of Fourier integral operators U k : L ( Y ) −→ L ( R d ) , associated with the canonical transformation Ψ k ( k = 0 , . . . n − ), andsuch that the pseudodifferential operator U ∗ k U k satisfies U ∗ k U k ˆ χ k = ˆ χ k + O ( ~ ∞ ) and ˆ χ k =ˆ χ k U ∗ k U k + O ( ~ ∞ ) (where the O is to be understood in the L ( Y ) -operator norm). Note thatthe operators U k depend on the sequence ( α , α , . . . , α n − ) , but (in the geometric situationdescribed above) we can assume that their symbols have derivatives of all orders boundedindependently of ( α , α , . . . , α n − ) and of k . We take ˆ P k = U k +1 ˆ χ α k +1 e − iτ ~ H ~ ˆ χ α k U ∗ k . It is a Fourier integral operator L ( R d ) −→ L ( R d ) , associated with the canonical transforma-tion κ k = Ψ k +1 Φ τH Ψ − k , which by construction preserves the horizontal foliation. Theseoperators also satisfy all assumptions (H), hence we can apply to them Theorems 2.2 and2.3.We give a concrete example of application, used in [6] and [5]. Let G denote a non-compact connected simple Lie group with finite center. We choose a Cartan involution Θ for G , and let K < G be the Θ -fixed maximal compact subgroup. Let g = Lie ( G ) , and let θ denote the differential of Θ , giving the Cartan decomposition g = k ⊕ p with k = Lie ( K ) .Let S = G/K be the associated symmetric space. For a lattice Γ < G we write X = Γ \ G and Y = Γ \ G/K , the latter being a locally symmetric space of non-positive curvature.Fix now a maximal abelian subalgebra a ⊂ p . The dimension of a is called the real rankof G , and will be denoted by r in the sequel. We denote by a ∗ the real dual of a . Let g α = { X ∈ g , ∀ H ∈ a : ad ( H ) X = α ( H ) X } , ∆ = t ∆( a : g ) = { α ∈ a ∗ \ { } , g α = { }} andcall the latter the (restricted) roots of g with respect to a . For α ∈ ∆ , we denote by m α thedimension of g α . The subalgebra g is θ -invariant, and hence g = ( g ∩ p ) ⊕ ( g ∩ k ) . Bythe maximality of a in p , we must then have g = a ⊕ m where m = Z k ( a ) , the centralizerof a in k .A subset Π ⊂ ∆( a : g ) will be called a system of simple roots if every root can beuniquely expressed as an integral combination of elements of Π with either all coefficientsnon-negative or all coefficients non-positive. Fixing a simple system Π we get a notionof positivity. We will denote by ∆ + the set of positive roots, by ∆ − = − ∆ + the set ofnegative roots. For n = ⊕ α> g α and ¯ n = Θ n = ⊕ α< g α we have g = n ⊕ a ⊕ m ⊕ ¯ n .Let N, A < G be the connected subgroups corresponding to the subalgebras n , a ⊂ g respectively, and let M = Z K ( a ) . Then m = Lie ( M ) , though M is not necessarilyconnected.On T ∗ S , consider the algebra H of smooth G -invariant hamiltonians, that are polynomialin the fibers of the projection T ∗ S −→ S . The structure theory of semisimple Lie algebrasshows that H is isomorphic to a polynomial ring in r generators. Moreover, the elementsof H commute under the Poisson bracket. Thus, we have on T ∗ S a family of r independentcommuting Hamiltonian flows H , ..., H r . Since all these flows are G -equivariant, theydescend to the quotient T ∗ Y .We apply the discussion above to Y and H ∈ H . We assume that O is such that thedifferentials ( dH , . . . , dH r ) are everywhere independent on O . It is known that any givenregular common energy layer { H = E , . . . , H r = E r } ⊂ T ∗ Y may naturally be identified(in a G -equivariant way) with G/M . We thus have an equivariant map O −→ R r × G/M which is a diffeomorphism onto its image. In all that follows, we identify O with an opensubset of R r × G/M . Under this identification, the action of Φ tH is transported to ( E , . . . , E r , ρM ) ( E , . . . , E r , ρe ta E , ··· ,Er M ) , where a E ,...,E r ∈ a depends smoothly on E , . . . , E r , and linearly on H – see [14, 6] formore explanations. The foliation F is invariant under Φ tH , and can be described as follows :the leaf of ( E , . . . , E r , ρM ) ∈ R r × G/M is ( E , . . . , E r ) × { ρa ¯ nM, a ∈ A, ¯ n ∈ ¯ N } . RODUCTS OF FOURIER INTEGRAL OPERATORS 15
We assume that each O k is small enough so that, for any given ( E , . . . , E r , ρM ) ∈ O k ,the map R r × n × a × ¯ n −→ R r × G/M ( ε , . . . , ε r , X, Y, Z ) ( ε , . . . , ε r , ρe X e Y e Z M ) is a local diffeomorphism from a neighbourhood of ( E , . . . , E r , , , onto O k . In suchcoordinates, the leaves of the foliation F are then given by the equations ( ε , . . . , ε r , X ) = cst .Let ( α , . . . , α n − ) be such that O α ∩ Φ − τH ( O α ) ∩ . . . ∩ Φ − ( n − τH ( O α n − ) = ∅ . We can thentake ρ = ρ k and ( E , . . . , E r ) such that ( E , . . . , E r , ρ k M ) ∈ O α k and ( E , . . . , E r , ρ k +1 M ) =Φ τH ( E , . . . , E r , ρ k M ) . As explained in the previous paragraph, fixing ρ k allows to identify O α k with a subset of R r × n × a × ¯ n ; and we denote ( ε , . . . , ε r , X, Y, Z ) these coordinates.Denote by d the dimension of S (note that d = r + dim n = r + P α ∈ ∆ + m α ). By theDarboux-Lie theorem (see [20]), we can find some coordinate system Ψ k = ( x k , . . . , x kd , ξ k , . . . , ξ kd ) : O α k −→ R d mapping ω to ω o , and such that ( ξ k , . . . , ξ kd ) = ( ε , · · · , ε r , X ) . The canonicaltransformations κ k = Ψ k +1 Φ τH Ψ − k preserve the horizontal foliation of R d , hence they areof the form κ k : ( x, ξ ) ( x ′ , ξ ′ = p k ( ξ )) . It turns out, in this particular case, that themaps p k are all the same, and of the particular form p k ( ε , · · · , ε r , X ) = ( ε , . . . , ε r , Ad ( e τa ε , ··· ,εr ) .X ) , where a ε ,...,ε r ∈ a depends smoothly on ε , . . . , ε r . The linear maps Ad ( e τa ε ,...,εr ) actingon n are all simultaneously diagonalizable, the eigenspaces being the root spaces g α , witheigenvalue e τα ( a ε ,...,εr ) . We are thus in a case of application of Theorem 2.3 provided that α ( a ε ,...,ε r ) ≤ for all α ∈ ∆ + . If we fix J ⊂ ∆ + arbitrarily, the map κ k preserves thecoisotropic foliation by the leaves ( ε , . . . , ε r , X J ) = cst (where any X ∈ n is decomposedinto X = P α ∈ ∆ + X α , X α ∈ g α , and X J is defined by X J = P α ∈ J X α ). Corollary 5.1.
Assume that H and O are such that α ( a ε ,...,ε r ) ≤ δ ≤ for all ( ε , . . . , ε r , ρM ) ∈ O and all α ∈ ∆ + . Fix a subset J ⊂ ∆ + of the sets of roots.Fix K > arbitrary. Then, for n = K| log ~ | , and for every sequence ( α , . . . , α n − ) , k n − Y k =0 ˆ χ α k +1 e − iτ ~ H ~ ˆ χ α k k ≤ π ~ ) ( d − r − P α ∈ J m α ) / Y α ∈ ∆ + \ J e nδ for ~ > small enough. Remark 5.2.
In the situation of [6], we actually do not have α ( a ε ,...,ε r ) ≤ δ ≤ , but α ( a ε ,...,ε r ) ≤ δ , where δ > can be made arbitrarily small by conveniently choosing theset O . Once K is given, we can choose δ small enough (and O ) so that the proof of Section4.2 still works for n = K| log ~ | .In the special case G = SO o ( d, , Y is a hyperbolic manifold of dimension d . We have r = 1 , and H is generated by the laplacian △ . Taking H = − △ , J = ∅ , and keeping the same notations as before, we obtain in this case k n − Y k =0 ˆ χ α k +1 e − iτ ~ H ~ ˆ χ α k k ≤ π ~ ) ( d − / e − nτ ( d − (1 − η ) if we assume that the symbols of the pseudodifferential operators ˆ χ k are all supported in {k ξ k ∈ [1 − η, η ] } for some small η > . The result proved in [1, 3], which was onlybased on the idea of Theorem 2.2, was k n − Y k =0 ˆ χ α k +1 e − iτ ~ H ~ ˆ χ α k k ≤ π ~ ) d/ e − nτ ( d − (1 − η ) . We see that Theorem 2.3 allows to improve the prefactor π ~ ) d/ to π ~ ) ( d − / , as neededin [5]. Remark 5.3.
Versions of the hyperbolic dispersion estimate have also been proved for moregeneral uniformly hyperbolic dynamical systems [1, 3, 16, 18], and even for certain non-uniformly hyperbolic systems [19]. We refer the reader to [15] for an expository paper. Itis not clear to me whether the new presentation (and improvement) introduced here can beused for those systems. Indeed, there is in general no smooth lagrangian foliations preservedby the hamiltonian flow, and so one cannot hope that the symplectic changes of coordinates Ψ k used above will have uniformly bounded derivatives. On the other hand, control ofhigh order derivatives is crucial when one applies the techniques of semiclassical analysis(method of stationary phase, integration by parts,...) It is a drawback of semiclassicalanalysis that it cannot deal with symplectic transformations of low regularity : I don’tknow if this obstacle can be overcome.6. A few definitions
For the reader’s convenience, we clarify the terminology used in this paper. This is ofcourse not an exhaustive tutorial on semiclassical analysis : for this we refer the reader to[13].Let M and N be two smooth manifolds of the same dimension d . Their cotangent bundles T ∗ M and T ∗ N are respectively equipped with the canonical symplectic forms ω M and ω N .Let κ be a diffeomorphism from an open relatively compact subset O M ⊂ T ∗ M onto anopen subset O N ⊂ T ∗ N , sending ω M to ω N (such a κ is called a symplectic diffeomorphism,of a canonical transformation).In this paper, we say that an operator ˆ P = ˆ P ~ : L ( M ) −→ L ( N ) is a (semiclassical)Fourier integral operator associated with κ , if it is a finite sum of operators of the form ˆ Qf ( x ′ ) = 1(2 π ~ ) d + m Z x ∈ R d ,θ ∈ R m e iS ( x,x ′ ,θ ) ~ a ( x, x ′ , θ, ~ ) f ( x ) dxdθ, where • m ≥ is an integer and S is a smooth function on M × N × R m . More properly, a family of operators depending on ~ > RODUCTS OF FOURIER INTEGRAL OPERATORS 17 • For a given ~ > , the function ( x, x ′ , θ ) a ( x, x ′ , θ, ~ ) is of class C ∞ and hascompact support, independent of ~ ; • When ~ −→ , a ( x, x ′ , θ, ~ ) has an asymptotic expansion a ( x, x ′ , θ, ~ ) ∼ ∞ X k =0 ~ k a k ( x, x ′ , θ ) , valid up to any order and in all the C ℓ norms. • for any ( x, x ′ , θ ) in the support of a , and for any ( x, ξ ) ∈ T ∗ x M , ( x ′ , ξ ′ ) ∈ T ∗ x ′ N , wehave (cid:2) ξ = − ∂ x S n ( x, x ′ , θ ) , ξ ′ = ∂ x ′ S n ( x, x ′ , θ ) , ∂ θ S n ( x, x ′ , θ ) = 0 (cid:3) = ⇒ (cid:2) ( x, ξ ) ∈ O M , ( x ′ , ξ ′ ) = κ ( x, ξ ) (cid:3) . If ˆ P is a Fourier integral operator associated with κ and ˆ P ′ is a Fourier integral operatorassociated with κ ′ , then ˆ P ′ ◦ ˆ P is a Fourier integral operator associated with κ ′ ◦ κ. Also,the adjoint ˆ P ∗ is a Fourier integral operator associated with κ − . In the case M = N , a pseudodifferential operator is a Fourier integral operator associatedwith κ = the identity.If ˆ P is a pseudodifferential operator on M = R d , we define its full symbol as the function a ~ ( x, ξ ) , defined on R d × R d , by a ~ ( x , ξ ) = e − i h ξ ,x i ~ ˆ P ( e i h ξ , •i ~ ) ⌉ x . By the stationary phase method, the function a ~ can be shown to have an asymptoticexpansion, valid in all C k -norms on R d , a ~ ( x, ξ ) ∼ X j ∈ N ~ j a j ( x, ξ ) . The term a is called the principal symbol. We say that the full symbol of ˆ P vanishes on Ω ⊂ R d × R d if all the a j vanish on Ω . References [1] N. Anantharaman,
Entropy and the localization of eigenfunctions , Ann. of Math. (2) 168 (2008), no.2, 435–475.[2] N. Anantharaman,
A hyperbolic dispersion estimate, with applications to the linear Schrödinger equa-tion , to appear in Proceedings of the ICM 2010.[3] N. Anantharaman, S. Nonnenmacher,
Half-delocalization of eigenfunctions for the laplacian on anAnosov manifold , Festival Yves Colin de Verdière. Ann. Inst. Fourier (Grenoble) 57 (2007), no. 7,2465–2523.[4] N. Anantharaman, H. Koch and S. Nonnenmacher,
Entropy of eigenfunctions , New Trends of Math-ematical Physics, selected contributions of the 15th International Congress on Mathematical Physics,Springer (2009), 1–22.[5] N. Anantharaman, G. Rivière,
Dispersion and controllability for the Schrödinger equation on negativelycurved manifolds , preprint 2010[6] N. Anantharaman, L. Silberman,
Asymptotic distribution of eigenfunctions on locally symmetric spaces ,work in progress. [7] N. Burq, C. Guillarmou, A. Hassell,
Strichartz estimates without loss on manifolds with hyperbolictrapped geodesics , preprint, arXiv:0907.3545.[8] H. Christianson,
Semiclassical non-concentration near hyperbolic orbits , J. Funct. Anal. 262 (2007),145–195; ibid,
Dispersive estimates for manifolds with one trapped orbit,
Comm. PDE 33 (2008), 1147–1174.[9] H. Christianson,
Cutoff resolvent estimates and the semilinear Schrödinger equation , Proc. AMS 136(2008), 3513–3520.[10] H. Christianson,
Applications to cut-off resolvent estimates to the wave equation , Math. Res. Lett.Vol. 16 (2009), no. 4, 577–590.[11] K. Datchev,
Local smoothing for scattering manifolds with hyperbolic trapped sets , Comm. Math. Phys.286, no. 3, 837–850.[12] S.–I. Doi,
Smoothing effects of Schrödinger evolution groups on Riemannian manifolds , Duke Math.J. 82( 1996), 679–706.[13] L.C. Evans, M. Zworski
Lectures on semiclassical analysis (version 0.3) avalaible athttp://math.berkeley.edu/ ∼ zworski/semiclassical.pdf (2003)[14] J. Hilgert An ergodic Arnold-Liouville theorem for locally symmetric spaces.
Twenty years ofBialowieza: a mathematical anthology, 163–184, World Sci. Monogr. Ser. Math., 8, World Sci. Publ.,Hackensack, NJ, 2005.[15] S. Nonnenmacher,
Entropy of chaotic eigenstates , Notes of the minicourse given at the workshop “Spec-trum and dynamics”, Centre de Recherches Mathématiques, Montréal, April 2008, arXiv:1004.4964.[16] S. Nonnenmacher, M. Zworski,
Quantum decay rates in chaotic scattering , Acta Mathematica, Volume203, Number 2, December 2009, 149–233.[17] S. Nonnenmacher, M. Zworski,
Semiclassical resolvent estimates in chaotic scattering , Appl. Math.Res. Express (2009) 74–86.[18] G. Rivière,
Entropy of semiclassical measures in dimension 2 , to appear in Duke Math. J.[19] G. Rivière,
Entropy of semiclassical measures for nonpositively curved surfaces , preprint.[20] I. Vaisman,
Basics of Lagrangian foliations , Publ. Mat. 33 (1989), no. 3, 559–575.
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