On a Class of h-Fourier Integral Operators

Abstract In this paper, we study the L2-boundedness and L2-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
Since 1970, numerous mathematicians are interested in these types of operators: like [6,12,1,9,7,18]. The integral operators (1.1) appear naturally in the expression of the solutions of the semiclassical hyperbolic partial differential equations and when expressing the C ∞ solution of the associated Cauchy's problem. Two C ∞ functions appear in (1.1): the phase function φ (x, y, θ) = S (x, θ) − yθ and the amplitude a.
In 1974 Melin and Sjostrand [15] studied an extension of the computation of the Fourier integral in the case where the phase functions assume complex values.
Our work consist a spectral study the L 2 -boundedness and L 2 -compactness of a class of h-Fourier integral operators with the complex phase; we're more particularly interested in continuity studies and on compactness on L 2 (R n ).
It was proven in [1] by a very elaborate demonstration and under certains conditions (relatively strong) on the phase function φ and the amplitude a that all operators of the form: (I (a, φ; h) ψ) (x) = (2πh) −n R n y R N θ e i h φ(x,θ,y) a (x, θ, y) ψ (y) dydθ are bounded on L 2 , where ψ ∈ S (R n ) (the Schwartz space), x ∈ R n , n ∈ N * and N ∈ N . The used technique is to show that I (a, φ) I * (a, φ), I * (a, φ) I (a, φ) are h−pseudodifferential and apply the Calderòn-Vaillancourt's theorem (here I * (a, φ) is the adjoint of I (a, φ)).
In this paper, we will apply the same technic of [1] to establish L 2 -boundedness and L 2 -compactness of form (1.1) operators. That's why we will give brief demonstrations .

C. Harrat
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 to zero. Using the estimate given in [17,19] for h−pseudodifferential (h−admissible) operators, we also establish an L 2 -estimate of F h .
We note that if the amplitude a is just bounded, the Fourier integral operator F is not necessarily bounded on L 2 (R n ).

A general class of h-Fourier integral operators with the complex phase
We consider the following integral transformations In general, the integral (2.1) is not absolutely convergent, so we use the technique of the oscillatory integral developed by Hörmander [13]. The phase function and the amplitude a are assumed to satisfy the following hypothesis: where λ (x, θ, y) = 1 + |x| 2 + |θ| 2 + |y| (H3) There exists K 1 , K 2 > 0, such that: (H3) * There exists K * 1 , K * 2 > 0, such that: For any open subset Ω of R n x × R N θ × R n y , µ ∈ R and ρ ∈ [0, 1] ; we set: When Ω = R n x × R N θ × R n y , we denote Γ µ ρ (Ω) = Γ µ ρ . To give a meaning to the right hand side of (2.1) , 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( S(R n )) (resp. L( S ′ (R n )) is the space of bounded linear mapping from S(R n ) to S(R n ) (resp. S ′ (R n ) to S ′ (R n )) and S ′ (R n ) the space of all distributions with temperate growth on R n ).
Proof. Let η ∈ C ∞ (R n ) such that suppη ⊆ [−1, 2] and η ≡ 1 on [0, 1] . For all ǫ > 0, we set The hypothesis (H3) implies that there exsits C > 0 such that we have on the support of ω ǫ Therefore, there exists ε 0 and a constant C 0 , such that ∀ε ≤ ε 0 we have on the support of ω ǫ In the sequel, we fix ǫ = ǫ 0 . Then it is immediate that I (ω ǫ a σ , φ; h) ψ is an absolutely convergent integral and we have Using (H2) we prove also that I (ω ǫ a, φ; h) ψ is a continuous operator from S(R n ) into itself. To study lim Clearly we have Let Ω 0 be the open subset of R n × R N × R n defined by We need the following lemma.
We have from (2.3), ∀q ≥ 0 So it results from (2.5),(2.7) and using Lebesgue's theorem we have where q > n + N + µ. From (2.2)and(2.8) we can prove the first part of the theorem. Now let us show that I ((1 − ω ǫ 0 )a σ , φ; h) is continuons. Taking account of (2.4)and(2.8) , we get . On the other hand, we have We deduce from (2.9)and(2.10) that, for all q > n + N + µ + |α| + |β| , there exists a constant C α,β,q such that which proves the continuity of
The distribution kernel of F F * is We obtain from(3.2)that if  We have We will study separately the kernels K 1,ε and K 2,ε . The study of K 2,ε . We shall show that for all h, we have Indeed, let Then L is a linear partial differential operator L of order 1 such that 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 ∀α, α ′ , β, β ′ ∈ N n , ∃ C α,α ′ ,β,β ′ > 0; sup Next, we study K 1,ε .This is more difficult and depends on the choice of the parameter ε. It follows from Taylor's formula that We define the vectorial function: We have ξ ε (x, x, θ) = ξ (x, x, θ) on suppb 1,ε , Moreover, for ε sufficiently small, Let us consider the mapping for which Jacobian matrix is  Thus, using that supp ω ′ ⊂ supp ω ⊂ ]−1, 1[ and ∂λ ∂θi (x,x, θ) ≤ 1, we obtain Now it follows from (G2), (4.6) and Taylor's formula that From (4.8) and (4.9) , there exists a positive constant C 7 > 0, such that .., n} . (4.10) If ε < δ0 2 C , then(4.10) and (G4) yields the estimate If ε is such that (4.6) and (4.11) are true, then the mapping given in (4.7) 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 Applying the stationary phase theorem (cf. [20,17]) to (4.13), we obtain the expression of the symbol of the h−pseudodifferential operatorF F * : where θ (x, x, ξ) is the inverse of the mapping θ → ∂ x f (x, θ) = ξ. Thus By (4.2) and (4.3) we have: ×a(x, θ) a x, θ v θ dθdx, ∀v ∈ S (R n ) .
The distribution kernel of the integral operator F(F * F )F −1 is Observe that we can deduce K(x, x) from K(θ, θ) 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 (cf. [14]) 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), (G3) and (G4). Then, we have: 1) If m ≤ 0, F h can be extended to a bounded linear mapping on L 2 (R n ) . 2) If m < 0, F h can be extended to 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 (cf. [3,17,19]) 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 for all u ∈ S(R n ) Thus F h is also a bounded linear operator on L 2 (R n ).
2) If m < 0, lim |x|+|θ|→+∞ λ m (x, θ) = 0, and the compactness theorem (see. [17,19]) show that the operator F * h F h can be extended to a compact 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 On a Class Of h-Fourier Integral Operators With The Complex Phase

11
Since F h is bounded, we have for all l ∈ L 2 (R n ) < ϕ j , l > ϕ j . Hence