-boundedness and-compactness of a class of Fourier integral operators.

Electronic Journal of Differential Equations , Vol . 2006 ( 2006 ) , No . 26 , pp . 1 – 1 2 . ISSN : 1 72 - 6691 . URL : http : / / ejde . math . txstate . edu or http : / / ej de . math . unt . edu

ftp ejde . math . txstate . edu ( login : ftp )

L2_{− BOUNDEDNESS AND L}2_{− COMPACTNESS OF A}

CLASS OF FOURIER INTEGRAL OPERATORS

BEKKAI MESSIRDI , ABDERRAHMANE SENOUSSAOUI

Abstract . In this paper , we study the L2_{−boundedness and L}2_{−compactness}

of a class of Fourier integral operators . These operators are bounded ( respec -t ively compac-t ) if -the weigh-t of -the ampli-tude is bounded ( respec-tively -tends

to 0 ) .

1 . Introduction

For ϕ ∈ S(Rn)( the Schwartz space ) , the integral operators

F ϕ(x) = Z

eiS(x,θ)a(x, θ)F ϕ(θ)dθ (1.1)

appear naturally in the expression of the solutions of the hyperbolic partial differ -ential equations and in the expression of the C∞− solution of the associate Cauchy ’ s problem ( see [ 5 , 1 0 ] ) .

If we write formally the Fourier transformation F ϕ(θ) in ( 1 . 1 ) , we obtain the following Fourier integral operators

F ϕ(x) = Z Z

ei(S(x,θ)−yθ)a(x, θ)ϕ(y)dydθ (1.2) in which appear two C∞− functions , the phase function φ(x, y, θ) = S(x, θ) − yθ and the amplitude a.

Since 1 970 , many efforts have been made by several authors in order to study these type of operators ( see , e . g . , [ 1 , 4 , 6 , 7 , 8 , 1 5 ] ) . The first works on Fourier inte - gral operators deal with lo cal properties . On the other hand , Asada and Fuj iwara have studied for the first time a class of Fourier integral operators defined on

R n

.

For the Fourier integral operators , an interesting question is under which condi -tions on a and S these operators are bounded on L2_{or are compact on L}2_.

It has been proved in [ 1 ] by a very elaborated proof and with some hypothesis on the phase function φ and the amplitude a that all operators of the form ( 2 . 1 ) ( see below ) are bounded on L2_{. 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 Cald ´e ron - Vaillancourt ’ s theorem ( here I(a, φ)∗ is the adjoint of I(a, φ)).

2000 Mathematics Subject Classification . 35 S 30 , 35 S 5 , 47 A 10 , 35 P 5 .

Key words and phrases . Fourier integral operators ; pseudodifferential operators ; symbol and phase ; boundedness and compactness .

circlecopyrt − c2006Texas State University - San Marcos . Submitted June 10 , 2005 . Published March 9 , 2006 .

2 B . MESSIRDI , A . S ENOUSSAOUI EJDE - 2 6 / 2 6

In this work , we apply the same technique of [ 1 ] to est ablish the boundedness and the compactness of the operators ( 1 . 2 ) . To this end we give a brief and simple proof for a result of [ 1 ] in our framework .

We mainly prove the continuity of the operator F on L2

(Rn_{) when the weight of}

the amplitude a is bounded . Moreover , F is compact on L2

(Rn_{) if this weight tends to}

zero . Using the estimate given in [ 1 2 ] for h− pseudodifferential (h− admissible )

operators , we also establish an L2_{− estimate of k F k .}

We note that if the amplitude a is j uste bounded , the Fourier integral operator F is not necessarily bounded on L2

(Rn_{). Recently , Hasanov [ 6 ] and Messirdi}

-Senoussaoui [ 1 1 ] constructed a class of unbounded Fourier integral operators with an amplitude in the H ¨o rmander ’ s class S0

1,1 and in

T

0<ρ<1Sρ,10 .

To our knowledge , this work constitutes a first attempt to diagonalize the Fourier integral operators on L2_(Rn)( relying on the compactness of these operators ) .

Let us now describe the plan of this article . In the second section we recall the continuity of some general class of Fourier integral operators on S(Rn) and on S0_(Rn). The assumptions and preliminaries results are given in the third section . The last section is devoted to prove the main result .

2 . _{A general class of Fourier integral operators} If ϕ ∈ S(Rn_{), we consider the following integral transformations}

(I(a, φ)ϕ)(x) = Z Z

eiφ(x,θ,y)a(x, θ, y)ϕ(y)dydθ (2.1) Rny × R

N θ

where , x ∈ Rn

, n ∈ N∗ and N ∈ N( if N = 0, θ doesn ’ t appear in ( 2 . 1 ) ) .

In general the integral ( 2 . 1 ) is not absolutely convergent , so we use the technique of the oscillatory integral developed by H ¨o rmander in [ 8 ] . The phase function φ and the amplitude a are assumed to satisfy the following hypothesis :

( H 1) φ ∈ C∞_(Rn_x× RN θ × R n y, R)(φ is a real function ) ( H 2 ) _{For all (α, β, γ) ∈ N}n× NN × Nn_{, there exists C} α,β,γ > 0 such that | ∂γy∂ β θ∂ α xφ(x, θ, y) |≤ Cα,β,γλ(2−|α|−|β|−|γ|)+ (x, θ, y)

where λ(x, θ, y) = (1+ | x |2_{+ | θ |}2_{+ | y |}2₎1/2 _{is called the weight and}

(2− | α | − | β | − | γ |)+= max(2− | α | − | β | − | γ |, 0)

( H 3 ) There exist K1, K2> 0 such that

K1λ(x, θ, y) ≤ λ(∂yφ, ∂θφ, y) ≤ K2λ(x, θ, y), ∀(x, θ, y) ∈ Rnx× R N θ × R

n y

( H 3∗) There exist K₁∗, K₂∗> 0 such that

K₁∗λ(x, θ, y) ≤ λ(x, ∂θφ, ∂xφ) ≤ K2∗λ(x, θ, y), ∀(x, θ, y) ∈ R n x× R N θ × R n y.

For any open subset Ω of Rnx× RNθ × R n y, µ ∈ R and ρ ∈ [0, 1], we set Γµ_ρ(Ω) = {a ∈ C∞_{(Ω) : ∀(α, β, γ) ∈ N}n_{× N}N_{× N}n, ∃Cα,β,γ > 0 : | ∂yγ∂ β θ∂ α xa(x, θ, y) |≤ Cα,β,γλµ−ρ(|α|+|β|+|γ|)(x, θ, y)} When Ω = Rnx× RNθ × R n y, we denote Γµρ(Ω) = Γµρ.

EJDE −206/26 L2− BOUNDEDNESS AND L2−COMPACTNESS 3 To give a meaning

to the right hand side of ( 2 . 1 ) , we consider g ∈ S(Rnx× RNθ×

Rny), g(0) = 1. If a ∈ Γ µ

0, we define

aσ(x, θ, y) = g(x/σ, θ/σ, y/σ)a(x, θ, y), σ > 0.

Now we are able to st ate the following result .

Theorem 2 . 1 . If φ satisfies ( H 1 ) , ( H 2 ) , ( H 3 ) and ( H 3∗), and if a ∈ Γµ₀, th en

1 . _{For al l ϕ ∈ S(R}n_{), lim}

σ→+∞[I(aσ, φ)ϕ](x) exists for every point x ∈ Rn

and is independent of the choice of the function g. We define

(I(a, φ)ϕ)(x) :=

(I(a

lim

→σ+∞

σ, φ)ϕ)(x) 2. I(a, φ) ∈ L(S(Rn_{)) and I(a, φ) ∈ L(S}0

(Rn_{)) ( here L(E) is}

the space of

bounded lin ear mapping from E to E and S0_(Rn_{) the space of al l distrib utions with}

t emperate growth on Rn_).

The proof of the above theorem can be found in [ 7 ] or in [ 1 2 , propostion II . 2 ] . Example 2 . 2 . Let us give two examples of operators of the form ( 2 . 1 ) which satisfy

(H1) − (H3∗) : ( 1 ) The Fourier transform F ψ(x) =R

Rne −ixy ψ(y)dy, ψ ∈ S(Rn_{), ( 2 )} Pseudod-ifferential operators Aψ(x) = (2π)−n Z R2n

ei(x−y)θa(x, y, θ)ψ(y)dydθ, withψ ∈ S(Rn), a ∈ Γµ_0(R3n).

3 . Assumptions and Preliminaries In this paper we consider the special form of the phase function

φ(x, y, θ) = S(x, θ) − yθ (3.1)

where S satisfies

(G1) S ∈ C∞_(Rn_x× Rn θ, R),

( G 2 ) _{For each (α, β) ∈ N}2n_{, there exist C}

α,β > 0, such that | ∂α x∂ β θS(x, θ) |≤ Cα,βλ(x, θ) (2−|α|−|β|)+_,

( G 3 ) There exists C1> 0 such that | x |≤ C1λ(θ, ∂θS), for all (x, θ) ∈ R2n,

( G 3∗) There exists C2 > 0, such that | θ |≤ C2λ(x, ∂xS), for all (x, θ) ∈ R2n.

Proposition 3 . 1 . Let ’ s assume that S satisfies ( G 1 ) , ( G 2 ) , ( G 3 ) and ( G 3∗). Then

the function φ(x, y, θ) = S(x, θ) − yθ satisfies ( H 1 ) , ( H 2 ) , ( H 3 ) and ( H 3∗). Proof . ( H 1 ) and ( H 2 ) are trivially satisfied . The condition ( G 3 ) implies

λ(x, θ, y) ≤ λ(x, θ) + λ(y) ≤ C3(λ(θ, ∂θS) + λ(y)), C3> 0.

λ(θ, ∂θS) = λ(∂yφ, ∂θφ + y) ≤ 2λ(∂yφ, ∂θφ, y),

which finally gives for some C4> 0,

λ(x, θ, y) ≤ C3(2λ(∂yφ, ∂θφ, y) + λ(y)) ≤

1 C4

λ(∂yφ, ∂θφ, y)

The second inequality in ( H 3 ) is a consequence of the assumption ( G 2 ) . By a similar argument we can show ( H 3∗). _

4 B . MESSIRDI , A . S ENOUSSAOUI EJDE - 2 6 / 2 6 We now introduce the assumption

( G 4 ) There exists δ0> 0 such that

inf

θ_,x∈Rn| det

∂2_S

∂x∂θ(x, θ) |≥ δ0. We note that if φ(x, y, θ) = S(x, θ) − yθ, then

D(φ)(x, θ, y) = ( ∂

2_φ

∂x∂yline − phi_∂θ∂y∂2 (x, (x, θ_,θ,yy )₎ ∂ 2_φ phi − line∂2∂θ∂θ ∂x∂θ (x, (x, θθ,_, yy )₎) = (0_−In ∂2S ∂∂x∂θ2 S ∂θ∂θ (x, (x, θθ )₎) and

| det D(φ)(x, θ, y)| = | det ∂

2_S

∂x∂θ(x, θ)| ≥ δ0.

Remark 3 . 2 . By the global implicit function theorem ( cf . [ 1 4 ] , [ 3 , theorem 4 . 1 . 7 ] )

and using ( G 1 ) , ( G 2 ) and ( G 4 ) , we can easily see that the mappings h1 and

h2defined by

h1: (x, θ) → (x, ∂xS(x, θ)), h2: (x, θ) → (θ, ∂θS(x, θ))

are global diffeomorphism of R2n. Indeed ,

h0₁(x, θ) = (I n₀ ∂2S ∂∂two−x 2_S ∂x∂θ (x, (x, θ ) θ)), h 0 2(x, θ) = ( 0 I n ∂2S ∂∂two−x∂_∂θ2 S θ (x, (x, θ) θ )).

and | det h0₁(x, θ) |=| det h0₂(x, θ) |=| det _∂x∂θ∂2S (x, θ) |≥ δ0> 0, for all (x, θ) ∈ R2n. Then

k (h0 1(x, θ))−1k = 1 | det ∂2_S ∂x∂θ(x, θ) | k tA(x,θ) k k (h0 2(x, θ))−1k = 1 | det ∂2_S ∂x∂θ(x, θ) | k tB(x,θ) k,

where A(x, θ), B(x, θ) are respectively the cofactor matrix of h0₁(x, θ), h0₂(x, θ). By ( G 2 ) , we know that k tA(x,θ) k and k tB(x,θ) k are uniformly bounded .

Let ’ s now assume that S satisfies the following condition which is stronger than ( G 2 ) .

( G 5 ) _{For all (α, β) ∈ N}n_{× N}n_{, there exist C}

α,β> 0, such that

| ∂α x∂

β

θS(x, θ) |≤ Cα,βλ(x, θ)(2−|α|−|β|).

Lemma 3 . 3 . If S satisfies ( G 1 ) , ( G 4 ) and ( G 5 ) , then S satisfies ( G 3 ) and ( G 3∗). Also there exists C5> 0 such that for al l (x, θ), (x0, θ0) ∈ R2n,

| x − x0 | + | θ − θ0|≤ C5[| (∂θS)(x, θ) − (∂θS)(x0, θ0) | + | θ − θ0|] (3.2)

Proof . The mappings

Rn3 θ → fx(θ) = ∂xS(x, θ), Rn3 x → gθ(x) = ∂θS(x, θ)

are global diffeomorphisms of R n

. F rom ( G 4 ) and ( G 5 ) , it follows that k (f

−1 x )0k,

k (g−1_θ )0k and k (h−1₂ )0k _{are uniformly bounded on R}2n_{. Thus ( G 5 ) and the Taylor}

’ s theorem lead to the following estimates : There exist M, N > 0, such that for all

(x, θ), (x0, θ0_{) ∈ R}2n, | θ |=| fx−1(fx(θ)) − fx−1(fx(0)) |≤ M | ∂xS(x, θ) − ∂xS(x, 0) |≤ C6λ(x, ∂xS),

EJDE −206/26 L2−BOUNDEDNESS AND L2−COMPACTNESS 5 withC6> 0; | x |=| g_θ−1(gθ(θ)) − g−1_θ (gθ(0)) |≤ N | ∂θS(x, θ) − ∂θS(0, θ) |≤ C7λ(∂θS, θ), withC7> 0; | (x, θ) − (x0, θ0) |=| h−1₂ (h2(x, θ)) − h−12 (h2(x0, θ0)) | ≤ C5| (θ, ∂θS(x, θ)) − (θ0, ∂θS(x0, θ0)) |  When θ = θ0 in ( 3 . 2 ) , there exists C5> 0, such that for all (x, x0, θ) ∈ R3n,

| x − x0|≤ C5| (∂θS)(x, θ) − (∂θS)(x0, θ) | . (3.3)

Proposition 3 . 4 . If S satisfies ( G 1 ) and ( G 5 ) , then there exists a constant 0> 0 such that the phase function φ given in ( 3 . 1 ) belongs to Γ21(Ωφ,0) where

Ωφ,0 = {(x, θ, y) ∈ R

3n

; | ∂θS(x, θ) − y |2< 0(| x |2+ | y |2+ | θ |2)}.

Proo f − period We have to show that : There exists 0> 0, such that for all

α, β, γ ∈ Nn_, thereexistCα,β,γ > 0 : | ∂xα∂ β θ∂ γ yφ(x, θ, y) |≤ Cα,β,γλ(x, θ, y)(2−|α|−|β|−|γ|), ∀(x, θ, y) ∈ Ωφ,0. (3.4) If | γ |= 1, then | ∂α x∂ β θ∂ γ yφ(x, θ, y) |=| ∂ α x∂ β θ(−θ) |= ( 0 if | α |= 0 | ∂_θβ(−θ) | ifα = 0; If | γ |> 1, then | ∂α_x∂_θβ∂_yγφ(x, θ, y) |= 0. Hence the estimate ( 3 . 4 ) is satisfied .

If | γ |= 0, then for all α, β ∈ Nn_{; | α | + | β |≤ 2, there exists C}

α,β> 0 such that | ∂α x∂ β θφ(x, θ, y) |=| ∂ α x∂ β θS(x, θ) − ∂ α x∂ β θ(yθ) |≤ Cα,βλ(x, θ, y)(2−|α|−|β|).

If | α | + | β |> 2, one has ∂_xα∂β_θφ(x, θ, y) = ∂_xα∂_θβS(x, θ). In Ωφ,0 we have

| y |=| ∂θS(x, θ) − y − ∂θS(x, θ) |≤

√

0(| x |2+ | y |2+ | θ |2)1/2+ C8λ(x, θ),

with C8> 0. For 0sufficiently small , we obtain a constant C9> 0 such that

| y |≤ C9λ(x, θ), ∀(x, θ, y) ∈ Ωφ,0. (3.5)

This inequality leads to the equivalence

λ(x, θ, y) ' λ(x, θ) inΩφ,0 (3.6)

thus the assumption (G5) and ( 3 . 6 ) give the estimate (3.4). _

Using ( 3 . 6 ) , we have the following result . Proposition 3 . 5 . If (x, θ) → a(x, θ) belongs to Γm_{k (R}n_x× Rn

θ), then (x, θ, y) →

a(x, θ) belo ngs to Γm_k(Rnx× Rnθ× R n

6 B . MESSIRDI , A . S ENOUSSAOUI EJDE - 2 6 / 2 6

4. L2− boundedness and L2

− compactness of F The main result is as follows .

Theorem 4 . 1 . Let F be the integral operator of distribution kernel

K(x, y) = Z

Rn

ei(S(x,θ)−yθ)a(x, θ)db_{θ (4.1)}

where dθ = (2π)b −ndθ, a ∈ Γm_k(R2n_{x ,θ}), k = 0, 1 and S satisfies (G1), ( G 4 ) and ( G 5 ) .

Then F F∗ _{and F}∗_{F are pseudodifferential operators with symbol in Γ}2m k (R 2n_), k = 0, 1, given by σ(F F∗)(x, ∂xS(x, θ)) ≡| a(x, θ) |2| (det ∂2_S ∂θ∂x) −1_{(x, θ) |} σ(F∗F )(∂θS(x, θ), θ) ≡| a(x, θ) |2| (det ∂2S ∂θ∂x) −1_{(x, θ) |}

we denote here a ≡ b for a, b ∈ Γ2p_{k (R}2n_{) if (a − b) ∈ Γ}2p−2

k (R

2n_{) and σ s tands for}

the symbol . Proof . _{If u ∈ S(R}n_{), then F u(x) is given by}

F u(x) = Z Rn K(x, y)u(y)dy == Z RnR_Rn Rnθ)a(x,θ)( R Rne −iyθ_u(y)dy) Rei(S(x,θ)−yθ) a(x, eiS(x, θ)u(y)dy b d_θ b dθ (4.2) = Z Rn

eiS(x,θ)a(x, θ)F u(θ)db_θ.

Here F is a continuous linear mapping from S(Rn) to S(Rn)( by Theorem 2 . 1 ) . Let

v ∈ S(Rn), then hF u, viL2 (Rn) = Z Rn ( Z Rn

eiS(x,θ)a(x, θ)F u(θ)db_{θ) v(x)dx}

= Z Rn F u(θ)( Z Rn eiS−(x,θ) a(x,θ)v(x)dx) bdθ thus hF u(x), v(x)iL2 (Rn) = (2π)−nhF u(θ), F ((F∗v))(θ)iL2_(Rn) where F ((F∗v))(θ) = Z Rn e−iS(x−e,θ) _a( e x,θ)v(x)de xe. (4.3) Hence , for all v ∈ S(Rn),

(F F∗v)(x) = Z Rn Z Rn ei(S(x,θ)−S(x−e,θ))_{a(x, θ) a(} e x,θ)dx bedθ. (4.4) The main idea to show that F F∗ _{is a pseudodifferential operator , is to use the fact}

after considering the change of variables (x,_ex,θ) → (x,ex,ξ = ξ(x,xe,θ)). The distribution kernel of F F∗ is

K(x, ˜x) = Z

Rn

ei(S(x,θ)−S(˜x,θ))_{a(x, θ) a(˜x}

EJDE −206/26 L2−BOUNDEDNESS AND L2−COMPACTNESS 7 We obtain from ( 3

. 3 ) that if | x −x |≥_e ₂λ(x,x_e,θ)( where  > 0 is sufficiently small )

then

| (∂θS)(x, θ) − (∂θS)(ex,θ) |≥  2C5

λ(x,x_e,θ). (4.5)

Choosing ω ∈ C∞_{(R) such that}

ω(x) ≥ 0, _{∀x ∈ R} ω(x) = 1 ifx ∈ [−1 2, 1 2] suppω ⊂] − 1, 1[ and setting b(x, ˜x,θ) := a(x, θ) a(˜x,θ) = b1,(x, ˜x,θ) + b2,(x, ˜x,θ) b1,(x, ˜x,θ) = ω( | x − ˜x | λ(x, ˜x,θ) )b(x, ˜x,θ) b2,(x, ˜x,θ) = [1 − ω( | x − ˜x | λ(x, ˜x,θ) )]b(x, ˜x,θ). We have K(x,_ex) = K1,(x,x) + K_e 2,(x,_ex), where Kj,(x, ˜x) = Z Rn ei(S(x,θ)−S(˜x,θ))_b j,(x, ˜x,θ)dbθ, j = 1, 2.

We will study separately the kernels K1, and K2,.

On the support of b2,, inequality ( 4 . 5 ) is satisfied and we have

K2,(x,_ex) ∈ S(Rn× Rn).

Indeed , using the oscillatory integral method , there is a linear partial differential operator L of order 1 such that

L(ei(S(x,θ)−S(˜x,θ))_{) = e}i(S(x,θ)−S(˜x,θ))

where n L = −i | (∂θS)(x, θ) − (∂θS)(ex,θ) | −2X [(∂θlS)(x, θ) − (∂θlS)(xe,θ)]∂θl. l = 1 The transpose operator of L is

n tL= X Fl(x,ex,θ)∂θl+ G(x,xe,θ) l = 1 whereFl(x,x_e,θ) ∈ Γ−10 (Ω), G(x,_ex,θ) ∈ Γ−20 (Ω), Fl(x,_ex,θ) = i | (∂θS)(x, θ) − (∂θS)(x_e,θ) |−2 ((∂θlS)(x, θ) − (∂θlS)(ex,θ)), n G(x,x_e,θ) = i X ∂θl[| (∂θS)(x, θ) − (∂θS)(xe,θ) | −2 _((∂ θlS)(x, θ) − (∂θlS)(xe,θ))], l = 1 Ω= {(x, ˜x,θ) ∈ R3n:| ∂θS(x, θ) − ∂θS(˜x,θ) |>  2C5 λ(x, ˜x,θ)}.

On the other hand we prove by induction on q that (tL)qb2,(x, ˜x,θ) = X gγ,q(x, ˜x,θ)∂θγb2,(x, ˜x,θ), γg(q)∈ Γ −q 0 (Ω), | γ |≤ q, γ ∈ Nn

8 B . MESSIRDI , A . S ENOUSSAOUI EJDE - 2 6 / 2 6 and so , K2,(x, ˜x) = Z Rn ei(S(x,θ)−S(˜x,θ))₍t_L)q_b 2,(x, ˜x,θ)dbθ.

Using Leibnitz ’ s formula , ( G 5 ) and the form (t_L)q_{, we can choose q large enough}

such that for all α, α0, β, β0∈ Nn_{, ∃C}

α,α0,β, β0> 0, sup ,x−e x ∈Rn | xα e xα_∂0 x β_∂β0 x−eK2,(x,_ex) |≤ Cα,α0,β, β0.

Next , we study K₁ : this is more difficult and depends on the choice of the parameter . It follows from Taylor ’ s formula that

S(x, θ) − S(_ex,θ) = hx −xe,ξ(x,xe,θ)iRn, ξ(x,x_e,θ) =

Z 1

0

(∂xS)(ex + t(x −ex), θ)dt. We define the vectorial function

e ξ(x,x_e,θ) = ω( | x − ˜x | 2λ(x, ˜x,θ) ) ξ(x,_ex,θ) + (1 − ω( | x − ˜x | 2λ(x, ˜x,θ) ))(∂xS)(_ex,θ). We have e ξ(x,_ex,θ) = ξ(x,x_e,θ)onsuppb1,.

Moreover , for  sufficiently small ,

λ(x, θ) ' λ(x_e,θ) ' λ(x,ex,θ)onsuppb1,. (4.6) Let us consider the mapping

R3n3 (x,xe,θ) → (x,ex,ξe(x,ex,θ)) (4.7) for which Jacobian matrix is

  In 0 0 0 In 0 ∂xξe ∂x−eξe ∂θξe  . We have ∂ eξ,j ∂θi (x,x_e,θ) = ∂ 2_S ∂θi∂xj (_ex,θ) + ω( | x − ˜x | 2λ(x, ˜x,θ) ) (∂ξj ∂θi (x,x_e,θ) − ∂2S ∂θi∂xj (x_e,θ)) − | x − ˜x | 2λ(x, ˜x,θ) ∂λ ∂θi (x, ˜x,θ)λ−1(x, ˜x,θ)ω0( | x − ˜x | 2λ(x, ˜x,θ) ) (ξj(x,ex,θ) − ∂S ∂xj (x_e,θ)). Thus , we obtain

|∂ eξ,j ∂θi (x,x_e,θ) − ∂2_S ∂θi∂xj (_ex,θ)| ≤ |ω( | x − ˜x | 2λ(x, ˜x,θ) )||∂ξj ∂θi (x,x_e,θ) − ∂2S ∂θi∂xj (_ex,θ)| +λ−1(x, ˜x,θ)|ω0( | x − ˜x | 2λ(x, ˜x,θ) )||ξj(x,ex,θ) − ∂S ∂xj (x_e,θ)|.

EJDE −206/26 L2−BOUNDEDNESS AND L2−COMPACTNESS 9 Now it follows from

( G 5 ) , ( 4 . 6 ) and Taylor ’ s formula that

|∂ξj ∂θi (x,x_e,θ) − ∂2S ∂θi∂xj (_ex,θ)| ≤ Z 1 0 | ∂ 2_S ∂θi∂xj (_ex + t(x −_ex), θ) − ∂ 2_S ∂θi∂xj (_ex,θ)|dt ≤ C10| x −x | λe −1_{(x, ˜x} ,θ), C10> 0 ( 4 . 8 ) |ξj(x,x_e,θ) − ∂S ∂xj (_ex,θ)| ≤ Z 1 0 |∂S ∂xj (x + t(x −_{e e}x), θ) − ∂S ∂xj (x_e,θ)|dt (4.9) ≤ C11| x −x |,_e C11> 0.

From ( 4 . 8 ) and ( 4 . 9 ) , there exists a positive constant C12> 0 such that

| ∂ eξ,j ∂θi (x,x_e,θ) − ∂2_S ∂θi∂xj (x_e,θ) |≤ C12, ∀i, j ∈ {1, ..., n}. (4.10) If  < δ0

2C−e ,then ( 4 . 1 0 ) and ( G 4 ) yields the estimate

δ0/2 ≤ − eC+ δ0≤ − eC+ det

∂2_S

∂x∂θ(x, θ) ≤ det ∂θξe(x,xe,θ), (4.11) with eC > 0 If  is such that ( 4 . 6 ) and ( 4 . 1 1 ) hold , then the mapping given in ( 4 . 7 ) is a global diffeomorphism of R3n_{. Hence there exists a mapping}

θ : Rn× Rn × Rn_{3 (x,} e x,ξ) → θ(x,_ex,ξ) ∈ Rn such that e ξ(x,xe,θ(x,ex,ξ)) = ξ θ(x,x_e,ξe(x,xe,θ)) = x (4.12) ∂αθ(x,x_e,ξ) = O(1), ∀α ∈ N3n\ {0}

If we change the variable ξ by θ(x,_ex,ξ) in K1,(x,_ex), we obtain

K1,(x,x) =e Z Rn eihx−˜x,ξi_b 1,(x, ˜x,θ(x,xe,ξ))| det ∂θ ∂ξ(x,xe,ξ)| b d_{ξ. (4.13)}

From ( 4 . 1 2 ) we have , for k = 0, 1, that b1,(x, ˜x,θ(x,x_e,ξ)) | det ∂θ_∂ξ(x,x_e,ξ) | belongs

toΓ2m_{k (R}3n)ifa ∈ Γm_{k (R}2n).

Applying the stationary phase theorem ( c . f . [ 1 2 ] ) to 4 . 1 3 , we obtain the expres - sion of the symbol of the pseudodifferential operator F F∗,

σ(F F∗) = b1,(x, ˜x,θ(x,xe,ξ))| det ∂θ ∂ξ(x,ex,ξ)| | e − x = x + R(x, ξ) where R(x, ξ) belongs to Γ2m−2_{k (R}2n_{) if a ∈ Γ}m k (R 2n_{), k = 0, 1.}

For ˜x = x, we have b1,(x, ˜x,θ(x,xe,ξ)) =| a(x, θ(x, x, ξ)) |

2 _{where θ(x, x, ξ) is the}

inverse of the mapping θ → ∂xS(x, θ) = ξ. Thus

σ(F F∗)(x, ∂xS(x, θ)) ≡| a(x, θ) |2| det

∂2_S

∂θ∂x(x, θ)|

−1 .

1 0 B . MESSIRDI , A . S ENOUSSAOUI EJDE - 2 6 / 2 6 From ( 4 . 2 ) and ( 4 . 3 ) , we

obtain the expression of F∗_{F : ∀v ∈ S(R}n),

(F (F∗F )F−1)v(θ) = Z Rn e−iS(x,θ) a(x, θ)(F (F−1v))(x)dx = Z Rn e−iS(x,θ) a(x, θ)( Z Rn

eiS(x,theta−e)a(x, eθ)(F (F−1v))(θe₎debθ_)dx

= Z Rn Z Rn e−i(S(x,θ)−S(x, ˜ θ₎₎ a(x, θ)a(x, eθ)v( ˜ θ₎debθ_dx.

Hence the distribution kernel of the integral operator F (F∗F )F−1 is

e

K(θ, eθ) = Z

Rn

e−i(S(x,θ)−S(x,θ˜)) a(x, θ)a(x, ˜θ)db_x.

We remark that we can deduce eK(θ, eθ) from K(x,_ex) 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 pseudodifferential operator with symbol}

σ(F (F∗F )F−1)(θ, −∂θS(x, θ)) ≡| a(x, θ) |2| det

∂2_S

∂x∂θ(x, θ)|

−1 .

Thus the symbol of F∗F is given by ( c . f . [ 9 ] )

σ(F∗F )(∂θS(x, θ), θ) ≡| a(x, θ) |2| det ∂2_S ∂x∂θ(x, θ)| −1 . 

Corollary 4 . 2 . Let F be the integral operator with the distribution kernel K(x, y) =

Z

Rn

ei(S(x,θ)−yθ)a(x, θ)db_θ

where a ∈ Γm

0(R2nx,θ) and S satisfies ( G 1 ) , ( G 4 ) and ( G 5 ) . Then , we

have :

( 1 ) For any m such that m ≤ 0, F can be extended as a bounded lin ear mapping

onL2_(Rn)

( 2 ) For any m such that m < 0, F can be extended as a compact operator on L2_(Rn).

Proo f − period It follows from Theorem 4 . 1 that F∗F is a pseudodifferential operator with

symbolinΓ2m0 (R 2n

).

( 1 ) If m ≤ 0, the weight λ2m_{(x, θ) is bounded , so we can apply the Cald ´e}

ron - Vaillancourt theorem ( see [ 2 , 1 2 , 1 3 ] ) for F∗F and obtain the existence of a positive constant γ(n) and a integer k(n) such that

k (F∗F )u k L2_{(Rn) ≤ γ(n)Qk(n)}(σ(F F∗))ku k L2_(Rn), ∀u ∈ S(Rn₎

Qk(n)(σ(F F∗)) =| α | +X |β|≤ k(n)(θsup,x ) ∈ R 2 n|∂xα∂ β θσ(F F ∗_)(∂ θS(x, θ), θ)|

Hence , for all u ∈ S(Rn_),

k F u k L2

(Rn) ≤k F∗F k1/2_L(L2_(Rn₎₎k u k L

2

(Rn) ≤ (γ(n)Qk(n)(σ(F F∗)))1/2k u k L2(Rn).

Thus F is also a bounded linear operator on L2

(Rn_{). ( 2 ) If m < 0, lim}

|x|+|θ|→+∞λm(x, θ) =

0, and the compactness theorem ( see [ 1 2 ,

EJDE −206/26 L2−BOUNDEDNESS AND L2−COMPACTNESS 1 1 Thus , the Fourier

integral operator F is compact on L2_(Rn). Indeed , let (ϕj)j ∈ N

be an orthonormal basis of L2_(Rn), then

n k F∗F −Xhϕj, .iF∗F ϕjk→ 0 asn → +∞.

j = 1 Since F is bounded , for all ψ ∈ L2

(Rn_), n n n kF ψ −Xhϕj, ψiF ϕjk2 ≤ kF∗F ψ − X hϕj, ψiF∗F ϕjkkψ − X hϕj, ψiϕjk, j = 1 j = 1 j = 1 it follows that n k F −Xhϕj, .iF ϕj k→ 0 asn → +∞ j = 1  Example 4 . 3 . We consider the function given by

S(x, θ) = X Cα,βxαθβ, for(x, θ) ∈ R2n

| α | + | β |= 2, α, β ∈ Nn

where Cα,β are real constants . This function satisfies ( G 1 ) , ( G 4 ) and ( G 5 ) .

Acknowledgements . This paper was completed while the second author was vis -iting the ” U niversit´ Libr d B r − u xelles ” H wa nt t h − ta − n k Professo J . - P Gosse

for his valuable discussions . Investigations supported by University of Oran Es -senia , Algeria . CNEPRU B 3 1 0 1 / 2 / 3 .

References

[ 1 ] K . Asada and D . Fujiwara , On some oscil latory transformation in L2_(Rn),Japan . J . Math . Vol . 4 ( 2 ) , 1 978 , p . 299 - 36 1 .

[ 2 ] A . P . Cald ´eron and R . Vaillancourt , On the boundedness of pseudodifferential operators , J . Math . Soc . Japan 23 , 1 972 , p . 374 - 378 .

[ 3 ] P . Drabek and J . Milota , Lectures on nonlinear analysis , Pleze nˇ,Czech Republic , 2004 .

[ 4 ] J . J . Duistermaat , Fourier integral operators , Courant Institute Lecture Notes , New -York 1 973 .

[ 5 ] Yu . V . Egorov and M . A . Shubin , Partial differential equations , Vol . 2 , Springer -Verlag , Berlin , 1 991 .

[ 6 ] M . Hasanov , A class of unbounded Fourier integral operators , J . Math . Analysis and appli - cation 225 , 1 998 , p . 641 - 65 1 .

[ 7 ] B . Helffer , Th e´orie spectrale pour des op´erateurs globalement ellip tiques , Soci ´et e´Math e´matiques de France , Ast ´erisque 1 1 2 , 1 984 .

[ 8 ] L . H o¨rmander , Fourier integral operators I , Acta Math . Vol . 1 27 , 1 971 , p . 33 - 57 . [ 9 ] L . H o¨rmander , The Weyl calculus of pseudodifferential operators , Comm . Pure . Appl . Math . 32 ( 1 979 ) , p . 359 - 443 . [ 10 ] B . Messirdi et A . Senoussaoui , Parametrix du probl e` me de Cauchy C∞hyperbolique muni d ’ un syst `eme ordres de Leray - Volev ˆıc , Journal

for Analysis and its Applications , Vol . 24 , No . 3 , ( 2005 ) , p . 58 1 - 592 . [ 1 1 ] B . Messirdi and A . Senoussaoui , A cl ass of unbounded Fourier integral operators with symbol

in T0<ρ<1S0ρ,1, Submitted 2005 .

[ 1 2 ] D . Robert , Autour de l ’ approximation s emi - classique , Birk ¨auser , 1 987 . [ 1 3 ] A . Senoussaoui , Op ´erateurs h - admissibles matriciels `asymboles ope´_{rateurs , African Diaspora}

Journal of Mathematics , Vol . 4 , No . 1 , ( 2005 ) . [ 14 ] J . T . Schwartz , Non linear functional analysis , Gordon and Breach Publishers , 1 969 . [ 1 5 ] M . Shubin , Pseudodifferential operators and spectral theory , Naukia Moscow , 1 978 .

1 2 B . MESSIRDI , A . S ENOUSSAOUI EJDE - 2 6 / 2 6

Bekkai Messirdi Universit ´e d ’ Oran Es - S e´nia , Facult ´e des S ciences , D e´ partement de Math e´matiques , B . P .

1 5 24 El - Mnaouer , Oran , Algeria

E - mail address : bmessirdi @u − niv - oran . dz

Abderrahmane Senoussaoui Universit e´d ’ Oran Es - S ´enia , Facult e´des S ciences , D ´epartement de Math ´ematiques , B . P .

1 5 24 El - Mnaouer , Oran , Algeria