\documentclass[reqno]{amsart} \usepackage{amssymb,hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2006(2006), No. 26, pp. 1--12.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2006 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2006/26\hfil $L^2$-boundedness and $L^2$-compactness] {$L^2$-boundedness and $L^2$-compactness of a class of Fourier integral operators} \author[B. Messirdi, A. Senoussaoui\hfil EJDE-2006/26\hfilneg] {Bekkai Messirdi, Abderrahmane Senoussaoui} % in alphabetical order \address{Bekkai Messirdi \newline Universit\'{e} d'Oran Es-S\'{e}nia, Facult\'{e} des Sciences, D\'{e}partement de\ Math\'{e}matiques, B.P. 1524 El-Mnaouer, Oran, Algeria} \email{bmessirdi@univ-oran.dz} \address{Abderrahmane Senoussaoui \newline Universit\'{e} d'Oran Es-S\'{e}nia, Facult\'{e} des Sciences, D\'{e}partement de Math\'{e}matiques, B.P. 1524 El-Mnaouer, Oran, Algeria} \email{asenouss@ulb.ac.be} \date{} \thanks{Submitted June 10, 2005. Published March 9, 2006.} \subjclass[2000]{35S30, 35S05, 47A10, 35P05} \keywords{Fourier integral operators; pseudodifferential operators; \hfill\break\indent symbol and phase; boundedness and compactness} \begin{abstract} In this paper, we study the $L^2$-boundedness and $L^2$-compactness of a class of Fourier integral operators. These operators are bounded (respectively compact) if the weight of the amplitude is bounded (respectively tends to $0$). \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} \newtheorem{corollary}[theorem]{Corollary} \section{Introduction} For $\varphi \in \mathcal{S}(\mathbb{R}^n)$ (the Schwartz space), the integral operators \begin{equation} F\varphi (x)=\int e^{iS(x,\theta ) }a(x,\theta )\mathcal{F}\varphi (\theta ) \,d\theta \label{1.1} \end{equation} appear naturally in the expression of the solutions of the hyperbolic partial differential equations and in the expression of the $C^{\infty }$ -solution of the associate Cauchy's problem (see \cite{EgSh,MeSe1}). If we write formally the Fourier transformation $\mathcal{F}\varphi (\theta )$ in \eqref{1.1}, we obtain the following Fourier integral operators \begin{equation} F\varphi (x)=\iint e^{i(S(x,\theta )-y\theta )}a(x,\theta )\varphi ( y)dy\,d\theta \label{1.2} \end{equation} in which appear two $C^{\infty }$-functions, the phase function $\phi (x,y,\theta )=S(x,\theta )-y\theta $ 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., \cite{AsFu,Du,Ha,He,Ho1,Sh}). The first works on Fourier integral operators deal with local properties. On the other hand, Asada and Fujiwara have studied for the first time a class of Fourier integral operators defined on $\mathbb{R}^n$. For the 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 \cite{AsFu} by a very elaborated proof and with some hypothesis on the phase function $\phi $ and the amplitude $a$ that all operators of the form \eqref{2.1} (see below) are bounded on $L^2$. The technique used there is based on the fact that the operators $I(a,\phi )I^{\ast }(a,\phi ),I^{\ast}(a,\phi )I(a,\phi )$ are pseudodifferential and it uses Cald\'{e}ron-Vaillancourt's theorem (here $I(a,\phi )^{\ast }$ is the adjoint of $I(a,\phi ))$. In this work, we apply the same technique of \cite{AsFu} to establish the boundedness and the compactness of the operators \eqref{1.2}. To this end we give a brief and simple proof for a result of \cite{AsFu} in our framework. We mainly prove the continuity of the operator $F$ on $L^2(\mathbb{R}^n)$ when the weight of the amplitude $a$ is bounded. Moreover, $F$ is compact on $L^2(\mathbb{R}^n)$ if this weight tends to zero. Using the estimate given in \cite{Ro} for $h$-pseudodifferential ($h$-admissible) operators, we also establish an $L^2$-estimate of $\|F\|$. We note that if the amplitude $a$ is juste bounded, the Fourier integral operator $F$ is not necessarily bounded on $L^2(\mathbb{R}^n)$. Recently, Hasanov \cite{Ha} and Messirdi-Senoussaoui \cite{MeSe2} constructed a class of unbounded Fourier integral operators with an amplitude in the H\"{o}rmander's class $S_{1,1}^{0}$ and in $\bigcap_{0<\rho <1}S_{\rho ,1}^{0}$. To our knowledge, this work constitutes a first attempt to diagonalize the Fourier integral operators on $L^2(\mathbb{R}^n)$ (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 $\mathcal{S}(\mathbb{R}^n)$ and on $\mathcal{S}'(\mathbb{R}^n)$. The assumptions and preliminaries results are given in the third section. The last section is devoted to prove the main result. \section{A general class of Fourier integral operators} If $\varphi \in \mathcal{S}(\mathbb{R}^n)$, we consider the following integral transformations \begin{equation} (I(a,\phi )\varphi )(x)=\underset{ \mathbb{R}_{y}^n\times \mathbb{R}_{\theta }^{N}}{\iint }e^{i\phi (x,\theta ,y)}a(x,\theta ,y)\varphi (y)dy\,d\theta \label{2.1} \end{equation} where, $x\in \mathbb{R}^n$, $n\in \mathbb{N}^{\ast }$ and $N\in \mathbb{N}$ (if $N=0$, $\theta $ doesn't appear in \eqref{2.1}). In general the integral \eqref{2.1} is not absolutely convergent, so we use the technique of the oscillatory integral developed by H\"{o}rmander in \cite{Ho1}. The phase function $\phi $ and the amplitude $a$ are assumed to satisfy the following hypothesis: \begin{itemize} \item[(H1)] $\phi \in C^{\infty }(\mathbb{R}_{x}^n\times \mathbb{R} _{\theta }^{N}\times \mathbb{R}_{y}^n,\mathbb{R})$ ($\phi$ is a real function) \item[(H2)] For all $(\alpha ,\beta ,\gamma )\in \mathbb{N} ^n\times \mathbb{N}^{N}\times \mathbb{N}^n$, there exists $C_{\alpha ,\beta,\gamma }>0$ such that \begin{equation*} |\partial _{y}^{\gamma }\partial _{\theta }^{\beta }\partial _{x}^{\alpha }\phi (x,\theta ,y)| \leq C_{\alpha ,\beta ,\gamma }\lambda ^{(2-|\alpha | -|\beta | -|\gamma | )_{+}}(x,\theta ,y) \end{equation*} where $\lambda (x,\theta ,y)=(1+|x| ^2+|\theta |^2+|y| ^2)^{1/2}$ is called the weight and \begin{equation*} (2-|\alpha | -|\beta| -|\gamma | )_{+} =\max (2-|\alpha | -|\beta | -|\gamma | ,0) \end{equation*} \item[(H3)] There exist $K_{1},K_{2}>0$ such that \begin{equation*} K_{1}\lambda (x,\theta ,y)\leq \lambda (\partial _{y}\phi ,\partial _{\theta }\phi ,y)\leq K_{2}\lambda (x,\theta ,y),\quad \forall (x,\theta ,y)\in \mathbb{R}_{x}^n\times \mathbb{R}_{\theta }^{N}\times \mathbb{R}_{y}^n \end{equation*} \item[(H3*)] There exist $K_{1}^{\ast },K_{2}^{\ast }>0$ such that \begin{equation*} K_{1}^{\ast }\lambda (x,\theta ,y)\leq \lambda (x,\partial _{\theta }\phi ,\partial _{x}\phi )\leq K_{2}^{\ast }\lambda (x,\theta ,y), \quad \forall (x,\theta ,y)\in \mathbb{R}_{x}^n\times \mathbb{R}_{\theta }^{N}\times \mathbb{R}_{y}^n. \end{equation*} \end{itemize} For any open subset $\Omega $ of $\mathbb{R}_{x}^n\times \mathbb{R}_{\theta }^{N}\times \mathbb{R}_{y}^n$, $\mu \in \mathbb{R}$ and $\rho \in [0,1] $, we set \begin{align*} \Gamma _{\rho }^{\mu }(\Omega ) &=\big\{a\in C^{\infty }(\Omega ): \forall (\alpha ,\beta ,\gamma )\in \mathbb{N}^n\times \mathbb{N}^{N}\times \mathbb{N}^n, \; \exists C_{\alpha ,\beta ,\gamma }>0: \\ &\quad |\partial _{y}^{\gamma }\partial _{\theta }^{\beta }\partial _{x}^{\alpha }a(x,\theta ,y)| \leq C_{\alpha ,\beta ,\gamma }\lambda ^{\mu -\rho (|\alpha | +|\beta | +|\gamma| )}(x,\theta ,y)\big\} \end{align*} When $\Omega =\mathbb{R}_{x}^n\times \mathbb{R}_{\theta }^{N}\times \mathbb{R}_{y}^n$, we denote $\Gamma _{\rho }^{\mu}(\Omega )=\Gamma _{\rho }^{\mu }$. To give a meaning to the right hand side of \eqref{2.1}, we consider $g\in \mathcal{S}(\mathbb{R}_{x}^n\times \mathbb{R}_{\theta }^{N}\times \mathbb{R}_{y}^n)$, $g(0)=1$. If $a\in \Gamma _{0}^{\mu }$, we define \begin{equation*} a_{\sigma }(x,\theta ,y)=g(x/\sigma ,\theta /\sigma ,y/\sigma )a(x,\theta ,y),\quad \sigma >0. \end{equation*} Now we are able to state the following result. \begin{theorem} \label{thm2.1} If $\phi $ satisfies (H1), (H2), (H3) and (H3*), and if $a\in \Gamma _{0}^{\mu }$, then 1. For all $\varphi \in \mathcal{S}(\mathbb{R}^n)$, $\lim_{\sigma \to +\infty } [ I(a_{\sigma },\phi)\varphi] (x)$ exists for every point $x\in \mathbb{R}^n$ and is independent of the choice of the function $g$. We define \begin{equation*} (I(a,\phi )\varphi )(x):=\lim_{\sigma \to +\infty} (I(a_{\sigma },\phi )\varphi )(x) \end{equation*} 2. $I(a,\phi )\in \mathcal{L}(\mathcal{S}(\mathbb{R}^n))$ and $I(a,\phi )\in \mathcal{L}(\mathcal{S}'(\mathbb{R}^n))$ (here $\mathcal{L}(E)$ is the space of bounded linear mapping from $E$ to $E$ and $\mathcal{S}'(\mathbb{R}^n)$ the space of all distributions with temperate growth on $\mathbb{R}^n$). \end{theorem} The proof of the above theorem can be found in \cite{He} or in \cite[propostion II.2]{Ro}. \begin{example} \label{exa2.2} \rm Let us give two examples of operators of the form \eqref{2.1} which satisfy (H1)-(H3*): \begin{enumerate} \item The Fourier transform $\mathcal{F}\psi (x)= \int_{\mathbb{R}^n} e^{-ixy}\psi (y)dy$, $\psi \in \mathcal{S} (\mathbb{R}^n)$, \item Pseudodifferential operators \[ A\psi (x)=(2\pi )^{-n}\int_{\mathbb{R}^{2n}} e^{i(x-y)\theta }a(x,y,\theta )\psi (y)dy\,d\theta , \] with $\psi \in \mathcal{S}(\mathbb{R}^n)$, $a\in \Gamma _{0}^{\mu }(\mathbb{R} ^{3n})$. \end{enumerate} \end{example} \section{Assumptions and Preliminaries} In this paper we consider the special form of the phase function \begin{equation} \phi (x,y,\theta )=S(x,\theta )-y\theta \label{3.1} \end{equation} where $S$ satisfies \begin{itemize} \item[(G1)] $S\in C^{\infty }(\mathbb{R}_{x}^n\times \mathbb{R}_{\theta }^n,\mathbb{R})$, \item[(G2)] For each $(\alpha ,\beta )\in \mathbb{N}^{2n}$, there exist $C_{\alpha ,\beta }>0$, such that \[ |\partial _{x}^{\alpha }\partial_{\theta }^{\beta }S(x,\theta )| \leq C_{\alpha ,\beta}\lambda (x,\theta )^{(2-|\alpha | -|\beta | )_{+}}, \] \item[(G3)] There exists $C_{1}>0$ such that $|x| \leq C_{1}\lambda (\theta ,\partial _{\theta }S)$, for all $(x,\theta )\in \mathbb{R}^{2n}$, \item[(G3*)] There exists $C_{2}>0$, such that $|\theta |\leq C_{2}\lambda (x,\partial _{x}S)$, for all $(x,\theta )\in \mathbb{R}^{2n}$. \end{itemize} \begin{proposition} \label{prop3.1} Let's assume that $S$ satisfies (G1), (G2), (G3) and (G3*). Then the function $\phi (x,y,\theta )=S(x,\theta )-y\theta $ satisfies (H1), (H2), (H3) and (H3*). \end{proposition} \begin{proof} (H1) and (H2) are trivially satisfied. The condition (G3) implies \begin{equation*} \lambda (x,\theta ,y)\leq \lambda (x,\theta )+\lambda (y)\leq C_{3}(\lambda (\theta ,\partial _{\theta }S)+\lambda (y)),\quad C_{3}>0. \end{equation*} Also, we have $\partial _{y_{j}}\phi =-\theta _{j}$ and $\partial _{\theta _{j}}\phi =\partial _{\theta _{j}}S-y_{j}$ and so \begin{equation*} \lambda (\theta ,\partial _{\theta }S)=\lambda (\partial _{y}\phi ,\partial _{\theta }\phi +y)\leq 2\lambda (\partial _{y}\phi ,\partial _{\theta }\phi ,y), \end{equation*} which finally gives for some $C_{4}>0$, \begin{equation*} \lambda (x,\theta ,y)\leq C_{3}(2\lambda (\partial _{y}\phi ,\partial _{\theta }\phi ,y)+\lambda (y))\leq \frac{1}{C_{4}}\lambda (\partial _{y}\phi ,\partial _{\theta }\phi ,y) \end{equation*} The second inequality in (H3) is a consequence of the assumption (G2). By a similar argument we can show (H3*). \end{proof} We now introduce the assumption \begin{itemize} \item[(G4)] There exists $\delta _{0}>0$ such that \begin{equation*} \inf_{x,\theta \in \mathbb{R}^n}|\det \frac{\partial ^2S}{ \partial x\partial \theta }(x,\theta )| \geq \delta _{0}. \end{equation*} \end{itemize} We note that if $\phi (x,y,\theta )=S(x,\theta )-y\theta $, then \[ D(\phi )(x,\theta ,y)= \begin{pmatrix} \frac{\partial ^2\phi }{\partial x\partial y}(x,\theta ,y)& \frac{\partial ^2\phi }{\partial x\partial \theta }(x,\theta ,y)\\ \frac{\partial ^2\phi }{\partial \theta \partial y}( x,\theta ,y)& \frac{\partial ^2\phi }{\partial \theta \partial \theta } (x,\theta ,y) \end{pmatrix} = \begin{pmatrix} 0 & \frac{\partial ^{\;2}S}{\partial x\partial \theta }(x,\theta ) \\ -I_{n} & \frac{\partial ^{\;2}S}{\partial \theta \partial \theta }(x,\theta ) \end{pmatrix} \] and \[ \big|\det D(\phi )(x,\theta ,y)\big| =\big|\det \frac{\partial ^2S}{\partial x\partial \theta }(x,\theta )\big| \geq \delta _{0}. \] \begin{remark} \label{rmk3.2} \rm By the global implicit function theorem (cf. \cite{Sc}, \cite[theorem 4.1.7]{DrJa}) and using (G1), (G2) and (G4), we can easily see that the mappings $h_{1}$ and $h_{2}$ defined by \begin{equation*} h_{1}:(x,\theta )\to (x,\partial _{x}S(x,\theta )),\quad h_{2}:(x,\theta )\to (\theta ,\partial _{\theta}S(x,\theta )) \end{equation*} are global diffeomorphism of $\mathbb{R}^{2n}$. Indeed, \begin{equation*} h_{1}'(x,\theta )= \begin{pmatrix} I_{n} & \frac{\partial ^2S}{\partial x^2}(x,\theta ) \\ 0 & \frac{\partial ^2S}{\partial x\partial \theta }(x,\theta ) \end{pmatrix},\quad h_{2}'(x,\theta )= \begin{pmatrix} 0 & \frac{\partial ^2S}{\partial x\partial \theta }(x,\theta ) \\ I_{n} & \frac{\partial ^2S}{\partial \theta ^2}(x,\theta ) \end{pmatrix}. \end{equation*} and $|\det h_{1}'(x,\theta )| =|\det h_{2}'(x,\theta )| =|\det \frac{\partial ^2S}{\partial x\partial \theta }(x,\theta )| \geq \delta _{0}>0$, for all $(x,\theta )\in \mathbb{R}^{2n}$. Then \begin{gather*} \|(h_{1}'(x,\theta )) ^{-1}\|=\frac{1}{|\det \frac{\partial ^2S}{\partial x\partial \theta }(x,\theta )| }\|^{t}A(x,\theta )\|\\ \|(h_{2}'(x,\theta )) ^{-1}\|=\frac{1}{|\det \frac{\partial ^2S}{\partial x\partial \theta }(x,\theta )| }\|^{t}B(x,\theta )\|, \end{gather*} where $A(x,\theta )$, $B(x,\theta )$ are respectively the cofactor matrix of $h_{1}'(x,\theta )$, $h_{2}'(x,\theta )$. By (G2), we know that $\|^{t}A(x,\theta )\|$ and $\|^{t}B(x,\theta )\|$ are uniformly bounded. \end{remark} Let's now assume that $S$ satisfies the following condition which is stronger than (G2). \begin{itemize} \item[(G5)] For all $(\alpha ,\beta )\in \mathbb{N} ^n\times \mathbb{N}^n$, there exist $ C_{\alpha ,\beta }>0$, such that \[ |\partial _{x}^{\alpha }\partial _{\theta }^{\beta }S(x,\theta )| \leq C_{\alpha ,\beta }\lambda (x,\theta )^{(2-|\alpha | -|\beta | )}. \] \end{itemize} \begin{lemma} \label{lem3.3} If $S$ satisfies (G1), (G4) and (G5), then $S$ satisfies (G3) and (G3*). Also there exists $ C_{5} >0$ such that for all $(x,\theta ),(x',\theta')\in \mathbb{R}^{2n}$, \begin{equation} |x-x'| +|\theta -\theta'| \leq C_{5}\big[ |( \partial _{\theta }S)(x,\theta )-(\partial _{\theta }S)(x',\theta ')|+|\theta -\theta '| \big] \label{3.2} \end{equation} \end{lemma} \begin{proof} The mappings \begin{equation*} \mathbb{R}^n\ni \theta \to f_{x}(\theta )=\partial _{x}S(x,\theta ),\quad \mathbb{R}^n\ni x\to g_{\theta }(x) =\partial _{\theta }S(x,\theta ) \end{equation*} are global diffeomorphisms of $\mathbb{R}^n$. From (G4) and (G5), it follows that $\|(f_{x}^{-1})'\|$, $\|(g_{\theta }^{-1})'\|$ and $\|(h_{2}^{-1})'\|$ are uniformly bounded on $\mathbb{R}^{2n}$. Thus (G5) and the Taylor's theorem lead to the following estimates: There exist $ M,N>0$, such that for all $(x,\theta ),(x',\theta ')\in \mathbb{R}^{2n}$, \begin{equation*} |\theta | =|\text{ }f_{x}^{-1}( f_{x}(\theta ))-f_{x}^{-1}(f_{x}( 0))| \leq M|\partial _{x}S( x,\theta )-\partial _{x}S(x,0)| \leq C_{6}\lambda (x,\partial _{x}S), \end{equation*} with $C_{6}>0$; \begin{equation*} |x| =|\text{ }g_{\theta }^{-1}( g_{\theta }(\theta ))-g_{\theta }^{-1}( g_{\theta }(0))| \leq N| \partial _{\theta }S(x,\theta )-\partial _{\theta }S(0,\theta )| \leq C_{7}\lambda (\partial _{\theta }S,\theta ), \end{equation*} with $C_{7}>0$; \begin{align*} |(x,\theta )-(x',\theta ')| &=|h_{2}^{-1}(h_{2}(x,\theta ))-h_{2}^{-1}( h_{2}(x',\theta '))| \\ &\leq C_{5}|(\theta ,\partial _{\theta }S\;(x,\theta ))-(\theta ',\partial _{\theta }S\;(x',\theta'))| \end{align*} \end{proof} When $\theta =\theta '$ in \eqref{3.2}, there exists $C_{5}>0$, such that for all $(x,x',\theta )\in \mathbb{R}^{3n}$, \begin{equation} |x-x'| \leq C_{5}|(\partial _{\theta }S)(x,\theta )-(\partial _{\theta }S)(x',\theta )| . \label{3.3} \end{equation} \begin{proposition} \label{prop3.4} If $S$ satisfies (G1) and (G5), then there exists a constant $\epsilon_{0}>0$ such that the phase function $\phi$ given in \eqref{3.1} belongs to $\Gamma _{1}^2(\Omega _{\phi ,\epsilon _{0}})$ where \begin{equation*} \Omega _{\phi ,\epsilon _{0}} =\big\{ (x,\theta ,y)\in \mathbb{R} ^{3n};\;\;|\partial _{\theta }S(x,\theta ) -y| ^2<\epsilon _{0}\;(| x| ^2+|y| ^2+|\theta | ^2)\big\} . \end{equation*} \end{proposition} \begin{proof} We have to show that: There exists $\epsilon _{0}>0$, such that for all $\alpha ,\beta ,\gamma \in \mathbb{N}^n$, there exist $C_{\alpha ,\beta ,\gamma }>0$: \begin{equation} |\partial _{x}^{\alpha }\partial _{\theta }^{\beta }\partial _{y}^{\gamma }\phi (x,\theta ,y)| \leq C_{\alpha ,\beta ,\gamma }\lambda (x,\theta ,y)^{(2-| \alpha | -|\beta | -| \gamma | )},\quad \forall (x,\theta ,y)\in \Omega _{\phi ,\epsilon _{0}}. \label{3.4} \end{equation} If $|\gamma | =1$, then $$ |\partial _{x}^{\alpha }\partial _{\theta }^{\beta }\partial _{y}^{\gamma }\phi (x,\theta ,y)| =|\partial _{x}^{\alpha }\partial _{\theta }^{\beta }(-\theta ) | =\begin{cases} 0&\text{if }|\alpha | \neq 0 \\ |\partial _{\theta }^{\beta }(-\theta )|&\text{if } \alpha =0; \end{cases} $$ If $|\gamma | >1$, then $| \partial _{x}^{\alpha }\partial _{\theta }^{\beta }\partial _{y}^{\gamma }\phi (x,\theta ,y)| =0$.\\ Hence the estimate \eqref{3.4} is satisfied. If $|\gamma | =0$, then for all $\alpha ,\beta \in \mathbb{N}^n$; $|\alpha |+|\beta | \leq 2$, there exists $C_{\alpha ,\beta }>0$ such that \begin{equation*} |\partial _{x}^{\alpha }\partial _{\theta }^{\beta }\phi (x,\theta ,y)| =|\partial _{x}^{\alpha }\partial _{\theta }^{\beta }S(x,\theta )-\partial _{x}^{\alpha }\partial _{\theta }^{\beta }(y\theta ) | \leq C_{\alpha ,\beta }\lambda (x,\theta ,y)^{(2-|\alpha | -|\beta | )}. \end{equation*} If $|\alpha | +|\beta | >2$, one has $\partial _{x}^{\alpha }\partial _{\theta }^{\beta }\phi (x,\theta ,y)=\partial _{x}^{\alpha }\partial _{\theta }^{\beta }S(x,\theta )$. In $\Omega _{\phi ,\epsilon _{0}}$ we have \begin{equation*} |y| =|\partial _{\theta }S(x,\theta )-y -\partial _{\theta }S(x,\theta )| \leq \sqrt{\epsilon _{0}}(|x| ^2+|y| ^2 +|\theta | ^2)^{1/2}+C_{8}\lambda (x,\theta), \end{equation*} with $C_{8}>0$. For $\epsilon _{0}$ sufficiently small, we obtain a constant $C_{9}>0$ such that \begin{equation} |y| \leq C_{9}\lambda (x,\theta ),\quad \forall (x,\theta ,y)\in \Omega _{\phi ,\epsilon _{0}\;}. \label{3.5} \end{equation} This inequality leads to the equivalence \begin{equation} \lambda (x,\theta ,y)\simeq \lambda (x,\theta )\quad \text{in }\Omega _{\phi ,\epsilon _{0}\;} \label{3.6} \end{equation} thus the assumption $(G5)$ and \eqref{3.6} give the estimate \eqref{3.4}. \end{proof} Using \eqref{3.6}, we have the following result. \begin{proposition} \label{prop3.5} If $(x,\theta )\to a(x,\theta )$ belongs to $\Gamma _{k}^{m}(\mathbb{R}_{x}^n\times \mathbb{R}_{\theta }^n)$, then $(x,\theta ,y)\to a(x,\theta )$ belongs to $\Gamma _{k}^{m}(\mathbb{R}_{x}^n\times \mathbb{R}_{\theta }^n\times \mathbb{R}_{y}^n)\cap \Gamma _{k}^{m}(\Omega _{\phi ,\epsilon _{0}\;})$, $k\in \{ 0,1\} $. \end{proposition} \section{$L^2$-boundedness and $L^2$-compactness of $F$} The main result is as follows. \begin{theorem} \label{thm4.1} Let $F$ be the integral operator of distribution kernel \begin{equation} K(x,y)=\int_{\mathbb{R}^n} e^{i(S(x,\theta ) -y\theta)}a(x,\theta )\widehat{d\theta } \label{4.1} \end{equation} where $\widehat{d\theta }=(2\pi )^{-n}\,d\theta$, $a\in\Gamma_{k}^{m}(\mathbb{R}_{x,\theta }^{\;2n})$, $k=0,1$ and $S$ satisfies $(G1)$, (G4) and (G5). Then $FF^{\ast }$ and $F^{\ast }F$ are pseudodifferential operators with symbol in $\Gamma_{k}^{2m}(\mathbb{R}^{2n})$, $k=0,1$, given by \begin{gather*} \sigma (FF^{\ast })(x,\partial _{x}S(x,\theta )) \equiv |a(x,\theta )| ^2|(\det \frac{\partial ^2S}{\partial \theta \partial x})^{-1}(x,\theta )| \\ \sigma (F^{\ast }F)(\partial _{\theta }S(x,\theta ),\theta ) \equiv |a(x,\theta )| ^2|(\det \frac{\partial ^2S }{\partial \theta \partial x})^{-1}(x,\theta )| \end{gather*} we denote here $a\equiv b$ for $a,b\in \Gamma _{k}^{2p}(\mathbb{R}^{2n})$ if $(a-b)\in \Gamma _{k}^{2p-2}(\mathbb{R}^{2n})$ and $\sigma $ stands for the symbol. \end{theorem} \begin{proof} If $u\in \mathcal{S}(\mathbb{R}^n)$, then $Fu(x)$ is given by \begin{equation} \begin{aligned} Fu(x) &=\int_{\mathbb{R}^n}K(x,y) u(y)\,dy \\ &=\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}e^{i(S(x,\theta ) -y\theta)}a(x,\theta )u(y)dy\widehat{d\theta } \\ &=\int_{\mathbb{R}^n}e^{iS(x,\theta )} a(x,\theta )\Big(\int_{\mathbb{R}^n}e^{-iy\theta}u(y)dy\Big)\widehat{d\theta } \\ &=\int_{\mathbb{R}^n}e^{iS(x,\theta )} a(x,\theta ) \mathcal{F} u(\theta )\widehat{d\theta }. \end{aligned} \label{4.2} \end{equation} Here $F$ is a continuous linear mapping from $\mathcal{S}(\mathbb{R}^n)$ to $\mathcal{S}(\mathbb{R}^n)$ (by Theorem \ref{thm2.1}). Let $v\in \mathcal{S}(\mathbb{R}^n)$, then \begin{align*} \langle Fu,v\rangle_{L^2(\mathbb{R}^n)} &=\int_{\mathbb{R}^n}\Big(\int_{ \mathbb{R}^n}e^{iS(x,\theta )}a(x,\theta )\mathcal{F}u(\theta ) \widehat{d\theta }\Big) \overline{v(x)}\,dx \\ &=\int_{\mathbb{R}^n}\mathcal{F}u(\theta )\Big(\int_{\mathbb{R} ^n}\overline{e^{-iS(x,\theta )} \overline{a(x,\theta )}v(x)\,dx}\Big) \widehat{d\theta } \end{align*} thus \begin{equation*} \langle Fu(x),v(x)\rangle _{L^2(\mathbb{R}^n)} =(2\pi )^{-n}\langle \mathcal{F}u(\theta ),\mathcal{F}(( F^{\ast }v))(\theta )\rangle_{L^2(\mathbb{R}^n)} \end{equation*} where \begin{equation} \mathcal{F}((F^{\ast }v))(\theta) =\int_{\mathbb{R}^n}e^{-iS(\widetilde{x},\theta )}\overline{a} (\widetilde{x},\theta ) v(\widetilde{x})d\widetilde{x}. \label{4.3} \end{equation} Hence, for all $v\in \mathcal{S}(\mathbb{R}^n)$, \begin{equation} (FF^{\ast }v)(x)=\int_{\mathbb{R} ^n}\int_{\mathbb{R}^n}e^{i(S(x,\theta )-S(\widetilde{x},\theta ))}a(x,\theta ) \overline{a}(\widetilde{x},\theta )d\widetilde{x} \widehat{d\theta }. \label{4.4} \end{equation} The main idea to show that $FF^{\ast }$ is a pseudodifferential operator, is to use the fact that $(S(x,\theta )-S(\widetilde{x},\theta ))$ can be expressed by the scalar product $\langle x-\widetilde{x},\xi (x,\widetilde{x},\theta )\rangle$ after considering the change of variables $(x,\widetilde{x},\theta)\to (x,\widetilde{x},\xi =\xi (x,\widetilde{x},\theta ))$. The distribution kernel of $FF^{\ast }$ is \begin{equation*} K(x,\tilde{x})=\int_{\mathbb{R}^n}e^{i( S(x,\theta )-S(\tilde{x},\theta ))}a(x,\theta )\overline{a}(\tilde{x},\theta )\widehat{d\theta }. \end{equation*} We obtain from \eqref{3.3} that if $|x-\widetilde{x}| \geq \frac{\epsilon }{2}\lambda (x,\widetilde{x},\theta )$ (where $\epsilon >0$ is sufficiently small) then \begin{equation} |(\partial _{\theta }S)(x,\theta )-( \partial _{\theta }S)(\widetilde{x},\theta ) | \geq \frac{\epsilon }{2C_{5}}\lambda ( x,\widetilde{x},\theta ). \label{4.5} \end{equation} Choosing $\omega \in C^{\infty }(\mathbb{R})$ such that \begin{gather*} \omega (x)\geq 0, \quad \forall x\in \mathbb{R} \\ \omega (x)=1 \quad \text{if } x\in [ -\frac{1}{2},\frac{1}{2}] \\ \mathop{\rm supp}\omega \subset ] -1,1[ \end{gather*} and setting \begin{gather*} b(x,\tilde{x},\theta ):= a(x,\theta )\overline{a} (\tilde{x},\theta )=b_{1,\epsilon }( x,\tilde{x},\theta )+b_{2,\epsilon }(x,\tilde{x},\theta ) \\ b_{1,\epsilon }(x,\tilde{x},\theta )=\omega (\frac{ |x-\tilde{x}| }{\epsilon \lambda (x,\tilde{x} ,\theta )})b(x,\tilde{x},\theta ) \\ b_{2,\epsilon }(x,\tilde{x},\theta )=[ 1-\omega (\frac{|x-\tilde{x}| }{\epsilon \lambda (x, \tilde{x},\theta )})] b(x,\tilde{x},\theta ). \end{gather*} We have $K(x,\widetilde{x})=K_{1,\epsilon }(x, \widetilde{x})+K_{2,\epsilon }( x,\widetilde{x})$, where \begin{equation*} K_{j,\epsilon }(x,\tilde{x})=\int_{\mathbb{R} ^n}e^{i(S(x,\theta )-S(\tilde{x},\theta )) }b_{j,\epsilon }(x,\tilde{x},\theta ) \widehat{d\theta },\quad j=1,2. \end{equation*} We will study separately the kernels $K_{1,\epsilon }$ and $K_{2,\epsilon } $. On the support of $b_{2,\epsilon }$, inequality \eqref{4.5} is satisfied and we have \begin{equation*} K_{2,\epsilon }(x,\widetilde{x})\in \mathcal{S}(\mathbb{R}^n\times \mathbb{R}^n). \end{equation*} Indeed, using the oscillatory integral method, there is a linear partial differential operator $L$ of order 1 such that \begin{equation*} L\big(e^{i(S(x,\theta )-S(\tilde{x},\theta )) }\big)=e^{i(S(x,\theta )-S(\tilde{x},\theta ))} \end{equation*} where \begin{equation*} L=-i|(\partial _{\theta }S) (x,\theta )-(\partial _{\theta }S)( \widetilde{x},\theta )| ^{-2}\sum_{l=1}^n\left[ (\partial _{\theta _{l}}S)(x,\theta )-(\partial _{\theta _{l}}S) (\widetilde{x},\theta )\right] \partial _{\theta_{l}}. \end{equation*} The transpose operator of $L$ is \begin{equation*} ^{t}L=\sum_{l=1}^nF_{l}(x,\widetilde{x},\theta) \partial _{\theta _{l}}+G(x,\widetilde{x},\theta ) \end{equation*} where $F_{l}(x,\widetilde{x},\theta )\in \Gamma _{0}^{-1}(\Omega _{\epsilon })$, $G(x,\widetilde{x},\theta )\in \Gamma _{0}^{-2}(\Omega _{\epsilon })$, \begin{gather*} F_{l}(x,\widetilde{x},\theta )=i|( \partial _{\theta }S)(x,\theta )-(\partial _{\theta }S)(\widetilde{x},\theta )| ^{-2}((\partial _{\theta _{l}}S)(x,\theta )-(\partial _{\theta _{l}}S) (\widetilde{x},\theta )), \\ G(x,\widetilde{x},\theta ) =i\sum_{l=1}^n\partial _{\theta _{l}}\left[ | (\partial _{\theta }S)(x,\theta )-(\partial _{\theta }S)(\widetilde{x},\theta )| ^{-2}((\partial _{\theta _{l}}S)(x,\theta)-(\partial _{\theta _{l}}S)( \widetilde{x},\theta ) )\right], \\ \Omega _{\epsilon }=\big\{ (x,\tilde{x},\theta )\in \mathbb{R}^{3n}:|\partial _{\theta }S(x,\theta )-\partial _{\theta}S(\tilde{x},\theta )| >\frac{\epsilon }{2C_{5}} \lambda (x,\tilde{x},\theta )\big\} . \end{gather*} On the other hand we prove by induction on $q$ that \begin{equation*} (^{t}L)^{q}b_{2,\epsilon }(x,\tilde{x},\theta ) =\sum_{|\gamma | \leq q ,\, \gamma \in \mathbb{N}^n} g_{\gamma ,q}(x,\tilde{x},\theta )\partial _{\theta }^{\gamma }b_{2,\epsilon }(x,\tilde{x},\theta ), \text{ }g_{\gamma }^{(q)}\in \Gamma _{0}^{-q}( \Omega _{\epsilon }), \end{equation*} and so, \begin{equation*} K_{2,\epsilon }(x,\tilde{x})=\int_{\mathbb{R} ^n}e^{i(S(x,\theta )-S(\tilde{x},\theta ))}( ^{t}L)^{q}b_{2,\epsilon }(x,\tilde{x},\theta )\widehat{d\theta }. \end{equation*} Using Leibnitz's formula, (G5) and the form $(^{t}L)^{q}$, we can choose $q$ large enough such that for all $\alpha ,\alpha ',\beta ,\beta '\in \mathbb{N} ^n,\exists C_{\alpha ,\alpha ',\beta ,\beta '}>0$, \begin{equation*} \sup_{x,\widetilde{x}\in \mathbb{R}^n}|x^{\alpha }\widetilde{x} ^{\alpha '}\partial _{x}^{\beta }\partial _{\widetilde{x}}^{\beta '}K_{2,\epsilon }( x,\widetilde{x})| \leq C_{\alpha ,\alpha ',\beta ,\beta '}. \end{equation*} Next, we study $K_{1}^{\epsilon }$: this is more difficult and depends on the choice of the parameter $\epsilon $. It follows from Taylor's formula that \begin{gather*} S(x,\theta )-S(\widetilde{x},\theta ) =\langle x- \widetilde{x},\xi (x,\widetilde{x},\theta ) \rangle_{\mathbb{R}^n} ,\\ \xi (x,\widetilde{x},\theta ) =\int_{0}^{1}(\partial _{x}S)(\widetilde{x}+t(x-\widetilde{x}) ,\theta )dt. \end{gather*} We define the vectorial function \begin{equation*} \widetilde{\xi }_{\epsilon }(x,\widetilde{x},\theta )=\omega \big(\frac{|x-\tilde{x}| }{2\epsilon \lambda (x,\tilde{x},\theta )}\big) \xi (x,\widetilde{x},\theta )+\big(1-\omega (\frac{|x-\tilde{x}| }{ 2\epsilon \lambda (x,\tilde{x},\theta )}) \big)(\partial _{x}S)(\widetilde{x},\theta). \end{equation*} We have \begin{equation*} \widetilde{\xi }_{\epsilon }(x,\widetilde{x},\theta )=\xi (x,\widetilde{x},\theta )\text{ on } \mathop{\rm supp} b_{1,\epsilon }. \end{equation*} Moreover, for $\epsilon $ sufficiently small, \begin{equation} \lambda (x,\theta )\simeq \lambda ( \widetilde{x},\theta )\simeq \lambda ( x,\widetilde{x},\theta )\text{ on } \mathop{\rm supp} b_{1,\epsilon }. \label{4.6} \end{equation} Let us consider the mapping \begin{equation} \mathbb{R}^{3n}\ni (x,\widetilde{x},\theta ) \to (x,\widetilde{x},\widetilde{\xi }_{\epsilon }(x,\widetilde{x},\theta ))\label{4.7} \end{equation} for which Jacobian matrix is \begin{equation*} \begin{pmatrix} I_{n} & 0 & 0 \\ 0 & I_{n} & 0 \\ \partial _{x}\widetilde{\xi }_{\epsilon } & \partial _{\widetilde{x}} \widetilde{\xi }_{\epsilon } & \partial _{\theta }\widetilde{\xi } _{\epsilon } \end{pmatrix}. \end{equation*} We have \begin{align*} &\frac{\partial \widetilde{\xi }_{\epsilon ,j}}{\partial \theta _{i}} (x,\widetilde{x},\theta )\\ &=\frac{\partial ^2S}{\partial \theta _{i}\partial x_{j}}(\widetilde{x},\theta ) +\omega \big(\frac{ |x-\tilde{x}| }{2\epsilon \lambda (x,\tilde{x} ,\theta )}\big)\big(\frac{\partial \xi _{j}}{\partial \theta _{i} }(x,\widetilde{x},\theta )-\frac{\partial ^2S}{\partial \theta _{i}\partial x_{j}}(\widetilde{x},\theta )\big) \\ &\quad -\frac{|x-\tilde{x}| }{2\epsilon \lambda (x, \tilde{x},\theta )}\frac{\partial \lambda }{\partial \theta _{i}} (x,\tilde{x},\theta )\lambda ^{-1}( x,\tilde{x},\theta )\omega '\big(\frac{|x-\tilde{x}| }{ 2\epsilon \lambda (x,\tilde{x},\theta )}\big) \big(\xi_{j}(x,\widetilde{x},\theta )-\frac{\partial S}{\partial x_{j}} (\widetilde{x},\theta )\big). \end{align*} Thus, we obtain \begin{align*} &\big|\frac{\partial \widetilde{\xi }_{\epsilon ,j}}{\partial \theta _{i}}(x,\widetilde{x},\theta ) -\frac{\partial ^2S}{\partial \theta _{i}\partial x_{j}}( \widetilde{x},\theta )\big|\\ &\leq \big|\omega (\frac{|x-\tilde{x}| }{ 2\epsilon \lambda (x,\tilde{x},\theta )})\big| \big|\frac{\partial \xi _{j}}{\partial \theta _{i}}(x,\widetilde{ x},\theta )-\frac{\partial ^2S}{\partial \theta _{i}\partial x_{j}} (\widetilde{x},\theta )\big| \\ &\quad + \lambda ^{-1}(x,\tilde{x},\theta )\big|\omega'(\frac{|x-\tilde{x}| }{2\epsilon \lambda (x,\tilde{x},\theta )}) \big| \big|\xi_{j}(x,\widetilde{x},\theta ) -\frac{\partial S}{\partial x_{j}} (\widetilde{x},\theta )\big| . \end{align*} Now it follows from (G5), \eqref{4.6} and Taylor's formula that \begin{equation} \begin{aligned} \big|\frac{\partial \xi _{j}}{\partial \theta _{i}}(x,\widetilde{ x},\theta )-\frac{\partial ^2S}{\partial \theta _{i}\partial x_{j}} (\widetilde{x},\theta )\big| &\leq \int_{0}^{1}\big|\frac{\partial ^2S}{\partial \theta _{i}\partial x_{j}}(\widetilde{x}+t( x-\widetilde{x}) ,\theta )-\frac{\partial ^2S}{\partial \theta _{i}\partial x_{j}} (\widetilde{x},\theta )\big| dt \\ &\leq C_{10}|x-\widetilde{x}| \lambda ^{-1}(x, \tilde{x},\theta ),\quad C_{10}>0 \end{aligned} \label{4.8} \end{equation} \begin{equation} \begin{aligned} \big|\xi _{j}(x,\widetilde{x},\theta )-\frac{\partial S}{ \partial x_{j}}(\widetilde{x},\theta )\big| &\leq \int_{0}^{1}\big|\frac{\partial S}{\partial x_{j}}(\widetilde{x}+t(x-\widetilde{x}),\theta )-\frac{\partial S}{\partial x_{j}}( \widetilde{x},\theta )\big| dt \\ &\leq C_{11}|x-\widetilde{x}| ,\quad C_{11}>0\,.\label{4.9} \end{aligned} \end{equation} From \eqref{4.8} and \eqref{4.9}, there exists a positive constant $C_{12}>0$ such that \begin{equation} |\frac{\partial \widetilde{\xi }_{\epsilon ,j}}{\partial \theta _{i}}(x,\widetilde{x},\theta ) -\frac{\partial ^2S}{\partial \theta _{i}\partial x_{j}}( \widetilde{x},\theta )| \leq C_{12}\epsilon ,\quad \forall i,j\in \{ 1,\dots ,n\} . \label{4.10} \end{equation} If $\epsilon <\frac{\delta _{0}}{2\widetilde{C}}$, then \eqref{4.10} and (G4) yields the estimate \begin{equation} \delta _{0}/2\leq -\widetilde{C}\epsilon +\delta _{0}\leq -\widetilde{C} \epsilon +\det \frac{\partial ^2S}{\partial x\partial \theta }(x,\theta )\leq \det \partial _{\theta }\widetilde{\xi }_{\epsilon }(x, \widetilde{x},\theta ), \label{4.11} \end{equation} with $\widetilde{C}>0$ If $\epsilon $ is such that \eqref{4.6} and \eqref{4.11} hold, then the mapping given in \eqref{4.7} is a global diffeomorphism of $\mathbb{R}^{3n}$. Hence there exists a mapping \begin{equation*} \theta :\mathbb{R}^n\times \mathbb{R}^n\times \mathbb{R}^n\ni (x, \widetilde{x},\xi )\to \theta ( x,\widetilde{x},\xi )\in \mathbb{R}^n \end{equation*} such that \begin{equation} \begin{gathered} \widetilde{\xi }_{\epsilon }(x,\widetilde{x},\theta (x, \widetilde{x},\xi ))= \xi \\ \theta (x,\widetilde{x},\widetilde{\xi }_{\epsilon }(x, \widetilde{x},\theta ))= x \\ \partial ^{\alpha }\theta (x,\widetilde{x},\xi )=\mathcal{O} (1), \quad \forall \alpha \in \mathbb{N}^{3n}\backslash \{0\} \end{gathered} \label{4.12} \end{equation} If we change the variable $\xi $ by $\theta (x,\widetilde{x},\xi)$ in $K_{1,\epsilon }(x,\widetilde{x})$, we obtain \begin{equation} K_{1,\epsilon }(x,\widetilde{x})=\int_{\mathbb{R} ^n}e^{i\langle x-\tilde{x},\xi \rangle}b_{1,\epsilon }( x,\tilde{x},\theta (x,\widetilde{x},\xi )) \big|\det \frac{\partial \theta }{\partial \xi }( x,\widetilde{x},\xi )\big| \widehat{d\xi }. \label{4.13} \end{equation} From \eqref{4.12} we have, for $k=0,1$, that $b_{1,\epsilon }(x,\tilde{x},\theta ( x,\widetilde{x},\xi ))|\det \frac{\partial \theta }{\partial \xi }(x,\widetilde{x},\xi )| $ belongs to $\Gamma _{k}^{2m}(\mathbb{R}^{3n})$ if $a\in \Gamma _{k}^{m}(\mathbb{R}^{2n})$. Applying the stationary phase theorem (c.f. \cite{Ro} ) to \ref{4.13}, we obtain the expression of the symbol of the pseudodifferential operator $FF^{\ast }$, \begin{equation*} \sigma (FF^{\ast })=b_{1,\epsilon }(x,\tilde{x},\theta (x, \widetilde{x},\xi ))\big|\det \frac{\partial \theta }{ \partial \xi }(x,\widetilde{x},\xi )\big| _{|\widetilde{x}=x }+R(x,\xi ) \end{equation*} where $R(x,\xi )$ belongs to $\Gamma _{k}^{2m-2}(\mathbb{R}^{2n})$ if $a\in \Gamma _{k}^{m}(\mathbb{R}^{2n})$, $k=0,1$. For $\tilde{x}=x$, we have $b_{1,\epsilon }(x,\tilde{x},\theta (x,\widetilde{x},\xi )) =|a(x,\theta (x,x,\xi ))| ^2$ where $\theta (x,x,\xi )$ is the inverse of the mapping $\theta \to \partial_{x}S(x,\theta )=\xi $. Thus \begin{equation*} \sigma (FF^{\ast })(x,\partial _{x}S(x,\theta ) )\equiv |a(x,\theta )| ^2\big|\det \frac{\partial ^2S}{\partial \theta \partial x}(x,\theta )\big| ^{-1}. \end{equation*} From \eqref{4.2} and \eqref{4.3}, we obtain the expression of $F^{\ast }F$: $\forall v\in \mathcal{S}(\mathbb{R}^n)$, \begin{align*} (\mathcal{F(}F^{\ast }F)\mathcal{F}^{-1})v( \theta ) &=\int_{\mathbb{R}^n}e^{-iS(x,\theta )}\overline{a} (x,\theta )(F(\mathcal{F}^{-1}v))(x)dx \\ &=\int_{\mathbb{R}^n}e^{-iS(x,\theta )}\overline{a} (x,\theta )\Big(\int_{\mathbb{R}^n}e^{iS(x,\widetilde{\theta } )}a(x,\widetilde{\theta })(\mathcal{F}(\mathcal{F} ^{-1}v))(\widetilde{\theta }) \widehat{d\widetilde{\theta }}\Big)dx\newline \\ &=\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}e^{-i( S(x,\theta )-S(x,\tilde{\theta}))\;} \overline{a}(x,\theta )\;a(x,\widetilde{\theta })v(\tilde{ \theta})\widehat{d\widetilde{\theta }}dx\text{.} \end{align*} Hence the distribution kernel of the integral operator $\mathcal{F(}F^{\ast }F)\mathcal{F}^{-1}$ is \begin{equation*} \widetilde{K}(\theta ,\widetilde{\theta }) =\int_{\mathbb{R}^n}e^{-i(S(x,\theta )-S(x,\tilde{\theta}) )}\overline{a}(x,\theta ) a(x,\tilde{\theta})\widehat{dx}. \end{equation*} We remark that we can deduce $\widetilde{K}(\theta ,\widetilde{\theta })$ from $K(x,\widetilde{x})$ by replacing $x$ by $\theta $. On the other hand, all assumptions used here are symmetrical on $x$ and $\theta $; therefore, $\mathcal{F(}F^{\ast }F)\mathcal{F}^{-1}$ is a nice pseudodifferential operator with symbol \begin{equation*} \sigma (\mathcal{F(}F^{\ast }F)\mathcal{F}^{-1})(\theta ,-\partial _{\theta }S(x,\theta ))\equiv | a(x,\theta )| ^2\big|\det \frac{\partial ^2S}{\partial x\partial \theta } (x,\theta )\big| ^{-1}. \end{equation*} Thus the symbol of $F^{\ast }F$ is given by (c.f. \cite{Ho2}) \begin{equation*} \sigma (F^{\ast }F)(\partial _{\theta }S(x,\theta ),\theta )\equiv |a(x,\theta )| ^2\big|\det \frac{\partial ^2S}{ \partial x\partial \theta }(x,\theta )\big| ^{-1}. \end{equation*} \end{proof} \begin{corollary} \label{coro4.2} Let $F$ be the integral operator with the distribution kernel \begin{equation*} K(x,y)=\int_{\mathbb{R}^n} e^{i(S(x,\theta )-y\theta)} a(x,\theta )\widehat{d\theta } \end{equation*} where $a\in \Gamma _{0}^{m}(\mathbb{R}_{x,\theta }^{2n})$ and $S$ satisfies (G1), (G4) and (G5). Then, we have: \begin{enumerate} \item For any $m$ such that $m\leq 0$, $F$ can be extended as a bounded linear mapping on $L^2(\mathbb{R}^n)$ \item For any $m$ such that $m<0$, $F$ can be extended as a compact operator on $L^2(\mathbb{R}^n)$. \end{enumerate} \end{corollary} \begin{proof} It follows from Theorem \ref{thm4.1} that $F^{\ast }F$ is a pseudodifferential operator with symbol in $\Gamma _{0}^{2m}(\mathbb{R}^{2n})$. \noindent (1) If $m\leq 0$, the weight $\lambda ^{2m}(x,\theta )$ is bounded, so we can apply the Cald\'{e}ron-Vaillancourt theorem (see \cite{CaVa,Ro,Se}) for $F^{\ast }F$ and obtain the existence of a positive constant $\gamma (n)$ and a integer $k(n)$ such that \begin{equation*} \|(F^{\ast }F)\;u\|_{L^2(\mathbb{R}^n)}\leq \gamma (n)\;Q_{k(n)}(\sigma (FF^{\ast })) \|u\|_{L^2(\mathbb{R}^n)},\quad \forall u\in \mathcal{S}(\mathbb{R}^n) \end{equation*} where \begin{equation*} Q_{k(n)}(\sigma (FF^{\ast })) =\sum_{|\alpha | +|\beta | \leq k(n)}\sup_{(x,\theta )\in \mathbb{R}^{2n}} \big|\partial _{x}^{\alpha }\partial _{\theta }^{\beta }\sigma (FF^{\ast })(\partial _{\theta }S(x,\theta ),\theta )\big| \end{equation*} Hence, for all $u\in \mathcal{S}(\mathbb{R}^n)$, \begin{equation*} \|Fu\|_{L^2(\mathbb{R}^n)}\leq \| F^{\ast }F\|_{_{\mathcal{L}( L^2(\mathbb{R}^n))}}^{1/2}\|u\| _{L^2(\mathbb{R}^n)}\leq (\gamma (n)\;Q_{k( n)}(\sigma (FF^{\ast }))) ^{1/2}\|u\|_{L^2(\mathbb{R}^n)}. \end{equation*} Thus $F$ is also a bounded linear operator on $L^2(\mathbb{R}^n)$. \noindent (2) If $m<0$, $\lim_{|x| +|\theta| \to +\infty } \lambda ^{m}(x,\theta )=0$, and the compactness theorem (see \cite{Ro,Se}) show that the operator $F^{\ast }F$ can be extended as a compact operator on $L^2(\mathbb{R}^n)$. Thus, the Fourier integral operator $F$ is compact on $L^2(\mathbb{R}^n)$. Indeed, let $(\varphi _{j})_{j\in \mathbb{N}}$ be an orthonormal basis of $L^2(\mathbb{R}^n)$, then \begin{equation*} \|F^{\ast }F-\sum_{j=1}^n \langle \varphi_{j},.\rangle F^{\ast } F\varphi _{j}\| \to 0 \quad \text{as } n\to +\infty. \end{equation*} Since $F$ is bounded, for all $\psi \in L^2(\mathbb{R}^n)$, \[ \big\|F\psi -\sum_{j=1}^n \langle \varphi_{j},\psi \rangle F\varphi _{j}\big\|^2 \leq \big\|F^{\ast }F\psi -\sum_{j=1}^n \langle \varphi _{j},\psi \rangle F^{\ast }F\varphi _{j}\big\| \big\|\psi -\sum_{j=1}^n \langle \varphi _{j},\psi \rangle \varphi _{j}\big\|, \] it follows that \begin{equation*} \|F-\sum_{j=1}^n \langle \varphi _{j},.\rangle F\varphi _{j}\| \to 0 \quad \text{as } n\to +\infty \end{equation*} \end{proof} \begin{example} \label{exa4.3} \rm We consider the function given by \begin{equation*} S(x,\theta )=\sum_{|\alpha| +|\beta | =2,\, \alpha ,\beta \in \mathbb{N}^n} C_{\alpha ,\beta }x^{\alpha }\theta ^{\beta },\quad \text{for }(x,\theta )\in \mathbb{R}^{2n} \end{equation*} where $C_{\alpha ,\beta }$ are real constants. This function satisfies (G1), (G4) and (G5). \end{example} \subsection*{Acknowledgements} This paper was completed while the second author was visiting the "Universit\'{e} Libre de Bruxelles". He wants to thank Professor J.-P. Gossez for his valuable discussions. Investigations supported by University of Oran Es-senia, Algeria. CNEPRU B3101/02/03. \begin{thebibliography}{99} \bibitem{AsFu} K. Asada and D. Fujiwara, \textit{On some oscillatory transformation in $L^2(\mathbb{R}^n)$}, Japan.\ J. Math.\ Vol. 4 (2), 1978, p. 299-361. \bibitem{CaVa} A. P. Cald\'{e}ron and R. Vaillancourt, \textit{On the boundedness of pseudodifferential operators}, J. Math. Soc. Japan 23, 1972, p. 374-378. \bibitem{DrJa} P. Drabek and J. Milota, \textit{Lectures on nonlinear analysis}, Pleze\v{n}, Czech Republic, 2004. \bibitem{Du} J. J. Duistermaat, \textit{Fourier integral operators}, Courant Institute Lecture Notes, New-York 1973. \bibitem{EgSh} Yu. V. Egorov and M. A. Shubin, \textit{Partial differential equations}, Vol. 2, Springer-Verlag, Berlin, 1991. \bibitem{Ha} M. Hasanov, \textit{A class of unbounded Fourier integral operators}, J. Math. Analysis and application 225, 1998, p. 641-651. \bibitem{He} B. Helffer, \textit{Th\'{e}orie spectrale pour des op\'{e}rateurs globalement elliptiques}, Soci\'{e}t\'{e} Math\'{e}matiques de France, Ast\'{e}risque 112, 1984. \bibitem{Ho1} L. H\"{o}rmander, \textit{Fourier integral operators I}, Acta Math. Vol. 127, 1971, p. 33-57. \bibitem{Ho2} L. H\"{o}rmander, \textit{The Weyl calculus of pseudodifferential operators}, Comm. Pure. Appl. Math. 32 (1979), p. 359-443. \bibitem{MeSe1} B. Messirdi et A. Senoussaoui, \textit{Parametrix du probl\`{e}me de Cauchy $C^{\infty }$ hyperbolique muni d'un syst\`{e}me ordres de Leray-Volev\^{\i}c}, Journal for Analysis and its Applications, Vol. 24, No. 3, (2005), p. 581-592. \bibitem{MeSe2} B. Messirdi and A. Senoussaoui, \textit{A class of unbounded Fourier integral operators with symbol in $\bigcap_{0<\rho<1}S_{\rho ,1}^{0}$}, Submitted 2005. \bibitem{Ro} D. Robert, \textit{Autour de l'approximation semi-classique}, Birk\"{a}user, 1987. \bibitem{Se} A. Senoussaoui, \textit{Op\'{e}rateurs h-admissibles matriciels \`{a} symboles op\'{e}rateurs}, African Diaspora Journal of Mathematics, Vol. 4, No. 1, (2005). \bibitem{Sc} J. T. Schwartz, \textit{Non linear functional analysis}, Gordon and Breach Publishers, 1969. \bibitem{Sh} M. Shubin, \textit{Pseudodifferential operators and spectral theory}, Naukia Moscow, 1978. \end{thebibliography} \end{document}