\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 88, pp. 1--7.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2009/88\hfil Generalized integral operators]
{A class of generalized integral operators}

\author[S. Bekkara, B. Messirdi, A. Senoussaoui \hfil EJDE-2009/88\hfilneg]
{Samir Bekkara, Bekkai Messirdi, Abderrahmane Senoussaoui} % in alphabetical order

\address{Samir Bekkara \newline
Universit\'{e} des Sciences et de la Technologie d'Oran,
Facult\'{e} des Sciences, D\'{e}partement de Math\'{e}matiques,
Oran, Algeria}
\email{sbekkara@yahoo.fr}

\address{Bekkai Messirdi, Abderrahmane Senoussaoui \newline
Universit\'{e} d'Oran Es-S\'{e}nia, Facult\'{e} des Sciences,
D\'{e}partement de Math\'{e}matiques. B.P. 1524 El-Mnaouer, Oran,
Algeria}
\email{bmessirdi@univ-oran.dz}
\email{senoussaoui.abdou@univ-oran.dz}

\thanks{Submitted February 12, 2009. Published July 27, 2009.}
\subjclass[2000]{35S30, 35S05, 47A10, 35P05}
\keywords{Integral operators; $L^{2}$-boundedness; \hfill\break\indent
unbounded Fourier integral operators}

\begin{abstract}
 In this paper, we introduce a class of generalized integral
 operators that includes Fourier integral operators.
 We  establish some conditions on these operators such that they
 do not have bounded extension on $L^{2}(\mathbb{R}^{n})$.
 This permit us in particular to construct a class of Fourier
 integral operators with bounded symbols in
 $S_{1,1}^{0}(\mathbb{R}^{n}\times \mathbb{R}^{n})$ and in
 $\bigcap_{0<\rho <1}S_{\rho ,1}^{0}(\mathbb{R}^{n}\times \mathbb{R}^{n})$
 which cannot be extended to bounded operators in
 $L^{2}( \mathbb{R}^{n})$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}

\section{Introduction}

The integral operators of type
\begin{equation}
A\varphi (x) =\int e^{iS(x,\theta)
}a(x,\theta) \mathcal{F}\varphi (\theta)
d\theta \label{1.1}
\end{equation}
appear naturally for solving the hyperbolic partial differential
equations and expressing the $C^{\infty }$-solution of the
associate Cauchy problem's (see e.g. \cite{MeRaSe,MeSe1}).

If we write formally the expression of the Fourier
transform $\mathcal{F}\varphi (\theta) $ in \eqref{1.1}, we
obtain the following Fourier integral operators, so-called
canonical transformations,
\begin{equation}
A\varphi (x) =\iint e^{i(S(x,\theta
) -y\theta) }a(x,y,\theta) \varphi
(y) dyd\theta \label{1.2}
\end{equation}
in which appear two $C^{\infty }$-functions, the phase function
$\phi (x,y,\theta) =S(x,\theta) -y\theta$ and the amplitude
$a$ called the symbol of the operator $A$.
In the particular case where $S(x,\theta) =x\theta $,
one recovers the notion of pseudodifferential operators (see e.g
\cite{Ho2,St}).

Since 1970, many of Mathematicians have been interested to these type of
operators:  Duistermaat \cite{Du},  H\"{o}rmander
\cite{Ho2,Ho3}  Kumano-Go \cite{Ku}, and
Fujiwara \cite{AsFu}. We mention also the works of  Hasanov
\cite{Ha}, and  the recent results of  Messirdi
Senoussaoui \cite{MeSe2} and Aiboudi-Messirdi-Senoussaoui
\cite{AiMeSe}.

In this paper we study a general class of integral operators
including the class of Fourier integral operators, specially we
are interested in their continuity on $L^{2}(\mathbb{R}^{n})$.

The continuity of the operator $A$ on $L^{2}(\mathbb{R}^{n}) $ is
guaranteed if the weight of the symbol
$a$ is bounded, if this weight tends to zero then $A$ is compact
on $L^{2}(\mathbb{R}^{n}) $ (see eg. \cite{MeSe2}).

If the symbol $a$ is only bounded the associated Fourier integral
operator $A$ is not necessary bounded on $L^{2}(\mathbb{R}^{n})$.
Indeed, in 1998  Hasanov \cite{Ha} constructed an example of
unbounded Fourier integral operators on $L^{2}(\mathbb{R}) $.

Aiboudi-Messirdi-Senoussaoui \cite{AiMeSe} constructed recently in
a class of Fourier integral operators with bounded symbols in
the H\"{o}rmander class
$\bigcap_{0<\rho <1}S_{\rho ,1}^{0}(\mathbb{R}^{n}\times \mathbb{R}^{n})$
that cannot be extended to be a
bounded operator in $L^{2}(\mathbb{R}^{n}) ,n\geq 1$.

These results of unboundedness was obtained by using the
properties of the operators
\begin{equation}
B\varphi (x)=\int_{\mathbb{R}^{n}}k(z)\varphi ((b(x)z+a(x))dz
\label{1.3}
\end{equation}
on $L^{2}(\mathbb{R}^{n}) ,n\geq 1$, where $k(z)\in S(\mathbb{R}^{n})$
 (the space of $C^{\infty }$-functions on
$\mathbb{R}^{n}$, whose derivatives decrease faster than any power
of $|x|$ as $|x| \to +\infty )$, $a(x)$ and $b(x)$ are real-valued,
measurable functions on $\mathbb{R}^{n}$. Operators of type
\eqref{1.3} was considered by  Hasanov \cite{Ha} and
a slightly different way by  Aiboudi  Messirdi Senoussaoui
\cite{AiMeSe}.

We also give in this paper a generalization of these results since
we consider a class of integral operators which is general than
thus of type \eqref{1.3}:
\begin{equation}
C\varphi (x)=\int_{\mathbb{R}^{n}}K(x,z)\varphi (F(x,z))dz
\label{1.4}
\end{equation}
where $K(x,z)$ and $F(x,z)$ are real-valued, measurable functions on
$\mathbb{R}^{2n}$. The generalized integral operator $C$  includes
Hilbert, Mellin and the Fourier-Bros-Iagolnitzer transforms which
they has been used by many authors and for many purposes,
in particular respectively by H\"{o}rmander \cite{Ho1} for the
analysis of linear partial
differential operators,  Robert \cite{Ro} about the functional
calculus of pseudodiffrential operators,  Sj\"{o}strand
\cite{Sj} in the area of microlocal and semiclassical analysis and
 Stein \cite{St} for the study of singular integral
operators.

The operators $C$ appears also in the study of the width of the
quantum resonances (see e.g. \cite{Me}).

We shall discuss in the second section bounded extension problems
for the class of operators type $C$. We give some technical
conditions on the functions $K(x,z)$ and $F(x,z)$ such that $C$ do
not admit a bounded extension on $L^{2}(\mathbb{R}^{n}) $.
We also indicate a connection between
transformations $C$ and Fourier integral operators.

In the third section, we construct an example of Fourier integral
with bounded symbols belongs respectively to
$S_{1,1}^{0}(\mathbb{R}^{n}) $, (the case $n=1$ is given in \cite{Ha}
and generalized for $n\geq 2$ in \cite{AiMeSe}), and
$\cap_{0<\rho <1}S_{\rho ,1}^{0}$ that cannot be extended as a bounded
operator on $L^{2}(\mathbb{R}^{n}) $, $n\geq 2$. In the case of
the H\"{o}rmander symbolic class $S_{1,1}^{0}(
\mathbb{R}^{n}) $ our constructions are direct and
technical.

\section{Unboundedness of the generalized integral
operators}

In this section we construct a class of operators $C$  that
cannot be extended to be a bounded operator in
$L^{2}(\mathbb{R}^{n})$, $n\geq 1$. We have first an easy
boundedness criterion of the operator $C$.

\begin{proposition}\label{prop2.1}
Let $F(x,.)\in C^{1}(\mathbb{R}^{n})$, and
$K(x,.)\in L^{2}( \mathbb{R}^{n})$ for all $x\in \mathbb{R}^{n}$.
Suppose that there exits a function $g(x)$ such that
\begin{gather*}
g(x)>0,\quad \forall x\in \mathbb{R}^{n} \\
|\det\big(\frac{\partial F(x,z)}{\partial z}\big)
|\geq g(x),\quad \forall x,z\in \mathbb{R}^{n} \\
\| K(x,.)\|_{L^{2}(\mathbb{R}^{n})} \big/ \sqrt{g(x)}
\in L^{2}(\mathbb{R}^{n})
\end{gather*}
then $C$ is a bounded operator on $L^{2}(\mathbb{R}^{n})$.
\end{proposition}

\begin{proof}
Using H\"{o}lder inequality and the change of variable $y=F(x,z)$,
it's inverse is denoted $z=G(x,y)$, we obtain for all $\varphi \in
L^{2}(\mathbb{R}^{n})$,
\begin{equation}  \label{2.1}
\begin{aligned}
\| C\varphi \| _{L^{2}(\mathbb{R}^{n})}^{2}
&= \int_{\mathbb{R}^{n}}\Big| \int_{\mathbb{R}^{n}}K(x,z)\varphi
 (F(x,z))dz\Big|^{2}dx  \\
&\leq  \int_{\mathbb{R}^{n}}\Big[ \int_{\mathbb{R}^{n}}|
K(x,z)\varphi (F(x,z))|dz\Big] ^{2}dx   \\
&\leq  \int_{\mathbb{R}^{n}}\Big[ \| K(x,.)\|
_{L^{2}( \mathbb{R}^{n})}^{2}\int_{\mathbb{R}^{n}}|
\varphi (F(x,z))|^{2}dz\Big] dx   \\
&= \int_{\mathbb{R}^{n}}\Big[ \| K(x,.)\| _{L^{2}(\mathbb{
R}^{n})}^{2}\int_{\mathbb{R}^{n}}|\varphi
(y)|^{2}\big|\det (\frac{\partial F(x,z)}{\partial z})_{(z=G(x,y))}
 \big| ^{-1}dy\Big] dx   \\
&\leq  \| \varphi \| _{L^{2}(\mathbb{R}^{n})}^{2}\int_{
\mathbb{R}^{n}}\frac{\| K(x,.)\| _{L^{2}(\mathbb{R}
^{n})}^{2}}{g(x)}dx
\end{aligned}
\end{equation}
hence $C$ is bounded operator on $L^{2}(\mathbb{R}^{n})$ with
$\|
C\| \leq M=\| \frac{\| K(x,.)\| _{L^{2}(
\mathbb{R}^{n})}}{\sqrt{g(x)}}\|
_{L^{2}(\mathbb{R}^{n})}$.
\end{proof}

Now we give the main result of this paper. We proof that under
some
conditions the operator $C$ do not admit a bounded extension on $L^{2}(
\mathbb{R}^{n})$ .

\begin{theorem}\label{MainTheorem}
Let $\delta \in ] 0,1[ $ and the
operator $C$ defined by \eqref{1.4} on $L^{2}(\mathbb{R}^{n}) $ for
$x=(x_{1},\dots ,x_{n})\in ] 0,\delta [ ^{n}$ such that:
\begin{itemize}
\item[(H1)]  For $\varepsilon >0$ and for all $x\in \mathbb{R}^{n}$
\begin{equation*}
\{z\in \mathbb{R}^{n}: |F(x,z)| \leq \varepsilon \}
=\prod_{i=1}^{n}[a_{i}^{-}(x,\varepsilon
),\; a_{i}^{+}(x,\varepsilon )]
\end{equation*}
where $a_{i}^{\pm }(x,t)$ are real-measurable functions on
$\mathbb{R}^{n}\times ]0,+\infty[$ satisfying
\\
1- for any $p\in \mathbb{N}^{\ast }$ and $i\in \{ 1,\dots ,n\} $,
\begin{equation*}
\lim_{x_{i}\to 0^{+}}a_{i}^{\pm }(px,x_{i})=\pm
\infty
\end{equation*}
2- for any $\lambda \in ]0,1[$, $i\in \{ 1,\dots ,n\} $
and $p\in \mathbb{N}^{\ast }$, the functions $a_{i}^{+}(px,\lambda )$
and $a_{i}^{-}(px,\lambda )$ are respectively decreasing and increasing
with respect to $x$ in $] 0,\delta[^{n}$.

\item[(H2)] There exists a constant $R>0$ such that for any $r\geq R$
and for all $x\in ] 0,\delta [ ^{n}$
\begin{equation*}
\big|\int_{[ -r,r]^{n}}K(x,z)dz\big|\geq \delta
\end{equation*}

\end{itemize}
Then the operator $C$ cannot be extended to a bounded operator on
$L^{2}(\mathbb{R}^{n}) $.
\end{theorem}

\begin{proof}
Let us define the generalized sequence of functions
\begin{equation}
\varphi _{\varepsilon }(x)=\begin{cases}
1, &\text{if }x \in [ -\varepsilon ,\varepsilon ]^{n} \\
0, &\text{otherwise }
\end{cases}
  \label{2.2}
\end{equation}
then $\varphi _{\varepsilon }\in L^{2}(\mathbb{R}^{n})$ for all
$\varepsilon >0$ and we have
\begin{equation*}
C\varphi _{\varepsilon
}(x)=\int_{\prod_{i=1}^{n}[a_{i}^{-}(x,\varepsilon
),a_{i}^{+}(x,\varepsilon )]}K(x,z)dz
\end{equation*}
Consequently,
\begin{equation}
C\varphi _{\varepsilon
_{j}}(x)=\int_{\prod_{i=1}^{n}[a_{i}^{-}(x,\varepsilon
_{j}),a_{i}^{+}(x,\varepsilon _{j})]}K(x,z)dz  \label{2.3}
\end{equation}
where $\varepsilon _{j}\geq 0$ and
$\lim_{j\to +\infty }\varepsilon _{j}=0$.

By condition $1$ of the the assumption $(H1)$, for any $p\in \mathbb{N}
^{\ast }$ there exists a number $\varepsilon _{p}\geq 0$ such that
\begin{equation}
a_{i}^{+}(p\Lambda _{p},\varepsilon _{p})\geq R  \label{2.4}
\end{equation}
and
\begin{equation}
a_{i}^{-}(p\Lambda _{p},\varepsilon _{p})\leq -R  \label{2.5}
\end{equation}
for $\Lambda _{p}=(\varepsilon _{p},\varepsilon _{p},\dots \varepsilon _{p})$, $
p\varepsilon _{p}\leq \delta <1$ and $i\in \left\{ 1,\dots ,n\right\}
$.

It follows from \eqref{2.4}, \eqref{2.5} and condition $2$ of the
assumption (H1) that for
$x\in ]0,p\varepsilon _{p}]^{n}$ and
$i\in \{ 1,\dots ,n\} $ we have
\begin{gather}
a_{i}^{+}(x,\varepsilon _{p})\geq a_{i}^{+}(p\Lambda
_{p},\varepsilon _{p})\geq R,  \label{2.6}
\\
a_{i}^{-}(x,\varepsilon _{p})\leq a_{i}^{-}(p\Lambda
_{p},\varepsilon _{p})\leq -R  \label{2.7}
\end{gather}
Finally using (H2), \eqref{2.3},
\eqref{2.6} and \eqref{2.7} we deduce
\begin{equation} \label{2.8}
\| C\varphi _{\varepsilon _{p}}\| _{L^{2}(\mathbb{R}^{n}) }^{2}
\geq \int_{]0,p\varepsilon _{p}]^{n}}|C\varphi
_{\varepsilon p}(x)|^{2}dx
\geq \delta ^{2}p^{n}\varepsilon _{p}^{n}
\end{equation}
If we consider that $C$ has a bounded extension to $L^{2}(\mathbb{R}
^{n}) $, then by virtue of \eqref{2.1} we obtain for
$\varphi =\varphi _{\varepsilon _{p}}\in L^{2}(\mathbb{R}^{n}) $
\begin{equation*}
\delta ^{2}p^{n}\varepsilon _{p}^{n}\leq \| C\varphi
_{\varepsilon _{p}}\| _{L^{2}(\mathbb{R}^{n})
}^{2}\leq M^{2}\varepsilon _{p}^{n}
\end{equation*}
and for any $p\in \mathbb{N}^{\ast }$
\begin{equation*}
p^{n}\leq \frac{M^{2}}{\delta ^{2}}
\end{equation*}
This is a contradiction. Consequently $A$ cannot be a bounded
operator in $L^{2}(\mathbb{R}^{n}) $.
\end{proof}

\begin{remark} \label{remk2.3} \rm
 (1) If in particular $K(x,z)=K(z)$ is independent on $x$ and $
F(x,z)=b(x)\circ z+a(x)$, where $K(z)$ is a real-valued measurable
function, $b(x),a(x)\in \mathbb{R}^{n}$ are measurable functions
on $\mathbb{R}^{n}$,
we obtain the so-called generalized Hilbert transforms introduced
in \cite{Ha}

(2) The operator $C$ is an Fourier integral operator for an
appropriate choice of the functions $K(x,z)$ and $F(x,z)$.
\begin{align*}
C\varphi (x) &= \int_{\mathbb{R}^{n}}K(x,z)\varphi (F(x,z))dz \\
&= \int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}e^{iz.\xi }
\mathcal{F}K(x,\xi )\varphi (F(x,z))d\xi dz,
\end{align*}
where $\mathcal{F}K(x,\xi )$ is the Fourier transform of the partial
function $z\to K(x,z)$. Setting $y=F(x,z)$ and $z=G(x,y)$, we have
\begin{equation*}
C\varphi (x)=\int_{\mathbb{R}^{n}}\int_{\mathbb{R}
^{n}}e^{iG(x,y).\xi }\mathcal{F}K(x,\xi )\varphi (y)|
\det (\frac{
\partial G}{\partial y})|d\xi dy
\end{equation*}
which is a Fourier integral operator with the phase function
$\phi(x,y,\xi )=G(x,y).\xi $ and the symbol
$p(x,y,\xi)=\mathcal{F}K(x,\xi )|\det (\frac{\partial G}{\partial
y})|$ if $K$ and $G$ are infinitely regular with
respect to $x,y$ and $\xi $.
\end{remark}

\section{A class of unbounded Fourier integral operators on
$L^{2}(\mathbb{R}^{n})$}

It follows from theorem \ref{MainTheorem} that with an appropriate choice
of $K(x,z)$ and $ F(x,z)$ we can construct a class of Fourier
integral operators which cannot be extended as bounded operators
on $L^{2}(\mathbb{R}^{n}) $.

An example of unbounded fourier integral operator with a symbol in
$S_{1,1}^{0}(\mathbb{R}\times \mathbb{R})$ and
$\bigcap_{0<\rho <1}S_{\rho ,1}^{0}(\mathbb{R}^{n}\times \mathbb{R}^{n}) $
was given respectively in \cite{Ha} and \cite{AiMeSe}, where if
$\rho \in \mathbb{R}$,
\begin{equation} \label{3.1}
\begin{aligned}
S_{\rho ,1}^{0}(\mathbb{R}^{n}\times \mathbb{R}^{n})
=\big\{& p\in C^{\infty }(\mathbb{R}^{n}\times \mathbb{R}^{n})
: \forall (\alpha ,\beta )\in \mathbb{N}^{n}\times
\mathbb{N}^{n} \exists
C_{\alpha ,\beta }>0; \\
&|\partial _{x}^{\alpha }\partial _{\theta
}^{\beta }p(x,\theta )|\leq   C_{\alpha ,\beta
}\lambda ^{-\rho |\beta |+|\alpha
|}(\theta )
\end{aligned}
\end{equation}

\subsection{A class with symbols in $S_{1,1}^{0}(\mathbb{R}^{n}\times
\mathbb{R}^{n})$}


Hre, we generalize the example given by  Hasanov on $\mathbb{R}$
 to high dimensions. Namely, in the same spirit of
\cite{Ku}. we have easily if we get
$K(z)\in \mathcal{S(}\mathbb{R}^{n})$ and
$b\in C^{\infty }( \mathbb{R}^{n},\mathbb{R})$.

\begin{proposition} \label{proposition}
If  $K(z)\in \mathcal{S(}\mathbb{R} ^{n})$ and
$b\in C^{\infty }(\mathbb{R}^{n},\mathbb{R}) $, then
for all $\alpha ,\beta \in \mathbb{N}^{n}$ there exists
$C_{\alpha \beta }>0$ such that
\begin{equation}
|\partial _{x}^{\alpha }\partial _{\xi }^{\beta
}K(b(x) \xi) |\leq C_{\alpha
\beta }(1+|\xi |) ^{|
\alpha |-|\beta |}  \label{3.2}
\end{equation}
for all $(x,\xi) \in [-1,1] ^{n}\times \mathbb{R}^{n}$.
\end{proposition}

\begin{proof}
It suffices to use the fact that $K\in \mathcal{S(}\mathbb{R}^{n})$ and
$\beta $ is bounded on $[ -1,1] ^{n}$.
\end{proof}

Let also $a=(a_{1},a_{2},\dots ,a_{n})\in C^{\infty }(\mathbb{R}^{n},
\mathbb{R}^{n}) $ such that $a,b,K$ satisfy (H1)
and (H2), with
\begin{equation}
\begin{gathered}
b(x) >0 \\
a_{i}^{\pm }(x,t)=\frac{\pm t+a_{i}(x)}{b(x) },\quad
t>0,\; x\in \mathbb{R}^{n}
\end{gathered}   \label{3.3}
\end{equation}
Then, for $q(x,\xi) =K(b(x) \xi) $ defined on
$[-1,1] ^{n}\times \mathbb{R}^{n}$, we have
\begin{equation*}
|\partial _{x}^{\alpha }\partial _{\xi }^{\beta }q(x,\xi) |
\leq C_{\alpha \beta }( 1+|\xi ) ^{|\alpha
|-|\beta |}
\end{equation*}
on $[ -1,1] ^{n}\times \mathbb{R}^{n}$,
$\alpha ,\beta \in\mathbb{N}^{n}$, $C_{\alpha \beta }$ being constants.

Thus, $q\in S_{1,1}^{0}([ -1,1] ^{n}\times \mathbb{R}^{n}) $,
in particular $q(x,\xi) $ is a well bounded symbol.
Take a function $\eta \in C_{0}^{\infty }(\mathbb{R}^{n}) $ with
$\mathop{\rm supp} \eta \subset [-1,1] ^{n}$ and
$\eta (x) =1$ for $x\in [-\delta ,\delta] ^{n}$, $\delta <1$.
It is now obvious to see that the function
$p(x,\xi ) =\eta (x) q(x,\xi) \in
S_{1,1}^{0}(\mathbb{R}^{n}\times \mathbb{R}^{n})$.

Now the Fourier integral operator defined by
\begin{align*}
C\varphi (x) &=\int_{\mathbb{R}^{2n}}e^{-i(a(x).\xi
+y.\xi) }p(x,\xi) \varphi (\xi) dyd\xi  \\
&=\int_{\mathbb{R}^{2n}}e^{-i(a(x) .\xi +y.\xi
) }\eta (x)K(b(x) \xi) \varphi
(\xi) dyd\xi
\end{align*}
is of the type \eqref{1.4}. Indeed, for
$s=b(x) \xi $ and $x\in] 0,\delta] ^{n}$
\begin{equation*}
C\varphi (x)=\int_{\mathbb{R}^{2n}}e^{-i\frac{(a(
x) +t) .s}{\beta (x) }}K(s)
\frac{1}{b^{n}(x) }\varphi (y) dyds
\end{equation*}
Finally, if we pose $\frac{a(x) +y}{b(x)}=z$, we have
\begin{equation*}
C\varphi (x) =\int \mathcal{F}K(z) \varphi
(b(x) z-a(x)) dz
\end{equation*}

By theorem \ref{MainTheorem}, we conclude that the operator $C$
cannot be extended as a bounded operator on $L^{2}(\mathbb{R}^{n})$.

\subsection{A class with symbols in $\bigcap_{0<\protect\rho <1}S_{
\protect\rho ,1}^{0}(\mathbb{R}^{n}\times
\mathbb{R}^{n}) $}

We describe in this section the results of
Aiboudi-Messirdi-Senoussaoui \cite{AiMeSe}, they constructed
a class of unbounded Fourier integral operators with a separated
variables phase function and a symbol in the H\"{o}rmander class
$\bigcap_{0<\rho <1}S_{\rho ,1}^{0}(\mathbb{R}^{n}\times \mathbb{R}^{n})$.

Precisely, let $K\in S(\mathbb{R)}$ with $K(t)=1$ on $[-\delta,\delta ]$
and $b(t)$ is continuous function on $[0,1]$ such that
\begin{equation} \label{3.4}
\begin{gathered}
b(t)\in C^{\infty }(]0,1]),\quad b(0)=0,\quad b'(t)>0\text{ in  }]0,1] \\
|b^{(k)}(t)|\leq \frac{C_{k}}{t^{k}}\text{  in  }
]0,1],\; k\in \mathbb{N}^{\ast },C_{k}>0
\end{gathered}
\end{equation}
$\chi (x),\psi (\xi )\in C^{\infty }(\mathbb{R}^{n},\mathbb{R)}$
homogeneous of degree $1$.
Thus the function
\begin{equation}
q(x,\xi )=e^{-i\chi (x)\psi (\xi
)}\prod_{j=1}^{n}K(b(|
x|)|x|\xi _{j}),\quad \xi =(\xi _{1},\dots ,\xi _{n})  \label{3.5}
\end{equation}
 belongs to $C^{\infty }([-1,1]^{n}\times \mathbb{R}^{n})$ and
satisfies, as in the proposition \ref{proposition}, the
following estimates

\begin{proposition} \label{prop3}
For all $\alpha ,\beta $ in $\mathbb{N}^{n}$,
\begin{equation}
|\partial _{x}^{\alpha }\partial _{\xi }^{\beta }q(x,\xi
)|\leq C_{\alpha \beta }\frac{(1+|\xi|)^{|\alpha |-|\beta |}}{
b((1+|\xi |)^{-1})^{|\beta
|}} \label{3.6}
\end{equation}
on $[-1,1]^{n}\times \mathbb{R}^{n}$ where $C_{\alpha \beta }>0$.
\end{proposition}

Now if $\phi (x)$ is a $C_{0}^{\infty }(\mathbb{R)}$-function
such that
\begin{gather*}
\phi (s)=1\quad \text{on  }[-\delta ,\delta ],\; \delta <1 \\
\mathop{\rm supp}\phi \subset [ -1,1]
\end{gather*}
define the global $C^{\infty }$ symbol on
$\mathbb{R}^{n}\times \mathbb{R}^{n}$ by
\begin{equation} \label{3.7}
\begin{gathered}
p(x,\xi ) = e^{-i\chi (x)\psi (\xi )}\prod_{j=1}^{n}\phi
(x_{j})K(b(|x|)|x|\xi_{j})
 \\
x =(x_{1},\dots x_{n}),\quad
\xi =(\xi _{1},\dots ,\xi_{n}).
\end{gathered}
\end{equation}
Then $p(x,\xi )\in \cap_{0<\rho <1}S_{\rho
,1}^{0}(\mathbb{R}^{n}\times \mathbb{R}^{n}) $ and the
corresponding Fourier
integral operator is
\begin{equation} \label{3.8}
\begin{aligned}
C\varphi (x) &= \int_{\mathbb{R}^{n}}e^{i\chi (x)\psi (\xi
)}p(x,\xi
)\mathcal{F}\varphi (\xi )d\xi    \\
&= \prod_{j=1}^{n}\phi
(x_{j})\int_{\mathbb{R}^{n}}K(b(|x|
)|x|\xi _{j})\mathcal{F}\varphi (\xi )d\xi
\end{aligned}
\end{equation}
By using an adequate change of variable in the integral \eqref{3.8},
we have
\begin{equation}
C\varphi (x)=\int_{\mathbb{R}^{n}}\varphi (b(|
x|)|x| z)\prod_{j=1}^{n}\mathcal{F}K(z_{j})d\xi ,\quad
z=(z_{1},\dots ,z_{n})  \label{3.9}
\end{equation}
which is of the form $C$ in theorem \ref{MainTheorem}
where the functions $F(x,z)=b(|x|)|x|z$ and
$K(x,z)=\prod_{j=1}^{n}\mathcal{F}K(z_{j})$ satisfy
(H1) and (H2). Consequently, the operator $C$
cannot be continuously extended on $L^{2}(\mathbb{R}^{n})$.

\begin{thebibliography}{00}

\bibitem{AiMeSe} M. Aiboudi, B. Messirdi, A. Senoussaoui;
\emph{An exampleof
unbounded Fourier integral operator on $L^{2}$ with symbol in $
\bigcap_{0<\rho <1}S_{\rho ,1}^{0}(
\mathbb{R}^{n}\times
\mathbb{R}^{n}) $}. Int. Journal of Math. Analysis. Vol. 1, 2007,
No. 18, 851-860.

\bibitem{AsFu} K. Asada, D. Fujiwara;
\emph{On some oscillatory transformations in
$L^{2}(\mathbb{R}^{n})$}. Japan J. Math. Vol.4(2), 1978,
299-361.

\bibitem{Du} J. J. Duistermaat;
\emph{Fourier integral operators}. Courant
Institute. Lecture Notes, New-York, 1973.

\bibitem{Ha} M. Hasanov;
\emph{A class of Unbounded Fourier Integral Operators}.
Journal of Mathematical Analysis and Applications, 225, 1998,
641-651.

\bibitem{Ho1} L. H\"{o}rmander;
\emph{The analysis of Linear Partial differential
Operators}. Springer-Verlag, 1983.

\bibitem{Ho2} L. H\"{o}rmander;
\emph{On $L^{2}$ continuity of pseudodifferential
operators}. Comm. Pure Appl. Math. 24, 1971, 529-535.

\bibitem{Ho3} L. H\"{o}rmander;
\emph{Fourier integral operators I}. Acta Math.
Vol. 127, 1971, 33-57.

\bibitem{Ku} H. Kumano-Go;
\emph{A problem of Nirenberg on pseudodifferential
operators}. Comm. Pure Appl. Math. 23, 1970, 151-121.

\bibitem{Me} B. Messirdi;
\emph{Asymptotique de Born-Oppenheimer pour la pr\'{e}
dissociation mol\'{e}culaire (cas de potentiels r\'{e}guliers)}.
Annales de l'I.H.P, Vol. 61, No. 3, 1994, 255-292.

\bibitem{MeRaSe} B. Messirdi-A. Rahmani-A. Senoussaoui;
\emph{Probl\`{e}me de Cauchy $C^{\infty }$ pour des syst\`{e}mes
d'\'{e}quations aux d\'{e}riv\'{e}
es partielles \`{a} caract\'{e}ristiques multiples de
multiplicit\'{e}
paire}. Bulletin Math. Soc. Sci. Math. Roumanie. Tome19/97,
No. 4, 2006, 345-358.

\bibitem{MeSe1} B. Messirdi, A. Senoussaoui:
\emph{Parametrixe du probl\`{e}me de
Cauchy $C^{\infty }$muni d'un syst\`{e}me d'ordre de
Leray-Volev\^{\i}c}. Journal for Anal. and its Appl. Vol. 24,
2005, 581-592\

\bibitem{MeSe2} B. Messirdi, A. Senoussaoui:
\emph{On the $L^{2}$-boundedness and $L^{2}$-compactness of a class
of Fourier integral operators}.
Electon. J. Diff. Equa. Vol. 2006 (2006), No. 26, 1-12.

\bibitem{Ro} D. Robert:
\emph{Autour de l'approximation semi-classique}. Birk\"{a}
user, 1987.

\bibitem{Sj} J. Sj\"{o}strand;
\emph{Singularit\'{e}s analytiques microlocales}.
Ast\'{e}risque 95, 1982.

\bibitem{St} E. M. Stein;
\emph{Harmonic Analysis: Real-Variable Methods,
Orthogonality, and Oscillatory Integrals}. Princeton Mathematical
Series 43, Princeton University Press, 1993.

\end{thebibliography}

\end{document}