Content-Type: multipart/mixed; boundary="-------------0405100335908" This is a multi-part message in MIME format. ---------------0405100335908 Content-Type: text/plain; name="04-149.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-149.comments" E-mail adresses of authors: bmessirdi@univ-oran.dz, asenouss@ulb.ac.be Classification AMS: 35S30, 35S05, 47A10, 35P05 ---------------0405100335908 Content-Type: text/plain; name="04-149.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-149.keywords" Fourier integral operators, Pseudodifferential operators, $L^{2}$-boundness and $L^{2}$-compactness, $C^{\infty }$ Cauchy problem ---------------0405100335908 Content-Type: application/x-tex; name="article.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="article.tex" %% This document created by Scientific Word (R) Version 2.5 %% Starting shell: mathart1 \documentclass[11pt,thmsa]{article} \usepackage{amsfonts} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \usepackage{sw20mitp} %TCIDATA{TCIstyle=article/art1.lat,mitp,mathart1} %TCIDATA{Created=Wed Jun 05 12:45:33 2002} %TCIDATA{LastRevised=Thu May 06 12:25:13 2004} %TCIDATA{Language=American English} \input{tcilatex} \begin{document} \begin{center} \bigskip {\LARGE \ On the L}$^{\text{{\large 2}}}-${\LARGE \ boundness and L}% $^{\text{{\large 2}}}-${\LARGE \ compactness}\\[0pt] {\LARGE \ of Fourier integral operators} ~\verb||{\LARGE \ } ~\verb||{\LARGE \ } \begin{tabular}[s]{cc} \multicolumn{2}{l}{} \\ \textbf{B. MESSIRDI} & \textbf{A. SENOUSSAOUI} \\ Universit\'{e} d'Oran Es-S\'{e}nia & Universit\'{e} Libre de Bruxelles \\ Facult\'{e} des Sciences & Boulvard du Triomphe \\ D\'{e}partement de\ Math\'{e}matiques & Campus Plaine CP 214 \\ Route d'Es-S\'{e}nia\ Es-S\'{e}nia\ (Oran)-ALGERIE & B1050 Brussels BELGIUM \end{tabular} \end{center} \textbf{Abstract: }In this paper, we study the $L^{2}$-continuity and $L^{2}$% -boundness of Fourier integral operators. These operators are bounded respectively compacts if the weight symbol is bounded respectively tends to $% 0$. \textbf{AMS Subj:} Classification 35S30, 35S05, 47A10, 35P05 \textbf{Key Words: }Fourier integral operators, Pseudodifferential operators, $L^{2}$-boundness and $L^{2}$-compactness, $C^{\infty }$ Cauchy problem \verb||{\LARGE \ } \textbf{0.} \textbf{Introduction} The Integral operators of type: \begin{equation} A\varphi \left( x\right) =\int e^{iS\left( x,\theta \right) }a\left( x,\theta \right) \mathcal{F}\varphi \left( \theta \right) d\theta \tag{0.1} \end{equation} appear naturally for solving the hyperbolic partial differential equations and expressing the $C^{\infty }$-solution of the associate Cauchy problem's. If we write formally the expression of the Fourier transformation $\mathcal{F% }\varphi \left( \theta \right) $ in $\left( 0.1\right) ,$ we obtain the following Fourier integral operators: \[ A\varphi \left( x\right) =\iint e^{i\left( S\left( x,\theta \right) -y\theta \right) }a\left( x,y,\theta \right) \varphi \left( y\right) dyd\theta \] in witch appear two $C^{\infty }$-functions, the phase function $\phi \left( x,y,\theta \right) =S\left( x,\theta \right) -y\theta $ and the amplitude $a$ called the symbol of the operator $A.$ Since 1970, many of Mathematicians are interested to these type of operators as J.J. Duistermaat [3], L. H\"{o}rmander [6] and Asada-Fujiwara [1] without forgeting the works of B. Helffer [5] and M. Hasanov [4]. In this paper we present a spectral study of a class of the Fourier integral operators, specially we are interested in their continuity and their compactness on $L^{2}\left( \Bbb{R}^{n}\right) .$ Mainly we shows the continuity of the operator $A$ on $L^{2}\left( \Bbb{R}% ^{n}\right) $ if the weight of the symbol $a$ is bounded, if this weight tends to zero then $A$ is compact on $L^{2}\left( \Bbb{R}^{n}\right) $. We give also an $L^{2}$-estimate of $\left\| A\right\| .$ If the symbol $a$ is only bounded the Fourier integral operator $A$ is not necessary bounded on $L^{2}\left( \Bbb{R}^{n}\right) .$ Indeed, in 1998 M. Hasanov [4] constructed a class of unbounded Fourier integral operators with amplitude in the symbolic class $S_{1,1}^{0}.$ Our result is an important generalization of the work of Asada-Fujiwara [1] using some weak assumptions on the phase function $\phi $ and making simple demonstrations. In our knowledge, this work constitutes also a first tentative of diagonalization of the Fourier integral operators on $L^{2}\left( \Bbb{R}% ^{n}\right) .$ \textbf{1. A general class of Fourier integral operators} If $\varphi \in \mathcal{S}\left( \Bbb{R}^{n}\right) ,$ $x\in \Bbb{R}^{n},$ we shall study the integral transformations of the following type: \begin{equation} \left( I\left( a,\phi \right) \varphi \right) \left( x\right) =\stackunder{% \Bbb{R}_{y}^{n}\times \Bbb{R}_{\theta }^{N}}{\iint }e^{i\phi \left( x,\theta ,y\right) }a\left( x,\theta ,y\right) \varphi \left( y\right) dyd\theta \tag{1.1} \end{equation} where $n\in \Bbb{N}^{*}$ and $N\in \Bbb{N}$ (if $N=0,$ $\theta $ doesn't appear in $(1.1)$). In general the integral $(1.1)$ is not absolutely convergent, so that we use the technique of the oscillatory integrals developed by L.H\"{o}rmander in [6]. The phase function $\phi $ and the amplitude function $a$ are assumed to satisfy the assumptions: $(H1)$ $\;\;\phi \in C^{\infty }\left( \Bbb{R}_{x}^{n}\times \Bbb{R}_{\theta }^{N}\times \Bbb{R}_{y}^{n},\Bbb{R}\right) $ ($\phi $ is a real function) $(H2)\;\;$ $\forall \left( \alpha ,\beta ,\gamma \right) \in \Bbb{N}% ^{n}\times \Bbb{N}^{N}\times \Bbb{N}^{n},$ $\exists C_{\alpha ,\beta ,\gamma }>0,$ such that \[ \left| \partial _{y}^{\gamma }\partial _{\theta }^{\beta }\partial _{x}^{\alpha }\phi \left( x,\theta ,y\right) \right| \leq C_{\alpha ,\beta ,\gamma }\lambda ^{\left( 2-\left| \alpha \right| -\left| \beta \right| -\left| \gamma \right| \right) _{+}}\left( x,\theta ,y\right) ,\text{ where } \] $\lambda \left( x,\theta ,y\right) =\left( 1+\left| x\right| ^{2}+\left| \theta \right| ^{2}+\left| y\right| ^{2}\right) ^{1/2},$ $\left( 2-\left| \alpha \right| -\left| \beta \right| -\left| \gamma \right| \right) _{+}=\sup \left( 2-\left| \alpha \right| -\left| \beta \right| -\left| \gamma \right| ,0\right) $ $(H3)$ $\exists K_{1},K_{2}>0,$ such that \[ K_{1}\lambda (x,\theta ,y)\leq \lambda (\partial _{y}\phi ,\partial _{\theta }\phi ,y)\leq K_{2}\lambda (x,\theta ,y),\text{ }\forall (x,\theta ,y)\in \Bbb{R}_{x}^{n}\times \Bbb{R}_{\theta }^{N}\times \Bbb{R}_{y}^{n} \] $(H3)^{*}$ $\,$\thinspace $\exists K_{1}^{*},K_{2}^{*}>0,$ such that \[ K_{1}^{*}\lambda (x,\theta ,y)\leq \lambda \left( x,\partial _{\theta }\phi ,\partial _{x}\phi \right) \leq K_{2}^{*}\lambda (x,\theta ,y),\text{ }% \forall (x,\theta ,y)\in \Bbb{R}_{x}^{n}\times \Bbb{R}_{\theta }^{N}\times \Bbb{R}_{y}^{n} \] For any open $\Omega $ of $\Bbb{R}_{x}^{n}\times \Bbb{R}_{\theta }^{N}\times \Bbb{R}_{y}^{n},$ $\mu \in \Bbb{R}$ and $\rho \in \left[ 0,1\right] ,$ we put $\Gamma _{\rho }^{\mu }\left( \Omega \right) =\left\{ \begin{array}{c} a\in C^{\infty }\left( \Omega \right) ;\text{ }\forall (\alpha ,\beta ,\gamma )\in \Bbb{N}^{n}\times \Bbb{N}^{N}\times \Bbb{N}^{n},\text{ } \\ \exists C_{\alpha ,\beta ,\gamma }>0,\text{ such that }\left| \partial _{y}^{\gamma }\partial _{\theta }^{\beta }\partial _{x}^{\alpha }a(x,\theta ,y)\right| \leq C_{\alpha ,\beta ,\gamma }\lambda ^{\mu -\rho (\left| \alpha \right| +\left| \beta \right| +\left| \gamma \right| )}(x,\theta ,y) \end{array} \right\} $ When $\Omega =\Bbb{R}_{x}^{n}\times \Bbb{R}_{\theta }^{N}\times \Bbb{R}% _{y}^{n},$ we note $\Gamma _{\rho }^{\mu }\left( \Omega \right) =\Gamma _{\rho }^{\mu }.$ For giving a definite meaning to the right hand side of $% \left( 1.1\right) ,$ we consider $g\in \mathcal{S}\left( \Bbb{R}% _{x}^{n}\times \Bbb{R}_{\theta }^{N}\times \Bbb{R}_{y}^{n}\right) ,$ $% g\left( 0\right) =1.$ If $a\in \Gamma _{0}^{\mu },$ we put \[ a_{\sigma }\left( x,\theta ,y\right) =g\left( x/\sigma ,\theta /\sigma ,y/\sigma \right) a\left( x,\theta ,y\right) ,\text{ }\sigma >0 \] \textbf{Theorem 1.1.} (see [7]) \textit{If }$\phi $ \textit{satisfies }$% (H1),(H2),(H3)$ and $(H3)^{*},$\textit{\ and if }$a\in \Gamma _{0}^{\mu },$% \textit{\ then} \textit{1. }$\forall \varphi \in \mathcal{S}\left( \Bbb{R}^{n}\right) ,% \stackunder{\sigma \rightarrow +\infty }{\lim }\left[ I\left( a_{\sigma },\phi \right) \varphi \right] \left( x\right) $\textit{\ exists at every point }$x\in \Bbb{R}^{n}$ \textit{and it's independent of particular choice of the function }$g$\textit{. We can get} \[ \left( I\left( a,\phi \right) \varphi \right) (x)\stackrel{def}{=}\text{ }% \stackunder{\sigma \rightarrow +\infty }{\lim }\left( I\left( a_{\sigma },\phi \right) \varphi \right) \left( x\right) \] \textit{2. }$I\left( a,\phi \right) $\textit{\ is a bounded linear mapping from }$\mathcal{S}(\Bbb{R}^{n})$\textit{\ to }$\mathcal{S}(\Bbb{R}^{n})$ \textit{and from }$\mathcal{S}^{\prime }(\Bbb{R}^{n})$ \textit{to }$\mathcal{% S}^{\prime }(\Bbb{R}^{n})$. \textbf{2. Study of a particular case} We consider the particular case where the phase function $\phi \left( x,y,\theta \right) $ has the form \begin{equation} \phi \left( x,y,\theta \right) =S(x,\theta )-y\theta \tag{2.1} \end{equation} and $S\;$satisfies the conditions $(G1)\ S\in C^{\infty }\left( \Bbb{R}_{x}^{n}\times \Bbb{R}_{\theta }^{n},% \Bbb{R}\right) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;% \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ $\left( G2\right) $ $\forall (\alpha ,\beta )\in \Bbb{N}^{n}\times \Bbb{N}% ^{n}\bigskip ,\ \exists C_{\alpha ,\beta }>0,\;\left| \partial _{x}^{\alpha }\partial _{\theta }^{\beta }S(x,\theta )\right| \leq C_{\alpha ,\beta }\;\lambda (x,\theta )^{(2-\left| \alpha \right| -\left| \beta \right| )_{+}} $ $(G3)$ $\exists C_{1}>0,\;\;\left| x\right| \leq $ $C_{1}\;\lambda (\theta ,\partial _{\theta }S),$ $\forall x,\theta \in \Bbb{R}^{n}$ $(G3)^{*}$ $\exists C_{2}>0,\;\left| \theta \right| \leq C_{2}\;\lambda (x,\partial _{x}S),\forall x,\theta \in \Bbb{R}^{n}$ \textbf{Proposition 2.1}. \textit{If the function }$S$\textit{\ satisfies }$% (G1),$ $(G2),$ $(G3)$ \textit{and }$(G3)^{*},$ \textit{then the phase function }$\phi \left( x,y,\theta \right) =S(x,\theta )-y\theta $ \textit{% satisfies }$(H1),$\textit{\ }$(H2),$\textit{\ }$(H3)$\textit{\ and}$% \;(H3)^{*}$\textit{.} \textit{Proof:} $(H1)$ and $(H2)$ are trivially satisfied. The condition $\left( G3\right) $ imply that \[ \lambda (x,\theta ,y)\leq \lambda (x,\theta )+\lambda (y)\leq C_{3}\left( \lambda (\theta ,\partial _{\theta }S)+\lambda (y)\right) ,\text{ }C_{3}>0 \] now, \[ \partial _{y_{j}}\phi =-\theta _{\;j}\;\text{and}\ \partial _{\theta _{j}}\phi =\partial _{\theta _{j}}S-y_{j} \] then \[ \lambda (\theta ,\partial _{\theta }S)=\lambda (\partial _{y}\phi ,\partial _{\theta }\phi +y)\leq 2\lambda (\partial _{y}\phi ,\partial _{\theta }\phi ,y) \] and \[ \lambda (x,\theta ,y)\leq C_{3}\left( 2\lambda (\partial _{y}\phi ,\partial _{\theta }\phi ,y)+\lambda (y)\right) \leq \frac{1}{C_{4}}\lambda (\partial _{y}\phi ,\partial _{\theta }\phi ,y),\;C_{4}>0 \] the second inequality in $\left( H3\right) $ is a consequence of the assumption $\left( G2\right) .$ By the same argument we obtain $(H3)^{*}% %TCIMACRO{\TeXButton{End Proof}{\endproof }} %BeginExpansion \endproof % %EndExpansion $ We also introduce the following assumptions: \begin{eqnarray*} \left( G4\right) \;\;\;\;\;\;\;\;\;\exists \delta _{0} &>&0,\text{ such that }\inf_{x,\theta \in \Bbb{R}^{n}}\left| \det \frac{\partial ^{2}S}{\partial x\partial \theta }\left( x,\theta \right) \right| \geq \delta _{0} \\ \left( H4\right) \;\;\;\;\;\;\;\;\;\exists \delta _{0} &>&0,\text{ such that }\inf_{x,\theta ,y\in \Bbb{R}^{n}}\left| \det D\left( \phi \right) \left( x,\theta ,y\right) \right| \geq \delta _{0} \end{eqnarray*} where $D(\phi )\;$is the matrix \[ D(\phi )\left( x,\theta ,y\right) =\left( \begin{array}{cc} \frac{\partial ^{2}\phi }{\partial x\partial y}\left( x,\theta ,y\right) & \frac{\partial ^{2}\phi }{\partial x\partial \theta }\left( x,\theta ,y\right) \\ \frac{\partial ^{2}\phi }{\partial \theta \partial y}\left( x,\theta ,y\right) & \frac{\partial ^{2}\phi }{\partial \theta \partial \theta }% \left( x,\theta ,y\right) \end{array} \right) \] \textbf{Proposition 2.2}. \textit{If }$S$\textit{\ satisfies }$\left( G4\right) ,$\textit{\ then the phase function }$\phi $\textit{\ defined in }$% \left( 2.1\right) $\textit{\ satisfies} $\left( H4\right) .$ \textit{Proof:} If $\phi \left( x,y,\theta \right) =S(x,\theta )-y\theta ,\;$% then \[ D(\phi )(x,\theta ,y)=\left( \begin{array}{cc} 0 & \frac{\partial ^{\;2}S}{\partial x\partial \theta }(x,\theta ) \\ -I_{n} & \frac{\partial ^{\;2}S}{\partial \theta \partial \theta }(x,\theta ) \end{array} \right) ,\text{ }I_{n}\;\text{is the unit matrix of size }n \] \[ \left| \det \;D(\phi )\;(x,\theta ,y)\right| =\left| \det \;\frac{\partial ^{2}S}{\partial x\partial \theta }\;(x,\theta )\right| \geq \delta _{0}% %TCIMACRO{\TeXButton{End Proof}{\endproof }} %BeginExpansion \endproof % %EndExpansion \] \textbf{Remark 2.3}\textit{.} By the global implicit function theorem we can easily see from $\left( G1\right) $ and $\left( G4\right) $ that the mappings $h_{1}$ and$\ h_{2}\;$are a global diffeomorphisms of $\Bbb{R}% ^{2n}, $ where \[ h_{1}:(x,\theta )\longrightarrow (x,\partial _{x}S\;(x,\theta )),\text{ }% h_{2}:(x,\theta )\longrightarrow (\theta ,\partial _{\theta }S\;(x,\theta )) \] because \[ h_{1}^{\prime }\left( x,\theta \right) =\left( \begin{array}{ll} I_{n} & \frac{\partial ^{2}S}{\partial x^{2}}\;(x,\theta ) \\ 0 & \frac{\partial ^{2}S}{\partial x\partial \theta }\;(x,\theta ) \end{array} \right) ,\text{ }h_{2}^{\prime }\left( x,\theta \right) =\left( \begin{array}{ll} 0 & \frac{\partial ^{2}S}{\partial x\partial \theta }\;(x,\theta ) \\ I_{n} & \frac{\partial ^{2}S}{\partial \theta ^{2}}\;(x,\theta ) \end{array} \right) \] and $\left| \det h_{1}^{\prime }\left( x,\theta \right) \right| =\left| \det h_{2}^{\prime }\left( x,\theta \right) \right| =\left| \det \;\frac{\partial ^{2}S}{\partial x\partial \theta }\;(x,\theta )\right| \geq \delta _{0}>0,$ $% \forall \left( x,\theta \right) \in \Bbb{R}^{2n}.$ We assume also the following assumption for $S$ (strong than $\left( G2\right) $): \[ \left( G5\right) \;\;\;\;\;\;\;\;\;\forall (\alpha ,\beta )\in \Bbb{N}% ^{n}\times \Bbb{N}^{n}\bigskip ,\ \exists C_{\alpha ,\beta }>0,\;\left| \partial _{x}^{\alpha }\partial _{\theta }^{\beta }S(x,\theta )\right| \leq C_{\alpha ,\beta }\;\lambda (x,\theta )^{(2-\left| \alpha \right| -\left| \beta \right| )} \] \textbf{Lemma 2.4.} \textit{If }$S\;$\textit{satisfies }$(G1),$\textit{\ }$% (G4)$\textit{\ and }$(G5),\;$\textit{then }$S\;$\textit{satisfies}$\;(G3)\;$% \textit{and }$(G3)^{*}$ \textit{and the estimate, }$\exists C_{5}>0,$ $% \forall (x,\theta ),$ $(x^{\prime },\theta ^{\prime })\in \Bbb{R}^{2n}$% \textit{:} \begin{equation} \left| x-x^{\prime }\right| +\left| \theta -\theta ^{\prime }\right| \leq C_{5}\left[ \left| \left( \partial _{\theta }S\right) (x,\theta )-\left( \partial _{\theta }S\right) (x^{^{\prime }},\theta ^{^{\prime }})\right| +\left| \theta -\theta ^{^{\prime }}\right| \right] \tag{2.2} \end{equation} \textit{Proof: }The mappings \[ \Bbb{R}^{n}\ni \theta \longrightarrow f_{x}\left( \theta \right) =\partial _{x}S\left( x,\theta \right) ,\text{ }\Bbb{R}^{n}\ni x\longrightarrow g_{\theta }\left( x\right) =\partial _{\theta }S\left( x,\theta \right) \] are a global diffeomorphisms of $\Bbb{R}^{n},$ since from $\left( G5\right) $ the matrixes $f_{x}^{\prime }\left( \theta \right) =g_{\theta }^{\prime }\left( x\right) =\frac{\partial ^{2}S}{\partial x\partial \theta }(x,\theta )$ and $\left( f_{x}^{-1}\right) ^{\prime }$ and $\left( g_{\theta }^{-1}\right) ^{\prime }$ are uniformly bounded on $\Bbb{R}^{2n}.$ Thus the Taylor's theorem leads to the estimates, $\exists M,N>0,$ $\forall (x,\theta )\in \Bbb{R}^{2n}$: \begin{eqnarray*} \left| \theta \right| &=&\left| \text{ }f_{x}^{-1}\left( f_{x}\left( \theta \right) \right) -f_{x}^{-1}\left( f_{x}\left( 0\right) \right) \right| \leq M\left| \partial _{x}S\left( x,\theta \right) -\partial _{x}S\left( x,0\right) \right| \\ &\leq &C_{6}\lambda (x,\partial _{x}S),\text{\ \ \ }C_{6}>0 \end{eqnarray*} \begin{eqnarray*} \left| x\right| &=&\left| \text{ }g_{\theta }^{-1}\left( g_{\theta }\left( \theta \right) \right) -g_{\theta }^{-1}\left( g_{\theta }\left( 0\right) \right) \right| \leq N\left| \partial _{\theta }S\left( x,\theta \right) -\partial _{\theta }S\left( 0,\theta \right) \right| \\ &\leq &C_{7}\lambda (\partial _{\theta }S,\theta ),\text{\ \ \ }C_{7}>0 \end{eqnarray*} Applying Taylor theorem to $h_{2}^{-1}$ we obtain, $\exists C_{5}>0,$ $% \forall (x,\theta ),(x^{\prime },\theta ^{\prime })\in \Bbb{R}^{2n},$% \[ \left| (x,\theta )-(x^{^{\prime }},\theta ^{^{\prime }})\right| =\left| h_{2}^{-1}\left( h_{2}(x,\theta )\right) -h_{2}^{-1}\left( h_{2}(x^{\prime },\theta ^{\prime })\right) \right| \leq C_{5}\left| (\theta ,\partial _{\theta }S\;(x,\theta ))-(\theta ^{\prime },\partial _{\theta }S\;(x^{\prime },\theta ^{\prime }))\right| %TCIMACRO{\TeXButton{End Proof}{\endproof }} %BeginExpansion \endproof % %EndExpansion \] \textbf{Corollary 2.5.}\textit{\ }In case $\theta =\theta ^{\prime }$ in $% \left( 2.2\right) ,$ we obtain \begin{equation} \exists C_{5}>0,\forall (x,x^{\prime },\theta )\in \Bbb{R}^{3n},\text{ }% \left| x-x^{\prime }\right| \leq C_{5}\left| \left( \partial _{\theta }S\right) (x,\theta )-\left( \partial _{\theta }S\right) (x^{\prime },\theta )\right| \tag{2.3} \end{equation} \textbf{Proposition 2.6.} \textit{If the function }$S\;$\textit{satisfies }$% (G1),$\textit{\ }$(G2),$\textit{\ }$(G4)$ and $(G5),\;$\textit{then there exists a constant }$\varepsilon _{0}>0,$\textit{\ such that} \[ \phi \in \Gamma _{1}^{2}(\Omega _{\phi ,\varepsilon _{0}}) \] \textit{where }$\phi \;$\textit{is the phase function associated to }$S$% \textit{\ in }$\left( 2.1\right) $ and\newline \textit{\ }$\Omega _{\phi ,\varepsilon _{0}}=\left\{ (x,\theta ,y)\in \Bbb{R}% ^{3n};\;\left| \partial _{\theta }\phi \left( x,\theta ,y\right) \right| ^{2}<\varepsilon _{0}\;(\left| x\right| ^{2}+\left| y\right| ^{2}+\left| \theta \right| ^{2})\right\} $ \textit{Proof:} We have to show the existence of $\varepsilon _{0}>0\;$such that $\forall \alpha ,\beta ,\gamma \in \Bbb{N}^{n}\bigskip ,\ \exists C_{\alpha ,\beta ,\gamma }>0,$% \begin{equation} \left| \partial _{x}^{\alpha }\partial _{\theta }^{\beta }\partial _{y}^{\gamma }\phi (x,\theta ,y)\right| \leq C_{\alpha ,\beta ,\gamma }\lambda (x,\theta ,y)^{(2-\left| \alpha \right| -\left| \beta \right| -\left| \gamma \right| )},\text{ }\forall (x,\theta ,y)\in \Omega _{\phi ,\varepsilon _{0}} \tag{2.4} \end{equation} If $\left\{ \begin{array}{c} \left| \gamma \right| =1,\text{ }\left| \partial _{x}^{\alpha }\partial _{\theta }^{\beta }\partial _{y}^{\gamma }\phi (x,\theta ,y)\right| =\left| \partial _{x}^{\alpha }\partial _{\theta }^{\beta }\left( -\theta \right) \right| =\left\{ \begin{array}{c} 0\;\;\;\;\;\;\;\;\;\;\;\;\text{if }\left| \alpha \right| \neq 0 \\ \left| \partial _{\theta }^{\beta }(-\theta )\right| \;\;\text{if }\alpha =0 \end{array} \right. \\ \left| \gamma \right| >1,\text{ \ \ \ \ \ \ \ }\left| \partial _{x}^{\alpha }\partial _{\theta }^{\beta }\partial _{y}^{\gamma }\phi (x,\theta ,y)\right| =0\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \end{array} \right. $ \newline then the estimate $\left( 2.4\right) $ is satisfied in the case $\left| \gamma \right| =1$ and $\left| \gamma \right| >1.$ If $\left| \gamma \right| =0,\forall \alpha ,\beta \in \Bbb{N}^{n}\bigskip ,\ \exists C_{\alpha ,\beta }>0$, such that \[ \left| \partial _{x}^{\alpha }\partial _{\theta }^{\beta }\phi (x,\theta ,y)\right| =\left| \partial _{x}^{\alpha }\partial _{\theta }^{\beta }S\left( x,\theta \right) -\partial _{x}^{\alpha }\partial _{\theta }^{\beta }\left( y\theta \right) \right| \leq C_{\alpha ,\beta }\lambda (x,\theta ,y)^{(2-\left| \alpha \right| -\left| \beta \right| )},\text{ when}\newline \text{ }\left| \alpha \right| +\left| \beta \right| \leq 2\newline \] for $\left| \alpha \right| +\left| \beta \right| >2,$ $\partial _{x}^{\alpha }\partial _{\theta }^{\beta }\phi (x,\theta ,y)=\partial _{x}^{\alpha }\partial _{\theta }^{\beta }S\left( x,\theta \right) ,$ then it is sufficient to prove the existence of $\varepsilon _{0}>0$ and $C_{\alpha ,\beta }>0,$ such that \[ \left| \partial _{x}^{\alpha }\partial _{\theta }^{\beta }S(x,\theta )\right| \leq C_{\alpha ,\beta }\;\lambda (x,\theta ,y)^{\left( 2-\left| \alpha \right| -\left| \beta \right| \right) },\text{ }\forall (x,\theta ,y)\in \Omega _{\phi ,\varepsilon _{0}\;} \] where $\Omega _{\phi ,\varepsilon _{0}\;}=\left\{ (x,\theta ,y)\in \Bbb{R}% ^{3n};\;\left| \partial _{\theta }S\left( x,\theta \right) -y\right| ^{2}<\varepsilon _{0}\;\left( \left| x\right| ^{2}+\left| y\right| ^{2}+\left| \theta \right| ^{2}\right) \right\} .$ We have in $\Omega _{\phi ,\varepsilon _{0}}$: \[ \left| y\right| =\left| \partial _{\theta }S\left( x,\theta \right) -y-\partial _{\theta }S\left( x,\theta \right) \right| \leq \sqrt{% \varepsilon _{0}}\left( \left| x\right| ^{2}+\left| y\right| ^{2}+\left| \theta \right| ^{2}\right) ^{1/2}+C_{8}\lambda \left( x,\theta \right) ,% \text{ }C_{8}>0 \] then for $\varepsilon _{0}$ sufficiently small, we obtain \begin{equation} \exists C_{9}>0,\text{ }\left| y\right| \leq C_{9}\lambda \left( x,\theta \right) ,\text{ }\forall (x,\theta ,y)\in \Omega _{\phi ,\varepsilon _{0}\;} \tag{2.5} \end{equation} and the equivalence \begin{equation} \lambda \left( x,\theta ,y\right) \simeq \lambda \left( x,\theta \right) \text{ in }\Omega _{\phi ,\varepsilon _{0}\;} \tag{2.6} \end{equation} thus the assumption $\left( G5\right) $ and $\left( 2.6\right) $ leads to the estimate $\left( 2.4\right) $% %TCIMACRO{\TeXButton{End Proof}{\endproof }} %BeginExpansion \endproof % %EndExpansion Applying $\left( 2.6\right) ,$ we have \textbf{Proposition 2.7.} \textit{1) If }$\left( x,\theta \right) \rightarrow a\left( x,\theta \right) $\textit{\ belongs to }$\Gamma _{0}^{m}\left( \Bbb{R}_{x}^{n}\times \Bbb{R}% _{\theta }^{n}\right) ,$\textit{\ then }$\left( x,\theta ,y\right) \rightarrow a\left( x,\theta \right) $\textit{\ belong to }$\Gamma _{0}^{m}\left( \Bbb{R}_{x}^{n}\times \Bbb{R}_{\theta }^{n}\times \Bbb{R}% _{y}^{n}\right) \cap \Gamma _{0}^{m}\left( \Omega _{\phi ,\varepsilon _{0}\;}\right) $ \textit{2) If }$\left( x,\theta \right) \rightarrow a\left( x,\theta \right) $\textit{\ belongs to }$\Gamma _{1}^{m}\left( \Bbb{R}_{x}^{n}\times \Bbb{R}% _{\theta }^{n}\right) ,$\textit{\ then }$\left( x,\theta ,y\right) \rightarrow a\left( x,\theta \right) $\textit{\ belongs to }$\Gamma _{1}^{m}\left( \Bbb{R}_{x}^{n}\times \Bbb{R}_{\theta }^{n}\times \Bbb{R}% _{y}^{n}\right) \cap \Gamma _{1}^{m}\left( \Omega _{\phi ,\varepsilon _{0}\;}\right) .$ Our fundamental result is: \textbf{Theorem 2.8.} \textit{Let }$F\;$\textit{be a integral operator of the kernel distribution:} \begin{equation} K(x,y)=\stackunder{\Bbb{R}^{n}}{\int }e^{i(S(x,\theta )-y\theta \;)}a(x,\theta )\widehat{d\theta } \tag{2.7} \end{equation} \textit{where }$\widehat{d\theta }=(2\pi )^{-n}d\theta ,\;a\in \Gamma _{j}^{m}(\Bbb{R}_{x,\theta }^{\;2n}),$ $j=0,1$\textit{\ and }$S\;$\textit{% satisfies }$(G1),$\textit{\ }$(G4)$\textit{\ and }$(G5).$\textit{\ Then }$% FF^{*}$\textit{\ and }$F^{*}F\;$\textit{are a pseudodifferential\ operators with a symbol functions in }$\Gamma _{j}^{2m}(\Bbb{R}^{2n}),$ $j=0,1.$% \textit{\newline The symbol functions of }$FF^{*}$\textit{\ and }$F^{*}F$\textit{\ are given by:} \begin{eqnarray*} \sigma (FF^{*})(x,\partial _{x}S(x,\theta )) &\equiv &\left| a(x,\theta )\right| ^{2}\left| (\det \frac{\partial ^{2}S}{\partial \theta \partial x}% )^{-1}(x,\theta )\right| \\ \sigma (F^{*}F)(\partial _{\theta }S(x,\theta ),\theta ) &\equiv &\left| a(x,\theta )\right| ^{2}\left| (\det \frac{\partial ^{2}S}{\partial \theta \partial x})^{-1}(x,\theta )\right| \end{eqnarray*} \textit{we denote here }$a\equiv b$ for $a,b\in \Gamma _{s}^{2p}(\Bbb{R}% ^{2n})$ \textit{if }$\left( a-b\right) \in \Gamma _{s}^{2p-2}(\Bbb{R}^{2n}).$ \textit{Proof:} If $u\in \mathcal{S}(\Bbb{R}^{n}),$ then $Fu(x)$ is given by: \begin{eqnarray} Fu(x) &=&\int\limits_{\Bbb{R}^{n}}K(x,y)\;u(y)\;dy=\int\limits_{\Bbb{R}% ^{n}}\int\limits_{\Bbb{R}^{n}}e^{i(S(x,\theta )-y\theta \;)}a(x,\theta )u(y)dy\widehat{d\theta } \nonumber \\ &=&\int\limits_{\Bbb{R}^{n}}e^{iS(x,\theta )}\;a(x,\theta )\;(\int\limits_{% \Bbb{R}^{n}}e^{-iy\theta \;}u(y)dy)\widehat{d\theta } \nonumber \\ &=&\int\limits_{\Bbb{R}^{n}}e^{iS(x,\theta )}\;a(x,\theta )\;\mathcal{F}% u(\theta )\widehat{d\theta }\text{ } \tag{2.8} \end{eqnarray} where $\mathcal{F}u$ is the Fourier transform of the function $u.$ $F$ is a continuous linear mapping from $\mathcal{S}(\Bbb{R}^{n})$ to $% \mathcal{S}(\Bbb{R}^{n})$ (by the theorem 1.1). Let $v\in \mathcal{S}(\Bbb{R}% ^{n}),$ then: \begin{eqnarray*} &<&Fu,v>_{L^{2}(\Bbb{R}^{n})}=\int\limits_{\Bbb{R}^{n}}(\int\limits_{\Bbb{R}% ^{n}}e^{iS(x,\theta )}\;a(x,\theta )\mathcal{F}u(\theta )\widehat{d\theta }% )\;\overline{v(x)}\;dx \\ &=&\int\limits_{\Bbb{R}^{n}}\mathcal{F}u(\theta )(\int\limits_{\Bbb{R}^{n}}% \overline{e^{-iS(x,\theta )}\;\overline{a(x,\theta )}\;v(x)\;dx})\widehat{% d\theta }\text{ } \end{eqnarray*} thus \[ _{L^{2}(\Bbb{R}^{n})}=_{L^{2}(\Bbb{R}^{n})}=(2\pi )^{-n}<\mathcal{% F}u\left( \theta \right) ,\mathcal{F}\left( \left( F^{*}v\right) \right) \left( \theta \right) >_{L^{2}(\Bbb{R}^{n})} \] where \begin{equation} \mathcal{F}\left( \left( F^{*}v\right) \right) \left( \theta \right) =\int\limits_{\Bbb{R}^{n}}e^{-iS(\widetilde{x},\theta )}\overline{a}\left( \widetilde{x},\theta \right) v(\widetilde{x})d\widetilde{x} \tag{2.9} \end{equation} Hence, $\forall v\in \mathcal{S}(\Bbb{R}^{n})$: \begin{equation} \left( FF^{*}v\right) \left( x\right) =\int\limits_{\Bbb{R}^{n}}\int\limits_{% \Bbb{R}^{n}}e^{i\left( S\left( x,\theta \right) -S\left( \widetilde{x}% ,\theta \right) \right) }a\left( x,\theta \right) \overline{a}\left( \widetilde{x},\theta \right) d\widetilde{x}\widehat{d\theta } \tag{2.10} \end{equation} Now we show that $FF^{*}$ is a pseudodifferential operator. To do this we use the fact that $\left( S\left( x,\theta \right) -S\left( \widetilde{x}% ,\theta \right) \right) $ can be expressed by the scalar product $$ and we consider the changes of variables $\left( x,\widetilde{x},\theta \right) \rightarrow \left( x,\widetilde{x},\xi =\xi \left( x,\widetilde{x},\theta \right) \right) .$ Let the kernel distribution of $FF^{*}$% \[ K\left( x,\tilde{x}\right) =\int\limits_{\Bbb{R}^{n}}e^{i\left( S(x,\theta )-S(\tilde{x},\theta )\right) }a(x,\theta )\overline{a}\left( \tilde{x}% ,\theta \right) \widehat{d\theta } \] we obtain from $\left( 2.3\right) $ that if \[ \left| x-\widetilde{x}\right| \geq \frac{\varepsilon }{2}\lambda \left( x,% \widetilde{x},\theta \right) \text{ (where }\varepsilon >0\text{ is sufficiently small}) \] then \begin{equation} \left| \left( \partial _{\theta }S\right) (x,\theta )-\left( \partial _{\theta }S\right) \left( \widetilde{x},\theta \right) \right| \geq \frac{% \varepsilon }{2C_{5}}\lambda \left( x,\widetilde{x},\theta \right) \tag{2.11} \end{equation} Let $\omega \in C^{\infty }\left( \Bbb{R}\right) $ such that \[ \left\{ \begin{array}{c} \omega \left( x\right) \geq 0,\text{ }\forall x\in \Bbb{R} \\ \omega \left( x\right) =1\text{ if }x\in \left[ -\frac{1}{2},\frac{1}{2}% \right] \\ \text{supp}\omega \subset \left] -1,1\right[ \end{array} \right. \] and put \[ b\left( x,\tilde{x},\theta \right) \stackrel{d\acute{e}f}{=}a(x,\theta )% \overline{a}\left( \tilde{x},\theta \right) =b_{1}^{\varepsilon }\left( x,% \tilde{x},\theta \right) +b_{2}^{\varepsilon }\left( x,\tilde{x},\theta \right) \; \] \[ \left\{ \begin{array}{c} b_{1}^{\varepsilon }\left( x,\tilde{x},\theta \right) =\omega \left( \frac{% \left| x-\tilde{x}\right| }{\varepsilon \lambda \left( x,\tilde{x},\theta \right) }\right) b\left( x,\tilde{x},\theta \right) \\ b_{2}^{\varepsilon }\left( x,\tilde{x},\theta \right) =\left[ 1-\omega \left( \frac{\left| x-\tilde{x}\right| }{\varepsilon \lambda \left( x,\tilde{% x},\theta \right) }\right) \right] b\left( x,\tilde{x},\theta \right) \end{array} \right. \] then the integral $K\left( x,\widetilde{x}\right) =K_{1}^{\varepsilon }\left( x,\widetilde{x}\right) +K_{2}^{\varepsilon }\left( x,\widetilde{x}% \right) ,$ where \[ K_{i}^{\varepsilon }\left( x,\tilde{x}\right) =\int\limits_{\Bbb{R}% ^{n}}e^{i\left( S(x,\theta )-S(\tilde{x},\theta )\right) }b_{i}^{\varepsilon }\left( x,\tilde{x},\theta \right) \widehat{d\theta },\text{ }i=1,2 \] We shall treat separately the kernels $K_{1}^{\varepsilon }$ and $% K_{2}^{\varepsilon }$. On the support of $b_{2}^{\varepsilon },$ the inequality $\left( 2.11\right) $ is satisfied and we have the following lemma \textbf{Lemma 2.9.} $K_{2}^{\varepsilon }\left( x,\widetilde{x}\right) \in \mathcal{S}\left( \Bbb{R}^{n}\times \Bbb{R}^{n}\right) .$ \textit{Proof:} Using the oscillatory integral method, we obtain a linear partial differential operator $L$ of order 1, such that \[ L\left( e^{i\left( S(x,\theta )-S(\tilde{x},\theta )\right) }\right) =e^{i\left( S(x,\theta )-S(\tilde{x},\theta )\right) }\text{ } \] \[ \text{where }L=-i\left| \left( \partial _{\theta }S\right) (x,\theta )-\left( \partial _{\theta }S\right) \left( \widetilde{x},\theta \right) \right| ^{-2}\sum\limits_{i=1}^{n}\left[ \left( \partial _{\theta _{i}}S\right) (x,\theta )-\left( \partial _{\theta _{i}}S\right) \left( \widetilde{x},\theta \right) \right] \partial _{\theta _{i}} \] then the transposed operator $^{t}L$ of $L$ is \[ ^{t}L=\sum\limits_{i=1}^{n}F_{j}\left( x,\widetilde{x},\theta \right) \partial _{\theta _{i}}+G\left( x,\widetilde{x},\theta \right) \text{ } \] where \[ \left\{ \begin{array}{c} F_{j}\left( x,\widetilde{x},\theta \right) =i\left| \left( \partial _{\theta }S\right) (x,\theta )-\left( \partial _{\theta }S\right) \left( \widetilde{x}% ,\theta \right) \right| ^{-2}\left( \left( \partial _{\theta _{i}}S\right) (x,\theta )-\left( \partial _{\theta _{i}}S\right) \left( \widetilde{x}% ,\theta \right) \right) \in \Gamma _{0}^{-1}\left( \Omega \right) \\ G\left( x,\widetilde{x},\theta \right) =i\sum\limits_{i=1}^{n}\partial _{\theta _{i}}\left[ \left| \left( \partial _{\theta }S\right) (x,\theta )-\left( \partial _{\theta }S\right) \left( \widetilde{x},\theta \right) \right| ^{-2}\left( \left( \partial _{\theta _{i}}S\right) (x,\theta )-\left( \partial _{\theta _{i}}S\right) \left( \widetilde{x},\theta \right) \right) \right] \in \Gamma _{0}^{-2}\left( \Omega \right) \\ \Omega =\left\{ \left( x,\tilde{x},\theta \right) \in \Bbb{R}^{3n};\;\left| \partial _{\theta }S(x,\theta )-\partial _{\theta }S\left( \tilde{x},\theta \right) \right| >\frac{\varepsilon }{2C_{5}}\lambda \left( x,\tilde{x}% ,\theta \right) \newline \right\} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \end{array} \right. \] On the other hand we prove by induction on $q,$ that \[ \left( ^{t}L\right) ^{q}b_{2}^{\varepsilon }\left( x,\tilde{x},\theta \right) =\sum\limits\Sb \left| \gamma \right| \leq q \\ \gamma \in \Bbb{N}% ^{n} \endSb g_{\gamma }^{\left( q\right) }\left( x,\tilde{x},\theta \right) \partial _{\theta }^{\gamma }b_{2}^{\varepsilon }\left( x,\tilde{x},\theta \right) ,\text{ }g_{\gamma }^{\left( q\right) }\in \Gamma _{0}^{-q} \] By the Leibnitz's formula, the form $\left( ^{t}L\right) ^{q},$ and integration by parts we obtain that $K_{2}^{\varepsilon }$ is a rapidly decreasing function: \[ \forall \alpha ,\alpha ^{\prime },\beta ,\beta ^{\prime }\in \Bbb{N}% ^{n},\exists C_{\alpha ,\alpha ^{\prime },\beta ,\beta ^{\prime }}>0,\text{ }% \sup_{x,\widetilde{x}\in \Bbb{R}^{n}}\left| x^{\alpha }\widetilde{x}^{\alpha ^{\prime }}\partial _{x}^{\beta }\partial _{\widetilde{x}}^{\beta ^{\prime }}K_{2}^{\varepsilon }\left( x,\widetilde{x}\right) \right| \leq C_{\alpha ,\alpha ^{\prime },\beta ,\beta ^{\prime }}% %TCIMACRO{\TeXButton{End Proof}{\endproof }} %BeginExpansion \endproof % %EndExpansion \] Next, we shall analyse $K_{1}^{\varepsilon }.$ This analyse is more difficult and dependent of the choice of the parameter $\varepsilon .$ It follows from Taylor's formula that \[ S\left( x,\theta \right) -S\left( \widetilde{x},\theta \right) =_{\Bbb{R}^{n}};% \text{ }\xi \left( x,\widetilde{x},\theta \right) =\int\limits_{0}^{1}\left( \partial _{x}S\right) \left( \widetilde{x}+t\left( x-\widetilde{x}\right) ,\theta \right) dt \] Let the vectorial function \[ \widetilde{\xi }^{\varepsilon }\left( x,\widetilde{x},\theta \right) =\omega \left( \frac{\left| x-\tilde{x}\right| }{2\varepsilon \lambda \left( x,% \tilde{x},\theta \right) }\right) \xi \left( x,\widetilde{x},\theta \right) +\left( 1-\omega \left( \frac{\left| x-\tilde{x}\right| }{2\varepsilon \lambda \left( x,\tilde{x},\theta \right) }\right) \right) \left( \partial _{x}S\right) \left( \widetilde{x},\theta \right) \] then \[ \widetilde{\xi }^{\varepsilon }\left( x,\widetilde{x},\theta \right) =\xi \left( x,\widetilde{x},\theta \right) \text{ on the support of }% b_{1}^{\varepsilon }\text{ } \] Moreover, for $\varepsilon $ sufficiently small we have on the support of $% b_{1}^{\varepsilon }$% \begin{equation} \lambda \left( x,\theta \right) \simeq \lambda \left( \widetilde{x},\theta \right) \simeq \lambda \left( x,\widetilde{x},\theta \right) \tag{2.12} \end{equation} Let us consider the mapping \begin{equation} \Bbb{R}^{3n}\ni \left( x,\widetilde{x},\theta \right) \rightarrow \left( x,% \widetilde{x},\widetilde{\xi }^{\varepsilon }\left( x,\widetilde{x},\theta \right) \right) \tag{2.13} \end{equation} with Jacobian matrix \[ \left( \begin{array}{lll} I_{n} & 0 & 0 \\ 0 & I_{n} & 0 \\ \partial _{x}\widetilde{\xi }^{\varepsilon } & \partial _{\widetilde{x}}% \widetilde{\xi }^{\varepsilon } & \partial _{\theta }\widetilde{\xi }% ^{\varepsilon } \end{array} \right) \] its Jacobian is equal to $\det \partial _{\theta }\widetilde{\xi }% ^{\varepsilon }\left( x,\widetilde{x},\theta \right) $. \[ \frac{\partial \widetilde{\xi }_{j}^{\varepsilon }}{\partial \theta _{i}}% \left( x,\widetilde{x},\theta \right) =\frac{\partial ^{2}S}{\partial \theta _{i}\partial x_{j}}\left( \widetilde{x},\theta \right) +\omega \left( \frac{% \left| x-\tilde{x}\right| }{2\varepsilon \lambda \left( x,\tilde{x},\theta \right) }\right) \left( \frac{\partial \xi _{j}}{\partial \theta _{i}}\left( x,\widetilde{x},\theta \right) -\frac{\partial ^{2}S}{\partial \theta _{i}\partial x_{j}}\left( \widetilde{x},\theta \right) \right) \] \[ -\frac{\left| x-\tilde{x}\right| }{2\varepsilon \lambda \left( x,\tilde{x}% ,\theta \right) }\frac{\partial \lambda }{\partial \theta _{i}}\left( x,% \tilde{x},\theta \right) \lambda ^{-1}\left( x,\tilde{x},\theta \right) \omega ^{\prime }\left( \frac{\left| x-\tilde{x}\right| }{2\varepsilon \lambda \left( x,\tilde{x},\theta \right) }\right) \left( \xi _{j}\left( x,% \widetilde{x},\theta \right) -\frac{\partial S}{\partial x_{j}}\left( \widetilde{x},\theta \right) \right) \] Thus, we obtain on the support of $\omega \left( \frac{\left| x-\tilde{x}% \right| }{2\varepsilon \lambda \left( x,\tilde{x},\theta \right) }\right) $% \begin{eqnarray*} \left| \frac{\partial \widetilde{\xi }_{j}^{\varepsilon }}{\partial \theta _{i}}\left( x,\widetilde{x},\theta \right) -\frac{\partial ^{2}S}{\partial \theta _{i}\partial x_{j}}\left( \widetilde{x},\theta \right) \right| &\leq &\left| \omega \left( \frac{\left| x-\tilde{x}\right| }{2\varepsilon \lambda \left( x,\tilde{x},\theta \right) }\right) \right| \left| \frac{\partial \xi _{j}}{\partial \theta _{i}}\left( x,\widetilde{x},\theta \right) -\frac{% \partial ^{2}S}{\partial \theta _{i}\partial x_{j}}\left( \widetilde{x}% ,\theta \right) \right| + \\ &&\left| \omega ^{\prime }\left( \frac{\left| x-\tilde{x}\right| }{% 2\varepsilon \lambda \left( x,\tilde{x},\theta \right) }\right) \right| \lambda ^{-1}\left( x,\tilde{x},\theta \right) \left| \xi _{j}\left( x,% \widetilde{x},\theta \right) -\frac{\partial S}{\partial x_{j}}\left( \widetilde{x},\theta \right) \right| \end{eqnarray*} Next it follows from $\left( G5\right) $ and Taylor's formula that \begin{eqnarray} \left| \frac{\partial \xi _{j}}{\partial \theta _{i}}\left( x,\widetilde{x}% ,\theta \right) -\frac{\partial ^{2}S}{\partial \theta _{i}\partial x_{j}}% \left( \widetilde{x},\theta \right) \right| &\leq &\int\limits_{0}^{1}\left| \frac{\partial ^{2}S}{\partial \theta _{i}\partial x_{j}}\left( \widetilde{x}% +t\left( x-\widetilde{x}\right) ,\theta \right) -\frac{\partial ^{2}S}{% \partial \theta _{i}\partial x_{j}}\left( \widetilde{x},\theta \right) \right| dt \nonumber \\ &\leq &C_{10}\left| x-\widetilde{x}\right| \lambda ^{-1}\left( \tilde{x}% ,\theta \right) ,\text{ }C_{10}>0 \tag{2.14} \end{eqnarray} and \begin{eqnarray} \left| \xi _{j}\left( x,\widetilde{x},\theta \right) -\frac{\partial S}{% \partial x_{j}}\left( \widetilde{x},\theta \right) \right| &\leq &\int\limits_{0}^{1}\left| \frac{\partial S}{\partial x_{j}}\left( \widetilde{x}+t\left( x-\widetilde{x}\right) ,\theta \right) -\frac{\partial S}{\partial x_{j}}\left( \widetilde{x},\theta \right) \right| dt \nonumber \\ &\leq &C_{11}\left| x-\widetilde{x}\right| ,\text{ }C_{11}>0\text{ } \tag{2.15} \end{eqnarray} from $\left( 2.14\right) $ and $\left( 2.15\right) ,$ there exists a positive constant $C_{12}>0,$ such that \begin{equation} \left| \frac{\partial \widetilde{\xi }_{j}^{\varepsilon }}{\partial \theta _{i}}\left( x,\widetilde{x},\theta \right) -\frac{\partial ^{2}S}{\partial \theta _{i}\partial x_{j}}\left( \widetilde{x},\theta \right) \right| \leq C_{12}\varepsilon ,\text{ }\forall i,j\in \left\{ 1,...,n\right\} \tag{2.16} \end{equation} if $\varepsilon <\frac{\delta _{0}}{2\widetilde{C}},$ then $\left( 2.16\right) $ and $\left( G4\right) $ yields the estimate \begin{equation} \delta _{0}/2\leq -\widetilde{C}\varepsilon +\delta _{0}\leq -\widetilde{C}% \varepsilon +\det \frac{\partial ^{2}S}{\partial x\partial \theta }% \;(x,\theta )\leq \det \partial _{\theta }\widetilde{\xi }^{\varepsilon }\left( x,\widetilde{x},\theta \right) ,\text{ }\widetilde{C}>0 \tag{2.17} \end{equation} If $\varepsilon $ is fixed such that $\left( 2.12\right) $ and $\left( 2.17\right) $ are true, then the mapping defined in $\left( 2.13\right) $ is a global diffeomorphism of $\Bbb{R}^{3n},$ hence there exists the mapping: \[ \theta :\Bbb{R}^{n}\times \Bbb{R}^{n}\times \Bbb{R}^{n}\ni \left( x,% \widetilde{x},\xi \right) \rightarrow \theta \left( x,\widetilde{x},\xi \right) \in \Bbb{R}^{n} \] such that \begin{equation} \left\{ \begin{array}{c} \widetilde{\xi }^{\varepsilon }\left( x,\widetilde{x},\theta \left( x,% \widetilde{x},\xi \right) \right) =\xi \\ \theta \left( x,\widetilde{x},\widetilde{\xi }^{\varepsilon }\left( x,% \widetilde{x},\theta \right) \right) =x \\ \partial ^{\alpha }\theta \left( x,\widetilde{x},\xi \right) =\mathcal{O}% \left( 1\right) ,\text{ }\forall \alpha \in \Bbb{N}^{3n}\backslash \left\{ 0\right\} \end{array} \right. \tag{2.18} \end{equation} If we use the change of variable $\xi \rightarrow \theta \left( x,\widetilde{% x},\xi \right) $ into $K_{1}^{\varepsilon }\left( x,\widetilde{x}\right) $, we obtain \begin{equation} K_{1}^{\varepsilon }\left( x,\widetilde{x}\right) =\int\limits_{\Bbb{R}% ^{n}}e^{i}b_{1}^{\varepsilon }\left( x,\tilde{x},\theta \left( x,\widetilde{x},\xi \right) \right) \left| \det \frac{\partial \theta }{\partial \xi }\left( x,\widetilde{x},\xi \right) \right| \widehat{d\xi } \tag{2.19} \end{equation} From $\left( 2.18\right) $ we have $b_{1}^{\varepsilon }\left( x,\tilde{x}% ,\theta \left( x,\widetilde{x},\xi \right) \right) \left| \det \frac{% \partial \theta }{\partial \xi }\left( x,\widetilde{x},\xi \right) \right| $ belongs to $\Gamma _{0}^{2m}\left( \Bbb{R}^{3n}\right) $ (respectively belongs to $\Gamma _{1}^{2m}\left( \Bbb{R}^{3n}\right) $) if $a\in \Gamma _{0}^{m}\left( \Bbb{R}^{2n}\right) $ (respectively if $a\in \Gamma _{1}^{m}\left( \Bbb{R}^{2n}\right) $). Applying the stationary phase theorem into $\left( 2.19\right) ,$ we obtain the symbol $\sigma (FF^{*})$ of the pseudodifferential operator $FF^{*}$% \[ \sigma (FF^{*})=b_{1}^{\varepsilon }\left( x,\tilde{x},\theta \left( x,% \widetilde{x},\xi \right) \right) \left| \det \frac{\partial \theta }{% \partial \xi }\left( x,\widetilde{x},\xi \right) \right| _{\left| \widetilde{% x}=x\right. }+R(x,\xi ) \] where $R(x,\xi )\;$belongs to$\;\Gamma _{\varrho }^{2m-2}\left( \Bbb{R}% ^{2n}\right) ,$ with $\varrho =0\;$if $a\in \Gamma _{0}^{2m}\left( \Bbb{R}% ^{2n}\right) \;$and $\varrho =1\;$if $a\in \Gamma _{1}^{m}\left( \Bbb{R}% ^{2n}\right) .$ When $\tilde{x}=x,$ $b_{1}^{\varepsilon }\left( x,\tilde{x},\theta \left( x,% \widetilde{x},\xi \right) \right) =\left| a\left( x,\theta \left( x,x,\xi \right) \right) \right| ^{2}$ where $\theta \left( x,x,\xi \right) $ is the inverse of the mapping $\theta \rightarrow \partial _{x}S\left( x,\theta \right) =\xi $ and thus $\det \frac{\partial \theta }{\partial \xi }\left( x,x,\xi \right) =\left( \det \frac{\partial ^{2}S}{\partial \theta \partial x% }\left( x,\theta \left( x,x,\xi \right) \right) \right) ^{-1},$ then \[ \sigma (FF^{*})\left( x,\partial _{x}S\left( x,\theta \right) \right) \equiv \left| a\left( x,\theta \right) \right| ^{2}\left| \det \frac{\partial ^{2}S% }{\partial \theta \partial x}\left( x,\theta \right) \right| ^{-1}\text{ (in }\Gamma _{\varrho }^{2m-2}\left( \Bbb{R}^{2n}\right) \text{) } \] (with $\varrho =0\;$if $a\in \Gamma _{0}^{2m}\left( \Bbb{R}^{2n}\right) \;$% and $\varrho =1\;$if $a\in \Gamma _{1}^{m}\left( \Bbb{R}^{2n}\right) $). From $\left( 2.8\right) $ and $\left( 2.9\right) $ we obtain the expression of $F^{*}F:$ $\forall v\in \mathcal{S}\left( \Bbb{R}^{n}\right) $% \begin{eqnarray*} \left( \mathcal{F(}F^{*}F)\mathcal{F}^{-1}\right) v\left( \theta \right) &=&\int\limits_{\Bbb{R}^{n}}e^{-iS\left( x,\theta \right) }\overline{a}% \left( x,\theta \right) \left( F(\mathcal{F}^{-1}v)\right) (x)dx \\ &=&\int\limits_{\Bbb{R}^{n}}e^{-iS\left( x,\theta \right) }\overline{a}% (x,\theta )(\int\limits_{\Bbb{R}^{n}}e^{iS\left( x,\widetilde{\theta }% \right) }a\left( x,\widetilde{\theta }\right) \left( \mathcal{F}(\mathcal{F}% ^{-1}v)\right) \left( \widetilde{\theta }\right) \widehat{d\widetilde{\theta }}dx\newline \\ &=&\int\limits_{\Bbb{R}^{n}}\int\limits_{\Bbb{R}^{n}}e^{-i\left( S\left( x,\theta \right) -S\left( x,\tilde{\theta}\right) \right) \;}\overline{a}% (x,\theta )\;a\left( x,\widetilde{\theta }\right) v\left( \tilde{\theta}% \right) \widehat{d\widetilde{\theta }}dx \end{eqnarray*} Hence the kernel distribution of the integral operator $\mathcal{F(}F^{*}F)% \mathcal{F}^{-1}$ is \[ \widetilde{K}(\theta ,\widetilde{\theta })=\int\limits_{\Bbb{R}% ^{n}}e^{-i\left( S\left( x,\theta \right) -S\left( x,\tilde{\theta}\right) \right) \;}\overline{a}(x,\theta )\;a\left( x,\tilde{\theta}\right) \widehat{% dx} \] We remark that we can obtain $\widetilde{K}(\theta ,\widetilde{\theta })$ from $K(x,\widetilde{x})$ by exchanging $x$ with $\theta $ and inversely. On the other hand, all the assumptions used here are symmetrical on $x$ and $% \theta ,$ therefore $\mathcal{F(}F^{*}F)\mathcal{F}^{-1}$ is a nice pseudodifferential operator with symbol function \begin{eqnarray*} \sigma (\mathcal{F(}F^{*}F)\mathcal{F}^{-1})\left( \theta ,-\partial _{\theta }S(x,\theta )\right) &\equiv &\left| a(x,\theta )\right| ^{2}\left| \det \frac{\partial ^{2}S}{\partial x\partial \theta }(x,\theta )\right| ^{-1} \\ \text{in }\Gamma _{\varrho }^{2m-2}(\Bbb{R}^{2n})\;(\varrho &=&0\;\text{or\ }% \varrho =1) \end{eqnarray*} Thus \begin{eqnarray*} \sigma (F^{*}F)(\partial _{\theta }S(x,\theta ),\theta ) &\equiv &\left| a(x,\theta )\right| ^{2}\left| \det \frac{\partial ^{2}S}{\partial x\partial \theta }(x,\theta )\right| ^{-1} \\ \text{in }\Gamma _{\varrho }^{2m-2}(\Bbb{R}^{2n})\;(\varrho &=&0\;\text{or\ }% \varrho =1) %TCIMACRO{\TeXButton{End Proof}{\endproof }} %BeginExpansion \endproof % %EndExpansion \end{eqnarray*} \textbf{Theorem 2.10.} \textit{Let }$F\;$\textit{be the integral operator with the kernel distribution} \[ K(x,y)=\stackunder{\Bbb{R}^{n}}{\int }e^{i(S(x,\theta )-y.\theta \;)}a(x,\theta )\widehat{d\theta } \] \textit{where }$a\in \Gamma _{0}^{m}(\Bbb{R}_{x,\theta }^{2n})\;$\textit{and }$S$\textit{\ satisfies }$(G1),$\textit{\ }$(G4)$\textit{\ and }$(G5).$% \textit{\ Then\newline 1)\ For any }$m,$ $m\leq 0,$\textit{\ }$F\;$\textit{can be extended as a bounded linear mapping on }$L^{2}\left( \Bbb{R}^{n}\right) $\textit{\newline 2)\ For any }$m,$ $m<0,\;F\;$\textit{can be extended as a compact operator on }$L^{2}\left( \Bbb{R}^{n}\right) $ \textit{Proof:} It follows from theorem 2.8 that $F^{*}F\;$is a pseudodifferential operator with symbol function in $\Gamma _{0}^{2m}\left( \Bbb{R}^{2n}\right) .$ 1)\ If $m\leq 0,\;$the weight $\lambda ^{2m}(x,\theta )\;$is bounded, we can then apply the Cald\'{e}ron-Vaillancourt theorem (see [2], [7], [8]) for $% F^{\ast }F\;$and obtain the existence of a positive constant $\gamma (n)$ and a integer $k\left( n\right) $ such that \[ \left\| (F^{\ast }F)\;u\right\| _{L^{2}(\Bbb{R}^{n})}\leq \gamma (n)\;\sup\Sb \left| \alpha \right| ,\left| \beta \right| \leq k(n) \\ (x,\theta )\in \Bbb{R}^{2n} \endSb \left| \partial _{x}^{\alpha }\partial _{\theta }^{\beta }\sigma (FF^{\ast })(\partial _{\theta }S(x,\theta ),\theta )\right| \;\left\| u\right\| _{L^{2}(\Bbb{R}^{n})},\text{ }\forall u\in \mathcal{S}(\Bbb{R}^{n}) \] \newline Hence, $\forall u\in \mathcal{S}(\Bbb{R}^{n})$ we have \begin{eqnarray*} \left\| Fu\right\| _{L^{2}(\Bbb{R}^{n})} &\leq &\left\| F^{\ast }F\right\| _{_{\mathcal{L}\left( L^{2}(\Bbb{R}^{n})\right) }}^{1/2}\left\| u\right\| _{L^{2}(\Bbb{R}^{n})} \\ &\leq &\left( \gamma (n)\;\sup\Sb \left| \alpha \right| ,\left| \beta \right| \leq k(n) \\ (x,\theta )\in \Bbb{R}^{2n} \endSb \left| \partial _{x}^{\alpha }\partial _{\theta }^{\beta }\sigma (FF^{\ast })(\partial _{\theta }S(x,\theta ),\theta )\right| \right) ^{1/2}\left\| u\right\| _{L^{2}(\Bbb{R}^{n})} \end{eqnarray*} thus $F$ is also a bounded linear operator on $L^{2}(\Bbb{R}^{n})$ 2) If $m<0,$ $\stackunder{\left| x\right| +\left| \theta \right| \rightarrow +\infty }{\;\lim }\lambda ^{m}(x,\theta )=0,\;$and by the compactness theorem (see [5], [7], [8]),\ $F^{*}F$ can be extended as a compact linear operator on $L^{2}(\Bbb{R}^{n}).$ Thus the following general lemma asserts that the Fourier integral operator $F\;$is compact on $L^{2}(\Bbb{R}^{n})$. \textbf{Lemma 2.11}.\ \textit{Assume that }$T\in \mathcal{L}$\textit{\ }$% (H)\;$\textit{and }$T^{*}T\;$\textit{is compact where }$H$\textit{\ is a separable Hilbert space. Then }$T\;$\textit{is compact.} \textit{Proof:} We know that $T^{*}T$ is the norm limit of a sequence of operators of finite rank. Let $(\varphi _{j})_{j\in \Bbb{N}}$ be an orthonormal basis of $H$,\ then \[ \stackrel{n}{\stackunder{j=1}{\sum }}<\varphi _{j},.>T^{*}T\;\varphi _{j}% \stackunder{n\rightarrow +\infty }{\longrightarrow }T^{*}T\; \] thus \[ T^{*}(\stackrel{n}{\stackunder{j=1}{\sum }}<\varphi _{j},.>T\;\varphi _{j})% \stackunder{n\rightarrow +\infty }{\longrightarrow }T^{*}T\;\;% \Leftrightarrow \;T^{*}(\stackrel{n}{\stackunder{j=1}{\sum }}<\varphi _{j},.>T\;\varphi _{j}-T)\stackunder{n\rightarrow +\infty }{\longrightarrow }% 0 \] Since $T\;$is bounded, we have also \[ TT^{*}(\stackrel{n}{\stackunder{j=1}{\sum }}<\varphi _{j},.>T\;\varphi _{j}-T)\stackunder{n\rightarrow +\infty }{\longrightarrow }T(0)=0 \] $TT^{*}$is positive, and invertible on its image, then \[ \stackrel{n}{\stackunder{j=1}{\sum }}<\varphi _{j},.>T\;\varphi _{j}% \stackunder{n\rightarrow +\infty }{\longrightarrow }T% %TCIMACRO{\TeXButton{End Proof}{\endproof }} %BeginExpansion \endproof % %EndExpansion \] \begin{center} \textbf{References} \end{center} [1] K. Asada, D. Fujiwara, \textit{On some oscillatory transformation in\ }$% L^{2}\left( \Bbb{R}^{n}\right) ,$\textit{\ Japan.\ J. Math.\ vol 4 (2), 1978, p299-361.} [2] A. P. Calderon, R. Vaillancourt, \textit{On the boundness of pseudodifferential operators. J. Math. Soc. Japan 23, 1972, p374-378.} [3] J.J. Duistermaat,\ \textit{Fourier integral operators, Courant Institute Lecture Notes, New-York 1973.} [4] M. Hasanov, \textit{A class of unbounded Fourier integral operators. J. Math. Analysis and application 225, 1998, p641-651.} [5] 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.} [6] L. H\"{o}rmander, \textit{Fourier integral operators I, Acta Math. vol 127, 1971, p33-57.} [7] D. Robert, \textit{Autour de l'approximation semi-classique, Birk\"{a}% user, 1987.} [8] A. Senoussaoui,\ \textit{Op\'{e}rateurs h-admissibles matriciels \`{a} symboles op\'{e}rateurs, Preprint, Universit\'{e} libre de Bruxelles, 2004.} \end{document} ---------------0405100335908--