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E-mail adresses of authors: bmessirdi@univ-oran.dz, asenouss@ulb.ac.be 
Classification AMS: 35S30, 35S05, 47A10, 35P05
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Fourier integral operators, Pseudodifferential 
operators, $L^{2}$-boundness and $L^{2}$-compactness, $C^{\infty }$ Cauchy problem
---------------0405100335908
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%TCIDATA{Created=Wed Jun 05 12:45:33 2002}
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\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 
\[
<Fu\left( x\right) ,v\left( x\right) >_{L^{2}(\Bbb{R}^{n})}=<u\left(
x\right) ,F^{*}v\left( x\right) >_{L^{2}(\Bbb{R}^{n})}=(2\pi )^{-n}<\mathcal{%
F}u\left( \theta \right) ,\mathcal{F}\left( \left( F^{*}v\right) \right)
\left( \theta \right) >_{L^{2}(\Bbb{R}^{n})} 
\]
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 $<x-%
\widetilde{x},\xi \left( x,\widetilde{x},\theta \right) >$ 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 
\[
S\left( x,\theta \right) -S\left( \widetilde{x},\theta \right) =<x-%
\widetilde{x},\xi \left( x,\widetilde{x},\theta \right) >_{\Bbb{R}^{n}};%
\text{ }\xi \left( x,\widetilde{x},\theta \right) =\int\limits_{0}^{1}\left(
\partial _{x}S\right) \left( \widetilde{x}+t\left( x-\widetilde{x}\right)
,\theta \right) dt 
\]
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<x-\tilde{x},\xi >}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.}

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