On a class of $h$-Fourier integral operators

In this paper, we study the $L^{2}$-boundedness and $L^{2}$-compactness of a class of $h$-Fourier integral operators. These operators are bounded (respectively compact) if the weight of the amplitude is bounded (respectively tends to $0)$.


Introduction
For ϕ ∈ S (R n ) (the Schwartz space), the integral operators appear naturally in the expression of the solutions of the semiclassical hyperbolic partial differential equations and in the expression of the C ∞ -solution of the associate Cauchy's problem. Which appear two C ∞ -functions, the phase function φ (x, y, θ) = S (x, θ) − yθ and the amplitude a.. Since 1970, many efforts have been made by several authors in order to study these type of operators (see, e.g., [2,7,8,5,9]). The first works on Fourier integral operators deal with local properties. On the other hand, K. Asada and D. Fujiwara ( [2]) have studied for the first time a class of Fourier integral operators defined on R n .
For the h-Fourier integral operators, an interesting question is under which conditions on a and S these operators are bounded on L 2 or are compact on L 2 .
It has been proved in [2] by a very elaborated proof and with some hypothesis on the phase function φ and the amplitude a that all operators of the form: (1.2) (I (a, φ) ϕ) (x) = R n y ×R N θ e iφ(x,θ,y) a (x, θ, y) ϕ (y) dydθ are bounded on L 2 where, x ∈ R n , n ∈ N * and N ∈ N (if N = 0, θ doesn't appear in (1.2)). The technique used there is based on the fact that the operators I (a, φ) I * (a, φ) , I * (a, φ) I (a, φ) are pseudodifferential and it uses Caldéron-Vaillancourt's theorem (here I (a, φ) * is the adjoint of I (a, φ)). In this work, we apply the same technique of [2] to establish the boundedness and the compactness of the operators (1.1). To this end we give a brief and simple proof for a result of [2] in our framework.
We mainly prove the continuity of the operator F h on L 2 (R n ) when the weight of the amplitude a is bounded. Moreover, F h is compact on L 2 (R n ) if this weight tends 1 to zero. Using the estimate given in [12,13] for h-pseudodifferential (h-admissible) operators, we also establish an L 2 -estimate of F h .
We note that if the amplitude a is juste bounded, the Fourier integral operator F is not necessarily bounded on L 2 (R n ) . Recently, M. Hasanov [7] and we [1] constructed a class of unbounded Fourier integral operators with an amplitude in the Hörmander's class S 0 1,1 and in 0<ρ<1 S 0 ρ,1 . To our knowledge, this work constitutes a first attempt to diagonalize the h-Fourier integral operators on L 2 (R n ) (relying on the compactness of these operators).

A general class of h-Fourier integral operators
If ϕ ∈ S(R n ), we consider the following integral transformations where, x ∈ R n , n ∈ N * and N ∈ N (if N = 0, θ doesn't appear in (2.3)).
In general the integral (2.3) is not absolutely convergent, so we use the technique of the oscillatory integral developed by Hörmander. The phase function φ and the amplitude a are assumed to satisfy the following hypothesis: where λ(x, θ, y) = (1 + |x| 2 + |θ| 2 + |y| 2 ) 1/2 called the weight and To give a meaning to the right hand side of (2.3), we consider g ∈ S(R n x × R N θ × R n y ), g(0) = 1. If a ∈ Γ µ 0 , we define a σ (x, θ, y) = g(x/σ, θ/σ, y/σ)a(x, θ, y), σ > 0. 2. I(a, φ; h) ∈ L(S(R n )) and I(a, φ; h) ∈ L(S ′ (R n )) (here L(E) is the space of bounded linear mapping from E to E and S ′ (R n ) the space of all distributions with temperate growth on R n ).
Proof. For all v ∈ S(R n ), we have: The main idea to show that F h F * h is a h-pseudodifferential operator, is to use the fact that (S(x, θ)− S( x, θ)) can be expressed by the scalar product x− x, ξ(x, x, θ) after considering the change of variables ( We obtain from (3.6) that if Choosing ω ∈ C ∞ (R) such that We will study separately the kernels K 1,ǫ and K 2,ǫ .
Proof. For all h, we have Indeed, using the oscillatory integral method, there is a linear partial differential operator L of order 1 such that L e The transpose operator of L is . On the other hand we prove by induction on q that and so, Using Leibnitz's formula, (G2) and the form ( t L) q , we can choose q large enough such that Next, we study K ǫ 1 : this is more difficult and depends on the choice of the parameter ǫ. It follows from Taylor's formula that We define the vectorial function We have ξ ε (x, x, θ) = ξ (x, x, θ) on supp b 1,ε . Moreover, for ε sufficiently small, Let us consider the mapping (4.14) for which Jacobian matrix is  Thus, we obtain Now it follows from (G2) , (4.13) and Taylor's formula that From (4.15) and (4.16) , there exists a positive constant C 7 > 0, such that If ε < δ0 2 C , then (4.17) and (G3) yields the estimate If ε is such that (4.13) and (4.18) are true, then the mapping given in (4.14) is a global diffeomorphism of R 3n . Hence there exists a mapping If we change the variable ξ by θ (x, x, ξ) in K 1,ε (x, x), we obtain: From (4.19) we have, for k = 0, 1, that b 1,ε (x,x, θ (x, x, ξ)) det ∂θ ∂ξ (x, x, ξ) belongs to Γ 2m k R 3n if a ∈ Γ m k R 2n . Applying the stationary phase theorem (c.f. [12], [13] ) to (4.20) , we obtain the expression of the symbol of the h-pseudodifferential operator F h F * h : If we change the variable ξ by θ(x, x, ξ) in K 1,ǫ (x, x), we obtain Applying the stationary phase theorem, we obtain the expression of the symbol of the h-pseudodifferential operator F h F * h , is The distribution kernel of the integral operator Remark that we can deduce K(θ, θ) from K(x, x) by replacing x by θ. On the other hand, all assumptions used here are symmetrical on x and θ, therefore F (F * F )F −1 is a nice h-pseudodifferential operator with symbol Thus the symbol of F * F is given by (c.f. [10]) Corollary 4.2. Let F h be the integral operator with the distribution kernel where a ∈ Γ m 0 (R 2n x,θ ) and S satisfies (G1), (G2) and (G3). Then, we have: (1) For any m such that m ≤ 0, F h can be extended as a bounded linear mapping on L 2 (R n ) (2) For any m such that m < 0, F h can be extended as a compact operator on L 2 (R n ).
Proof. It follows from theorem 4.1 that F * h F h is a h-pseudodifferential operator with symbol in Γ 2m 0 R 2n . 1) If m ≤ 0, the weight λ 2m (x, θ) is bounded, so we can apply the Caldéron-Vaillancourt theorem (see [4,12,13]) for F * h F h and obtain the existence of a positive constant γ(n) and a integer k (n) such that Hence, we have ∀u ∈ S(R n ) Thus F h is also a bounded linear operator on L 2 (R n ).
Thus, the Fourier integral operator F h is compact on L 2 (R n ). Indeed, let (ϕ j ) j∈N be an orthonormal basis of L 2 (R n ), then Since F h is bounded, we have ∀ψ ∈ L 2 (R n ) We consider the function given by where C α,β are real constants. This function satisfies (G1), (G2) and (G3).