Continuity properties for Born-Jordan operators with symbols in H{\"o
Maurice de Gosson
Joachim Toft
We show that the Weyl symbol of a Born-Jordan operator is in the same class as the Born-Jordan symbol, when H{\"ormander symbols and certain types of modulation spaces are used as symbol classes. We use these properties to carry over continuity and Schatten-von Neumann properties to the Born-Jordan calculus. %We study the effects of warm water on the local penguin population. The major %finding is that it is extremely difficult to induce penguins to drink warm %water, unless the penguins are in up and down position, and the glasses %are decorated by smiling jellyfishes brushing their canine teeth. The success %factor is approximately %$-e^{-i\pi-1$.
1702.05714.tex
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\author{Maurice de Gosson}

\address{Faculty of Mathematics, NuHAG,
University of Vienna, Vienna, Austria}

\email{maurice.degosson@gmail.com}

\author{Joachim Toft}

\address{Department of Mathematics,
Linn{\ae}us University, V{\"a}xj{\"o}, Sweden}

\email{joachim.toft@lnu.se}


\title{Continuity properties for Born-Jordan operators with symbols in
H{\"o}rmander classes and modulation spaces}




\begin{document}


\begin{abstract}
We show that the Weyl symbol of a Born-Jordan operator
is in the same class as the Born-Jordan symbol, when
H{\"o}rmander symbols and
certain types of modulation spaces are used as symbol
classes. We use these properties to carry over
continuity and Schatten-von Neumann properties to the
Born-Jordan calculus.
%We study the effects of warm water on the local penguin population. The major
%finding is that it is extremely difficult to induce penguins to drink warm
%water, unless the penguins are in up and down position, and the glasses
%are decorated by smiling jellyfishes brushing their canine teeth. The success
%factor is approximately
%$-e^{-i\pi}-1$.
\end{abstract}

\maketitle




%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}\label{sec0}
%%%%%%%%%%%%%%%%%%%%%%%%%

%The relation between the Weyl pseudo-differntial calculus and Schatten classes
%is well-known; see \emph{e.g}. Toft \cite{Toft1,Toft2}.

\par

A fundamental question in quantum mechanics concerns quantization. That is,
finding rules which takes observables $a$ in classical mechanics into
corresponding observables $T_a$ in quantum mechanics. Usually $a$ is a
function of the location $x\in \rr d$ and the momentum $\xi \in \rr d$, and $T_a$
is a linear operator which acts between suitable Hilbert spaces of functions on
$\rr d$.

\par

For any $t \in \mathbf R$, the rule where $a\in \mascS '(\rr {2d})$ is mapped into
the pseudo-differential operator ($t$-Shubin operator) $\op _t (a)$, given by
%%
\begin{equation*}
(\op _t (a)f)(x) = (2\pi )^{-d}\iint _{\rr {2d}} a((1-t )x+t y,\xi )f(y)
e^{i\scal {x-y}\xi}\, dyd\xi
\end{equation*}
%%
when $f\in \mascS (\rr d)$ can be considered as a candidate of
quantization. Here the right-hand side should be
interpreted in distribution sense when $a\notin L^1$ (see Section \ref{sec1} for
details and notations).

\par

We remark that for $t =0$,
the so-called normal or Kohn-Nirenberg representation
$$
a(x,D)=\op (a)=\op _{\KN}(a)=\op _0(a)
$$
is obtained. The latter was considered in the foundational papers by Kohn
and Nirenberg \cite{KoNi} and H{\"o}rmander \cite{Horm0}. If instead
$t =\frac 12$, then $\op _t (a)$ is equal to the Weyl operator
(or Weyl quantization) $\op ^w(a)$, introduced by H. Weyl in \cite{Weyl} as
a candidate of quantization.

\par

The action of $\op _t (a)$ can also in convenient ways be described in terms of
$t$-Wigner distributions. In fact, if $a$, $f$ and $g$ are suitable, then we have
%%
\begin{equation}\label{Eq:WignerPseudoRel}
(\op _t (a)f,g)_{L^2(\rr d)} = (2\pi )^{-d/2}(a,W_t (g,f))_{L^2(\rr {2d})},
\end{equation}
%%
where $W(f,g)$ is the $t$-Wigner distribution, given by
%%
\begin{equation}\label{wignertdef}
(W_{t}(f,g))(x,\xi ) = (2\pi )^{-d/2}\int _{\rr d} f(x+t y)
\overline{g(x-(1-t )y)}e^{-i\scal y\xi} \, dy.
\end{equation}
%%
Here $(f,\phi ) = (f,\phi )_{L^2(\rr d)}\equiv \scal f{\overline \phi}$, where
$\scal \cdo \cdo$ is the action between suitable test functions and their
distributions on $\rr d$.
In the Weyl case, $t =\frac 12$, we set
$W(f,g)=W_t (f,g)$ and call this the (standard) Wigner distribution of $f$ and $g$.

\par

In practice, some further rules need to be imposed in quantization. For example, it is
essential that $T_a$ above is self-adjoint when $a$ is real, and that the quantization
possess properties of symplectic invariance. By adding such conditions, only the Weyl
quantization $\op ^w(a)$ among the candidates above remains as a good candidate
of quantization. In fact, the symplectic invariance is emphasized already by
H{\"o}rmander in \cite{Hormweyl}, and by the definition it follows that the (formal)
adjoint $\op ^w(a)^*$ of $\op ^w(a)$ for admissible $a$ is given by
$\op ^w(\overline a)$.

\medspace

The Born-Jordan quantization $\op _{\BJ}(a)$ is obtained by taking the average
$$
\op _{\BJ}(a) \equiv \int _0^1\op _t (a)\, dt
$$
of $\op _t (a)$ over $t \in [0,1]$, provided the right-hand side is well-defined.
It can be proved that 
%%
\begin{equation}\label{Eq:WeylBJ1}
\op _{BJ}(a) = \op ^w(\Phi *a),
\qquad \text{where}\quad
\Phi (x,\xi ) = (2\pi )^{-d}\sinc \scal x\xi .
\end{equation}
%%
Here $\sinc$ is the sinc function, given by
$$
\sinc t=
\begin{cases}
1 &\text{when}\ t=0,
\\[0ex]
\frac {\sin t}t &\text{when}\ t\neq 0.
\end{cases}
$$
In view of the relation \eqref{Eq:WeylBJ1} and that $\op ^w(a)^*
= \op ^w(\overline a)$ we have
$$
\op_{\BJ}(a)^* = \op _{\BJ}(\overline a).
$$
Hence the Born-Jordan operators share with Weyl operators the property
of being (formally) self-adjoint
%if and only if
when
their symbol is real. This makes
Born-Jordan operators good alternative candidates for quantization.

\par


\par

If $a(x,\xi ) = x_j^m\xi _j^l$, for $m,l\ge 0$ being integers, then it
follows by straightforward computations that
%$$
%\op _{\BJ}(a) = \frac 1{l+1}\sum _{k=0}^l \op (\xi _j^{l-k})\circ \op (x_j^m)\circ \op (\xi _j^k).
%$$
$$
\op _{\BJ}(a) = \frac 1{l+1}\sum _{k=0}^l D _j^{l-k}\circ x_j^m\circ D _j^k.
$$
This is precisely the quantization rule, proposed in early days by Born and Jordan \cite{BJ}
(see \cite{Transam} for details and proofs), and is the reason for denomination
$\op _{\BJ}(a)$ as the Born-Jordan operator or Born-Jordan quantization of $a$.

\par

By \eqref{Eq:WignerPseudoRel} and \eqref{Eq:WeylBJ1}, it follows that
$$
(\op _{\BJ} (a)f,g)_{L^2(\rr d)} = (2\pi )^{-d/2}(a,W_{\BJ} (g,f))_{L^2(\rr {2d})}
$$
for suitable $a$, $f$ and $g$, where $W_{\BJ}$ is the transform $Q$ in
\cite{bogetal1} by Boggiatto, de Donno and Oliaro, given
by
$$
W_{\BJ} (f,g) = \Phi *W(f,g) = \int _0^1 W_t (f,g)\, dt .
$$

\par

During the last 10 years, Born-Jordan quantization has also been considered in
time-frequency analysis. In this field, time-frequency resolutions, i.{\,}e. simultaneously
localizations of the time and frequency
for signals, are essential. Such resolutions are often performed by considering
the so-called short-time Fourier transform of the involved signals. An equivalent way
is to replace these short-time Fourier transforms with $t$-Wigner distributions,
since they can be carried over to each others by simple manipulations. A problem with
these kind of resolutions concerns interpolating frequencies or so-called ghost frequencies.
These frequencies might be absent in the signal, but might appear in the graphs of
their resolutions. They originate from interferens of existing frequencies.

\par

Already in \cite{bogetal1,bogetal2} it is proposed that the transform $W_{\BJ}$ can be
used in place of the short-time Fourier transform and $t$-Wigner distributions.
Some properties of the $W_{BJ}$ were deduced in \cite{Tur}.
%\\[1ex]
%\textbf{Include here papers by Cordero-deGosson-Nicola}
%\\[1ex]
Especially we remark that in \cite{Tur}, Turunen shows that the time-frequency resolutions
usually becomes significantly more clear when using $W_{\BJ}$ transform instead of
$t$-Wigner distributions. For example, it is shown there that the ghost frequencies
miraculously almost disappear when using suitable resolutions based on the $W_{\BJ}$
transform.

\par

In the present paper we study some functional-analytic preparations of the
Born--Jordan
operators (\cite{BJ}). We focus in particular on their regularity properties
in the Schatten-von Neumann classes, which are generalizations of Hilbert--Schmidt
operators. In fact, there are several relations between pseudo-differential
calculi and Schatten-von Neumann classes (see e.{\,}g.
\cite{BuTo,Toft,Toft4}). Hence one can expect that some of these properties extend
Born-Jordan operators.

\par

We show that several continuity and compactness
properties for pseudo-differential calculus carry over to Born-Jordan operators.
Especially we prove that if the symbol of a Born-Jordan operator
belongs to suitable modulation spaces or in a suitable H{\"o}rmander
class $S(m,g)$, then
the corresponding Weyl symbol belongs to the same class. More precisely,
suppose $g$ is a Riemannian metric on $\rr {2d}$, which is strongly feasible in
the sense and splitof \cite{BuTo,Toft4}, and let $m$ be a $g$-continuous
and $(\sigma ,g)$-temperate weight function on $\rr {2d}$. If $a\in S(m,g)$,
then we prove that there is a unique $b\in S(m,g)$ such that
%%
\begin{equation}\label{Eq:BJWeylIdent}
\op _{\BJ}(a) = \op ^w(b).
\end{equation}
%%
We note that we may choose $m$ and $g$ in such ways that $S(m,g)$ agree with any
Shubin class or any H{\"o}rmander class $S^r_{\rho ,\delta}$ as well as any $\SG$-class.
In particular we are able to cover the continuity properties in \cite{CoGoNi} for Born-Jordan
operators with Shubin symbols.

\par

If instead $a\in M^{p,q}_{(\omega )}(\rr {2d})$ for some $p\in [1,\infty ]$, $q\in (0,\infty ]$
and $\omega (x,\xi ,\eta ,y)=\omega _0(\eta ,y)$ with $\omega _0$ being a moderate weight
on $\rr {2d}$, then \eqref{Eq:BJWeylIdent} still holds for some $b\in M^{p,q}_{(\omega )}(\rr {2d})$.
Consequently, if $a\in M^{p,q}(\rr {2d})$ with $q\le \min (p,p')$, then $\op _{\BJ}(a)$ is a
Schatten-von Neumann operator on $L^2(\rr d)$, since similar facts hold for Weyl operators
with symbols in modulation spaces (cf. e.{\,}g. \cite{Toft6,Toft}). In the same way, the continuity
and Schatten-von Neumann properties in \cite{BuTo,Toft4} carry over from Weyl
to Born-Jordan case.

\par

%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Acknowledgement} Maurice de Gosson has been
supported by the Austrian research agency FWF
(grant number P27773).
%%%%%%%%%%%%%%%%%%%%%%%%%%

\par

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Preliminaries}\label{sec1}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\par

\subsection{Gelfand-Shilov spaces and their duals}
%Next we recall the definition of $\Sigma _1(\rr d)$
%and its distribution space $\Sigma _1'(\rr d)$, which belong to the family of
%Gelfand-Shilov spaces and their duals (cf. e.{\,}g. \cite{GS}). 
Let $h\in \mathbf R_+$ and $s>0$ be fixed. Then $\mathcal S_{s,h}(\rr d)$
is the set of all $f\in C^\infty (\rr d)$ such that
%%
\begin{equation*}%\label{gfseminorm}
\nm f{\mathcal S_{s,h}}\equiv \sup \frac {|x^\beta \partial ^\alpha
f(x)|}{h^{|\alpha | + |\beta |}(\alpha !\, \beta !)^s}
\end{equation*}
%%
is finite. Here the supremum is taken over all $\alpha ,\beta \in
\mathbf N^d$ and $x\in \rr d$.

\par

Obviously $\mathcal S_{s,h}\subseteq
\mathscr S$ is a Banach space which increases with $h$ and $s$.
Furthermore, if $s>\frac 12$, then $\mathcal S_{s,h}$ and $\cup _{h>0}\mathcal S_{1/2,h}$
contain all finite linear combinations of Hermite functions $h_\alpha (x)=H_\alpha
(x)e^{-|x|^2/2}$, where $H_\alpha$ is the Hermite polynomial
of order $\alpha \in \nn d$.
Since such linear combinations are dense in $\mathscr S$, it follows
that the ($L^2$-)dual $(\mathcal S_{s,h})'(\rr d)$ of $\mathcal S_{s,h}(\rr d)$ is
a Banach space which contains $\mathscr S'(\rr d)$.

\par

The \emph{Gelfand-Shilov spaces} $\maclS _s(\rr d)$ and $\Sigma _s(\rr d)$ are
the inductive limit and projective limit, respectively
of $\mathcal S_{s,h}(\rr d)$ with respect to $h$. This implies that
%%
\begin{equation}\label{GSspacecond1}
\maclS _{s}(\rr d) =\bigcup _{h>0}\mathcal
S_{s,h}(\rr d)
\quad \text{and}\quad
\Sigma _{s}(\rr d) =\bigcap _{h>0}\mathcal
S_{s,h}(\rr d),
\end{equation}
%%
and that the latter space is a Fr{\'e}chet space with semi norms
$\nm \cdo{\mathcal S_{s,h}}$, $h>0$.

\medspace

The \emph{Gelfand-Shilov distribution spaces} $\maclS _s'(\rr d)$ and
$\Sigma _s'(\rr d)$ are the
projective limit and the inductive limit respectively of $\mathcal S_{s,h}'(\rr d)$
with respect to $h>0$.  Hence
%%
\begin{equation}\tag*{(\ref{GSspacecond1})$'$}
\maclS _s'(\rr d) =\bigcap _{h>0} \mathcal S_{s,h}'(\rr d)
\quad \text{and}\quad
\Sigma _s'(\rr d) =\bigcup _{h>0} \mathcal S_{s,h}'(\rr d).
\end{equation}
%%
We remark that  $\maclS _s'(\rr d)$ and $\Sigma _s'(\rr d)$ are the duals of
$\maclS _s(\rr d)$ and $\Sigma _s(\rr d)$, also in topological sense (cf. \cite{Pil1,Pil2}).

\par

From now on we let $\mathscr F$ be the Fourier transform,
given by
$$
(\mathscr Ff)(\xi ) = \widehat f(\xi ) \equiv (2\pi )^{-d/2}\int _{\rr
{d}} f(x)e^{-i\scal  x\xi }\, dx
$$
when $f\in L^1(\rr d)$. Here $\scal \cdo \cdo$ denotes the
usual scalar product on $\rr d$. The map $\mathscr F$ extends 
uniquely to homeomorphisms on $\mathscr S'(\rr d)$, $\maclS _s'(\rr d)$ and
$\Sigma _s'(\rr d)$, and restricts to homeomorphisms on
$\mathscr S(\rr d)$, $\maclS _s(\rr d)$ and $\Sigma _s(\rr d)$, and to a unitary
operator on $L^2(\rr d)$.

\medspace

Next we recall some mapping properties of Gelfand-Shilov
spaces under short-time Fourier transforms and $t$-Wigner distributions.
Let $\phi \in \mathscr S(\rr d)$ be fixed. For every $t \in \mathbf R$,
$f\in \mathscr S'(\rr d)$, the \emph{short-time Fourier transform} $V_\phi
f$ is the distribution on $\rr {2d}$ defined by the formula
%%
\begin{equation}\label{defstft}
(V_\phi f)(x,\xi ) =\mathcal F(f\, \overline{\phi (\cdo -x)})(\xi ) =
(f,\phi (\cdo -x)e^{i\scal \cdo \xi}).
\end{equation}
%%
The \emph{$t$-Wigner distribution} is given by
%%
\begin{equation}\label{wignertdef2}
W_{t} (f,\phi )(x,\xi ) \equiv \mathcal F (f(x+t\cdo
)\overline{\phi (x-(1-t )\cdo )} )(\xi ),
\end{equation}
%%
and we observe that this definition coincide with \eqref{wignertdef} when
$f,g\in \mascS (\rr d)$.

\par

The definition of short-time Fourier transforms and Wigner distributions extend in different
ways, and possess various kinds of continuity properties. In the context of test function
spaces and distribution spaces we have the following.

\par

\begin{prop}\label{Prop:WignerDistExt}
Let $s\ge \frac 12$ and let $T(t ,f,\phi )\equiv V_\phi f$ or $T(t ,f,\phi )
\equiv W_{t} (f,\phi )$ when $f,\phi \in \mascS (\rr d)$. Then the following
is true:
\begin{enumerate}
\item the map $(t ,f,\phi)\mapsto T(t ,f,\phi )$ is continuous from
$\mathbf R\times \mascS (\rr d)\times \mascS (\rr d)$ to
$\mascS (\rr {2d})$ and restricts to a continuous map from
$\mathbf R\times \maclS _s(\rr d)\times \maclS _s(\rr d)$ to
$\maclS _s(\rr {2d})$;

%\vrum

\item the map $(t ,f,\phi)\mapsto T(t ,f,\phi )$ from
$\mathbf R\times \maclS _s(\rr d)\times \maclS _s(\rr d)$ to
$\maclS _s(\rr {2d})$ extends uniquely to a continuous map
from $\mathbf R\times \maclS _s'(\rr d)\times \maclS _s'(\rr d)$ to
$\maclS _s'(\rr {2d})$ and from
$\mathbf R\times \mascS '(\rr d)\times \mascS '(\rr d)$ to
$\mascS '(\rr {2d})$.
%
%\item the map $(t ,f,\phi)\mapsto T(t ,f,\phi )$ from
%$\mathbf R\times \maclS _s(\rr d)\times \maclS _s(\rr d)$ to
%$\maclS _s(\rr {2d})$ extends uniquely to a continuous map
%from $\mathbf R\times \maclS _s'(\rr d)\times \maclS _s(\rr d)$ to
%$\maclS _s'(\rr {2d})\cap C^\infty (\rr {2d})$ and from
%$\mathbf R\times \mascS '(\rr d)\times \mascS '(\rr d)$ to
%$\mascS '(\rr {2d})\cap C^\infty (\rr {2d})$.
\end{enumerate}
The same holds true for $s>\frac 12$ when each $\maclS _s$ and $\maclS _s'$
are replaced by $\Sigma _s$ and $\Sigma _s'$, respectively.
\end{prop}

\par

Proposition \ref{Prop:WignerDistExt} is essentially available in the literature (see
e.{\,}g. \cite{CPRT10,Toft8}). Since in contrast we have included the parameter
$t$ as a variable, we here recall the arguments for the $t$-Wigner distribution.

\par

\begin{proof}
Let $\mascF _2F$ denote the partial Fourier of $F(x,y)\in \mascS _s'(\rr {2d})$ with
respect to the $y$-variable. By the definition we have
$$
T(t ,f,\phi) = \mascF _2\circ U_t \circ S,
$$
where
$$
(U_t F)(x,y) = F(x+t y,x-(1-t )y)
\quad \text{and}\quad
S(f,\phi )= f\otimes \overline{\phi}.
$$
Since it is evident that $S$ is continuous from $\mascS (\rr d)\times \mascS (\rr d)$
to $\mascS (\rr {2d})$,
that
$(t ,F)\mapsto U_t F$ is continuous from
$\mathbf R \times \mascS (\rr {2d})$ to $\mascS (\rr {2d})$,
and that
$\mascF _2$ is continuous on $\mascS (\rr {2d})$, it follows that $T$ is continuous
from $\mathbf R\times \mascS (\rr d)\times \mascS (\rr d)$ to
$\mascS (\rr {2d})$.

\par

The same holds true after each $\mascS$ is replaced by $\maclS _s$, $\Sigma _s$,
$\maclS _s'$, $\Sigma _s'$, or by $\mascS '$, and the other continuity assertions follow.
\end{proof}

\par

\begin{rem}\label{Rem:WignerDistExt}
By the previous proof it also follows that the mappings in
Proposition \ref{Prop:WignerDistExt} are in fact locally uniformly bounded. 
\end{rem}

\par

%We recall that if $T(f,\phi )\equiv V_\phi f$ or $T(f,\phi )\equiv W_{t} (f,\phi )$,
%%when $f,\phi \in \Sigma _1(\rr d)$,
%then $T$ restricts to a continuous map from $\mathscr S(\rr d)\times
%\mathscr S(\rr d)$
%to $\mathscr S(\rr {2d})$, and from $\Sigma _1(\rr d)\times \Sigma _1(\rr d)$
%to $\Sigma _1(\rr {2d})$.
We also recall that if $T$ is the same as in Proposition \ref{Prop:WignerDistExt},
then the mappings
%%
\begin{alignat*}{2}
T\, &:\, & \mathbf R\times \mascS '(\rr d)\times \mascS (\rr d) &\to
\mascS '(\rr {2d})\bigcap C^\infty (\rr {2d}),
\\[1ex]
T\, &:\, & \mathbf R\times \maclS _s'(\rr d)\times \maclS _s(\rr d) &\to
\maclS _s '(\rr {2d})\bigcap C^\infty (\rr {2d}),
\intertext{and}
T\, &:\, & \mathbf R\times \Sigma _s'(\rr d)\times \Sigma _s(\rr d) &\to
\Sigma _1'(\rr {2d})\bigcap C^\infty (\rr {2d})
\end{alignat*}
%%
are continuous (cf. \cite{CPRT10,Toft8, Toft10}). 
%We also note that $V_\phi f$ and $W_{t} (f,\phi )$ take the form
%%%
%\begin{align}
%V_\phi f(x,\xi ) &= (2\pi )^{-d/2}\int _{\rr d}f(y)\overline {\phi
%(y-x)}e^{-i\scal y\xi}\, dy\tag*{(\ref{defstft})$'$}
%\intertext{and}
%W_{t} (f,\phi ) &= (2\pi )^{-d/2}\int _{\rr d} f(x+ty)
%\overline{\phi (x-(1-t)y)}e^{-\scal y\xi} \, dy \tag*{(\ref{wignertdef})$'$}
%\end{align}
%%%
%respectively, when $f,\phi \in L^2(\rr d)$. 

\par

There are several ways to characterize Gelfand-Shilov
spaces and their distribution spaces. For example, they can
easily be characterized by Hermite functions and other
related functions (cf. e.{\,}g. \cite{GrPiRo,JaEi}). They can
also be characterized by suitable estimates of their Fourier
and Short-time Fourier transforms (cf.
\cite{ChuChuKim,GrZi,Toft8,Toft10}).

\par

\subsection{Pseudo-differential operators}

\par

Let $t \in \mathbf{R}$ be fixed. For any $a\in\mathscr {S}
(\mathbf{R}^{2d})$ (the symbol), the pseudo-differential operator
$\operatorname*{Op}_{t}(a)$ (of Shubin type) is the linear
and continuous operator on $\mathscr S(\rr d)$, defined by
%%
\begin{equation}
\op _{t}(a)f(x)
=
(2\pi)^{-d}\iint a((1-t )x+t y,\xi)f(y)e^{i\scal {x-y} \xi}
\, dyd\xi 
\label{opta1}
\end{equation}
%%
when $f\in \mathscr S(\rr d)$. By straightforward computations it follows that
%%
\begin{equation}\label{Eq:OpTauEquiv}
(\op _t (a)f,g)_{L^2(\rr d)} = (2\pi )^{-d/2}(a,W_t (g,f))_{L^2(\rr {2d})}
\end{equation}
%%
when $g\in \mathscr S(\rr d)$.

\par

If more generally $a\in \mathscr S'(\rr {2d})$ ($a\in \Sigma _1'(\rr {2d})$),
then $\operatorname*{Op}_{t}(a)$ is the linear and continuous operator from
$\mathscr S(\rr d)$ to $\mathscr S'(\rr d)$ (from $\Sigma _1(\rr d)$ to
$\Sigma _1'(\rr d)$) such that $(\op _t (a)f,g)$ is equal to the right-hand side
of \eqref{Eq:OpTauEquiv} when $f,g\in \mathscr S(\rr d)$ ($f,g\in \Sigma (\rr d)$).
This makes sense, in view of the continuity properties for the Wigner distribution,
described above.

\par

%\par
%
%For $t\in\mathbf{R}$ we define (formally) the Shubin operator
%$\operatorname*{Op}_{t}(a)$ associated with the symbol $a\in\mathscr {S}%
%(\mathbf{R}^{2d})$ by%
%%%
%\begin{equation}
%\operatorname*{Op}\nolimits_{t}(a)f(x)=(2\pi)^{-d}\iint e^{i\scal {x-y} \xi
%}a((1-t )x+t y,\xi)f(y)\, dyd\xi \label{opta1}%
%\end{equation}
%%%
%(we will give below a concise and rigorous definition using $t$-Wigner
%transforms). When $t=\frac{1}{2}$ one recovers the usual Weyl operator
%$\operatorname*{Op}\nolimits ^{w}(a)$; when $t=1$ we get the
%Kohn--Nirenberg operator with symbol $a$. In what follows we assume that
%$\operatorname*{Op}\nolimits_{t}(a):L^{2}(\mathbf{R}^{d})\to
%L^{2}(\mathbf{R}^{d})$.

%\section{Born--Jordan Calculus}
%
%\subsection{Definition and motivation}
%
%We begin by recalling the relation between the $t$-Wigner transform and
%Shubin operators (see de Gosson \cite{Transam} and the references therein).
%Let $f,g\in\mathcal{S}(\mathbf{R}^{d})$; one can then define the operator
%$\operatorname*{Op}\nolimits_{t}(a)$ by the formula
%%%
%\begin{equation}
%\langle \op _{t}(a)f|\overline{g}\rangle
%= (2\pi )^{-d/2} \langle\langle a,W_{t}(f,g)\rangle\rangle\label{opta2}
%\end{equation}
%%%
%where $W_{t}(f,g)$ is the $t$-cross-Wigner transform of $(f,g)$:%
%%%
%\begin{equation}
%W_{t}(f,g)(z)= (2\pi)^{-d/2}\int e^{-i \scal y\xi}f(x+(1-t )
%y)\overline{g(x-t y)}dy;\label{wtau}%
%\end{equation}
%%%
%here $\langle\cdot|\cdot\rangle$ and\ $\langle\langle\cdot,\cdot\rangle
%\rangle$ are the distributional brackets on $\mathbf{R}^{d}$ and
%$\mathbf{R}^{2d}$, respectively. The most obvious advantage of this definition
%compared to (\ref{opta1}) is that is that it allows to define rigorously
%$\operatorname*{Op}\nolimits_{t}(a)f$ for $a\in\mathcal{S}^{\prime
%}(\mathbf{R}^{d})$. In fact, one shows that%
%\[
%W_{t}:\mathscr{S}(\mathbf{R}^{d})\times\mathscr{S}(\mathbf{R}%
%^{d})\to\mathscr{S}(\mathbf{R}^{2d})
%\]
%hence the right-hand side of (\ref{opta2}) is well-defined, and so is
%therefore $\operatorname*{Op}\nolimits_{t}(a)f\in\mathscr{S}^{\prime
%}(\mathbf{R}^{d})$. (We note that (\ref{opta2}) is related to Proposition 4.11
%in Toft \cite{Toft}.) One easily verifies using this redefinition that the
%(formal) adjoint of $\operatorname*{Op}_{t}(a)$ is given by
%%%
%\begin{equation}
%\operatorname*{Op}\nolimits_{t}(a)^{\ast}=\operatorname*{Op}%
%\nolimits_{1-t}(\overline{a}).\label{opadj}%
%\end{equation}
%%%
%
%
%In \cite{bogetal1,bogetal2} Boggiatto and his collaborators define a transform
%$Q:\mathscr{S}(\mathbf{R}^{d})\times\mathscr{S}(\mathbf{R}^{d})\to
%\mathscr{S}(\mathbf{R}^{d})$ by integrating over $[0,1]$ the $t$-cross
%Wigner transforms(\ref{wtau}); we will use the notation $Q=W_{\text{BJ}}$;
%thus, for $f,g\in\mathscr{S}(\mathbf{R}^{d})$:
%%%
%\begin{equation}
%W_{\text{\textrm{BJ}}}(f,g)=\int_{0}^{1}W_{t}(f,g)dt.\label{wbj}%
%\end{equation}
%%%
%The function $W_{\text{\textrm{BJ}}}(f,g)$ has properties similar to those of
%the usual cross-Wigner transform. For $a\in\mathscr{S}^{\prime}(\mathbf{R}%
%^{2d})$ we define the Born--Jordan operator $\operatorname*{Op}%
%\nolimits_{\mathrm{BJ}}(a)$ by the formula%
%%%
%\begin{equation}
%\langle\operatorname*{Op}\nolimits_{\mathrm{BJ}}(a)f|\overline{g}%
%\rangle
%=
%(2\pi )^{-d/2} \langle\langle a,W_{\text{BJ}}(f,g)\rangle\rangle\label{abj1}%
%\end{equation}
%%%
%$f,g\in\mathscr{S}(\mathbf{R}^{d})$. Taking definition (\ref{wbj}) into
%account we can write%
%%%
%\begin{equation}
%\operatorname*{Op}\nolimits_{\mathrm{BJ}}(a)=\int_{0}^{1}\operatorname*{Op}%
%\nolimits_{t}(a)dt.\label{abj3}%
%\end{equation}
%%%
%
%
%The reason for the denomination \textquotedblleft Born--Jordan
%operators\textquotedblright\ is the following: assume that $a(z)=x_{j}^{m}%
%\xi_{j}^{\ell}$; then%
%%%
%\begin{equation}
%\operatorname*{Op}\nolimits_{\mathrm{BJ}}(a)=\frac{1}{\ell+1}\sum_{k=0}^{\ell
%}\widehat{\Xi_{j}}^{\ell-k}\widehat{X_{j}}^{m}\widehat{\Xi_{j}}^{k}%
%\label{abj4}%
%\end{equation}
%%%
%where $\widehat{X_{j}}$ is the operator of multiplication by $x_{j}$ and
%$\widehat{\Xi_{j}}=-i\partial/\partial x_{j}$. It turns out that this is
%precisely the \textquotedblleft quantization rule\textquotedblright\ proposed
%in the early days of quantum mechanics by the physicists Born and Jordan
%\cite{bj} (see de Gosson \cite{Transam} and de Gosson and Luef \cite{golu} for
%details and proofs). Our definition is thus a generalization of Born and
%Jordan's prescription to arbitrary symbols. 
%
%\subsection{Properties of Born--Jordan operators}
%
%Let us introduce the following notation and terminology: for $a\in
%\mathscr{S}^{\prime}(\mathbf{R}^{2d})$ we set
%%%
%\begin{equation}
%a_{\sigma}(z)=\mathcal F_{\sigma}a(z)=(2\pi )^{-d} \int
%e^{-i\sigma(z,z^{\prime})}a(z^{\prime})dz^{\prime}%
%\end{equation}
%%%
%($\mathcal F_{\sigma}$ is the symplectic Fourier transform of $a$). For $z_{0}%
%=(x_{0},\xi_{0})$ the Heisenberg--Weyl operator $\widehat{T}(z_{0})$ is
%defined by
%%%
%\begin{equation}
%\widehat{T}(z_{0})=e^{-\frac{i}{\hslash}\sigma(\hat{z},z_{0})};\label{thw}%
%\end{equation}
%%%
%its explicit action on $f\in\mathscr{S}^{\prime}(\mathbf{R}^{d})$ is given by
%%%
%\begin{equation}
%\widehat{T}(z_{0})f(x)=e^{i(\scal x{\xi_{0}}) -  \scal {x_0}{\xi _0}/2%
%)}f(x-x_{0})\label{HW}%
%\end{equation}
%%%
%(see de Gosson \cite{Birkbis}, Chapter 8 for a detailed study of these
%operators). One shows that the Weyl operator $\operatorname*{Op}%
%\nolimits^{w}(a)=\operatorname*{Op}\nolimits _{1/2}(a)$ can be
%represented as follows:%
%\[
%\operatorname*{Op}\nolimits^{w}(a)f=( 2\pi )
%^{-d}\int a_{\sigma}(z)\widehat{T}(z)fdz.
%\]
%
%
%In \cite{Transam} we proved the following:
%
%\begin{prop}
%The Born--Jordan operator $\operatorname*{Op}\nolimits_{\mathrm{BJ}}(a)$ is
%given by
%%%
%\begin{equation}
%\operatorname*{Op}\nolimits_{\mathrm{BJ}}(a)f=\int a_{\sigma}(z)\Theta
%(z)\widehat{T}(z)fdz\label{atheta}%
%\end{equation}
%%%
%where $\Theta$ is the real function defined by
%%%
%\begin{equation}
%\Theta(z)=\frac{\sin(2\pi \scal x\xi )}{2\pi \scal x \xi }.\label{theta}%
%\end{equation}
%%%
%\end{prop}
%
%It follows from the fact that $\Theta\in L^{\infty}(\mathbf{R}^{2d})$ that the
%operator $\operatorname*{Op}\nolimits_{\mathrm{BJ}}(a)$ is a continuous
%operator $\mathscr{S}(\mathbf{R}^{d})\to\mathscr{S}^{\prime
%}(\mathbf{R}^{d})$ for every $a\in\mathscr{S}^{\prime}(\mathbf{R}^{2d})$. 
%
%In view of the relation (\ref{opadj}) between a $t$-operator and its
%adjoint we have
%%%
%\begin{equation}
%\operatorname*{Op}\nolimits_{\mathrm{BJ}}(a)^{\ast}=\operatorname*{Op}%
%\nolimits_{\mathrm{BJ}}(\overline{a})\label{bjadj}%
%\end{equation}
%%%
%hence the Born--Jordan operators share with Weyl operators the property of
%being (formally) self-adjoint if and only if their symbol is real. This makes
%Born--Jordan operators good alternative candidates for quantization.

\par

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Born-Jordan operators with distribution symbols}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\par

In this section we deduce various kinds of mapping properties of $\op _{\BJ}(a)$
when $a$ belongs to suitable test-function or distribution spaces. In particular we
show that $\op _{\BJ}(a)$ makes sense as a continuous operator from
$\mascS (\rr d)$ to $\mascS '(\rr d)$ when $a\in \mascS '(\rr {2d})$.

\par

We begin with the following analogy of Proposition \ref{Prop:WignerDistExt}
in Born-Jordan situation.

\par

\begin{prop}\label{Prop:WignerBJDistExt}
Let $s\ge \frac 12$ and let $T(f,\phi )
\equiv W_{\BJ} (f,\phi )$ when $f,\phi \in \mascS (\rr d)$. Then the following
is true:
\begin{enumerate}
\item the map $(f,\phi)\mapsto T(f,\phi )$ is continuous from
$\mascS (\rr d)\times \mascS (\rr d)$ to
$\mascS (\rr {2d})$ and restricts to a continuous map from
$\maclS _s(\rr d)\times \maclS _s(\rr d)$ to
$\maclS _s(\rr {2d})$;

%\vrum

\item the map $(f,\phi)\mapsto T(f,\phi )$ from
$\maclS _s(\rr d)\times \maclS _s(\rr d)$ to
$\maclS _s(\rr {2d})$ extends uniquely to a continuous map
from $\maclS _s'(\rr d)\times \maclS _s'(\rr d)$ to
$\maclS _s'(\rr {2d})$ and from
$\mascS '(\rr d)\times \mascS '(\rr d)$ to
$\mascS '(\rr {2d})$.
%
%\item the map $(t ,f,\phi)\mapsto T(t ,f,\phi )$ from
%$\mathbf R\times \maclS _s(\rr d)\times \maclS _s(\rr d)$ to
%$\maclS _s(\rr {2d})$ extends uniquely to a continuous map
%from $\mathbf R\times \maclS _s'(\rr d)\times \maclS _s(\rr d)$ to
%$\maclS _s'(\rr {2d})\cap C^\infty (\rr {2d})$ and from
%$\mathbf R\times \mascS '(\rr d)\times \mascS '(\rr d)$ to
%$\mascS '(\rr {2d})\cap C^\infty (\rr {2d})$.
\end{enumerate}
The same holds true for $s>\frac 12$ when each $\maclS _s$ and $\maclS _s'$
are replaced by $\Sigma _s$ and $\Sigma _s'$, respectively.
\end{prop}

\par

\begin{proof}
We have
%%
\begin{equation}\label{Eq:WignerBJIntForm}
W_{BJ} (f,g) =\int _0^1W_t (f,g)\, dt ,
\end{equation}
%%
when $f,g\in \mascS (\rr d)$. By Proposition \ref{Prop:WignerDistExt}
and Remark \ref{Rem:WignerDistExt} it follows that the map
$(f,\phi )\mapsto W_{BJ} (f,g)$ is continuous from $\mascS (\rr d)\times \mascS (\rr d)$
to $\mascS (\rr {2d})$, and that the same holds true if each $\mascS$ is replaced by
$\maclS _s$ or by $\Sigma _s$.

\par

By Proposition \ref{Prop:WignerDistExt}
and Remark \ref{Rem:WignerDistExt} it follows that the map
$$
(f,g)\mapsto \int _0^1W_t (f,g)\, dt
$$
is well-defined and continuous from $\mascS '(\rr d)\times \mascS '(\rr d)$
to $\mascS '(\rr {2d})$, and that the same holds true if each $\mascS '$ is replaced by
$\maclS _s'$ or by $\Sigma _s'$. Hence, by letting $W_{\BJ}(f,g)$ be defined by
the right-hand side of \eqref{Eq:WignerBJIntForm} for such $f$ and $g$, the asserted
continuity of the extensions of the map $T$ follows.

\par

It remains to show the asserted uniqueness of the extensions of $T$. 
Let $f,g\in \maclS _s'(\rr d)$. By Proposition \ref{Prop:WignerDistExt}
and its proof, and Remark \ref{Rem:WignerDistExt}, it follows
that if
$$
\{ f_\ep \} _{\ep >0} \subseteq \maclS _s'(\rr d)
\quad \text{and}\quad
\{ g_\ep \} _{\ep >0} \subseteq \maclS _s'(\rr d),
$$
are such that
$$
\lim _{\ep \to 0+}f_\ep = f
\quad \text{and}\quad
\lim _{\ep \to 0+}g_\ep = g
$$
with convergence in $\maclS _s'(\rr d)$, then 
$$
\lim _{\ep \to 0+}W_t (f_\ep ,g_\ep ) = W_t (f,g)
$$
in $\maclS _s'(\rr {2d})$, locally uniformly with respect to $t$. Hence
$$
\lim _{\ep \to 0+}W_{\BJ} (f_\ep ,g_\ep ) = W_{\BJ} (f,g).
$$
The uniqueness assertions now follow from the facts that
we may choose $f_\ep$ and $g_\ep$ in $\maclS _s(\rr d)$, and that
$$
\mascS '\subseteq \Sigma _s' \subseteq \maclS _{1/2}',
$$
and the result follows.
\end{proof}

\par

We have now the following. Here $\mathscr L(V_1,V_2)$ is the set of all
linear and continuous mappings from the topological vector space $V_1$
into the topological vector space $V_2$.

\par

\begin{thm}\label{Thm:BJopsDistrMap}
Let $s\ge \frac 12$. Then the following is true:
\begin{enumerate}
\item if $a\in \maclS _s(\rr {2d})$, then $\op _{\BJ}(a)$ from
$\maclS _s(\rr d)$ to $\maclS _s'(\rr d)$ is uniquely extendable
to a continuous map from $\maclS _s'(\rr d)$ to $\maclS _s(\rr d)$;

\item the map $a\mapsto \op _{\BJ}(a)$ from $\maclS _s(\rr {2d})$
to $\mathscr L(\maclS _s(\rr d),\maclS _s'(\rr d))$ is uniquely extendable
from $\maclS _s'(\rr {2d})$ to $\mathscr L(\maclS _s(\rr d),\maclS _s'(\rr d))$.
\end{enumerate}
The same holds true if each $\maclS _s$ and $\maclS _s'$ are replaced by
$\Sigma _s$ and $\Sigma _s'$, respectively, or by $\mascS$ and $\mascS '$.
\end{thm}

\par

\begin{proof}
We only prove (1) and (2). The assertions involving $\mascS$ or $\Sigma _s$, and their
duals follow by similar arguments and are left for the reader.

\par

By Proposition \ref{Prop:WignerBJDistExt} it follows that the map
$$
(a,f,g)\mapsto (2\pi )^{-d/2}(a,W_t (g,f))_{L^2(\rr {2d})}
$$
from
$\maclS _s(\rr {2d})\times \maclS _s(\rr d)\times \maclS _s(\rr d)$ to $\mathbf C$
extends uniquely to continuous mappings from
$\maclS _s'(\rr {2d})\times \maclS _s(\rr d)\times \maclS _s(\rr d)$ to $\mathbf C$,
and from
$\maclS _s'(\rr {2d})\times \maclS _s(\rr d)\times \maclS _s(\rr d)$ to $\mathbf C$.
Hence, by letting $\op _{BJ}(a)f$ be defined by \eqref{Eq:WignerPseudoRel},
the asserted continuity follows. The uniqueness of these extensions follows by
similar arguments to those of Proposition \ref{Prop:WignerBJDistExt}. The details
are left to the reader.

\end{proof}

\par

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Born-Jordan operators with modulation space symbols}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\par

In this section we deduce that any Born-Jordan operator with symbol in
the modulation space $M^{p,q}_{(\omega )}(\rr {2d})$ is a pseudo-differential operator
with symbol in $M^{p,q}_{(\omega )}(\rr {2d})$, when $\omega (x,\xi ,\eta ,y)
=\omega _0(\eta ,y)$. We also deduce continuity and
Schatten-von Neumann properties for such operators. 

\par

We start with the following.
%by proving that Born-Jordan operators are pseudo-differential
%operators with symbols in a certain modulation space, when the
%Born-Jordan symbols belong to the same modulation space.

\par

\begin{thm}\label{BJModSymbols}
Let $p\in [1,\infty ]$, $q\in (0,\infty ]$, $t \in \mathbf R$, $\omega \in
\mascP _E(\rr {4d})$
be such that $\omega (x,\xi ,\eta ,y)=\omega _0(\eta ,y)$ for some
$\omega _0\in \mascP _E(\rr {2d})$, and let
$a\in M^{p,q}_{(\omega )}(\rr {2d})$. Then $\opBJ (a) =\opp {t } (b)$
for some $b\in M^{p,q}_{(\omega )}(\rr {2d})$.
\end{thm}

\par

%For the proof we need the following lemma, which essentially follows from
%Proposition 1.7 in \cite{Toft1}, and its proof.
%
%\par

%\begin{lemma}\label{LemmaProp1.7}
%Let $t \in [0,1]$, $\Phi \in \Sigma _1
%(\rr {2d})\setminus 0$ and $a,a_t \in \Sigma _1'(\rr {2d})$ such that
%$$
%\opp t (a) = \opp 0 (a_t ),
%$$
%then there is a bounded set $\Phi _t$, $t \in [0,1]$, in
%$\Sigma _1(\rr {2d})$ and a bounded set of real quadratic forms $Q_t$
%on $\rr {4d}$ such that
%%%
%\begin{equation}\label{STFTCalcTransf}
%(V_{\Phi}a_t )(x,\xi ,\eta ,y) = e^{iQ_t (x-t y ,\xi -t \eta ,\eta ,y)}
%(V_{\Phi _t}a)(x-t y,\xi -t \eta ,\eta ,y).
%\end{equation}
%%%
%\end{lemma}
%
%\par
%
%\begin{proof}
%The result follows from (1.13) in \cite{Toft1}, and its proof.
%The details are left for the reader.
%\end{proof}

\par

\begin{proof}
It suffices to prove the result in the case $t =0$.
We also assume that $p,q<\infty$. The cases when
$p=\infty$ or $q=\infty$ follow by similar arguments and
are left for the reader.

\par

Let $r=\min (1,q)$, $\Lambda _\ep =\ep \zz {2d}$ and
$\Lambda _\ep ^2=\Lambda _\ep\times \Lambda _\ep$
when $\ep >0$. By \cite{Toft9} there are
$v\in \mascP _E(\rr {4d})$ which is submultiplicative
such that $\omega$ is $v$-moderate,
$$
\Psi \in \Sigma _1(\rr {2d})
\quad \text{and}\quad
\Psi _{\! 0}\in M^r_{(v)}(\rr {2d})
$$
such that
%%
\begin{align}
a(X) &= \sum _{\mabfj ,\mabfk \in \Lambda _\ep} c(\mabfj ,\mabfk)\Psi (X-\mabfj )
e^{i\scal X{\rho (\mabfk )}},\label{Eq:GaborExp}
\intertext{where}
c(\mabfj ,\mabfk) &= (V_{\Psi _{\! 0}}a)(\mabfj ,\rho (\mabfk ))\label{Eq:GaborCoeff}
\end{align}
%%
when $a\in M^{p,q}_{(\omega )}(\rr {2d})$, provided $\ep >0$ is chosen small enough.
Here $\rho$ is the reflexion operator on $\rr {2d}$ given by
$\rho (x,\xi )=(\xi ,x)$ when $x,\xi \in \rr d$. We have
$$
\nm c{\ell ^{p,q}_{(\omega)}(\Lambda _\ep ^2)}\asymp \nm a{M^{p,q}_{(\omega )}},
$$
when $c$ is given by \eqref{Eq:GaborCoeff}.

\par

For $a\in M^{p,q}_{(\omega )}(\rr {2d})$ and $\Phi \in \Sigma _1(\rr {2d})\setminus 0$
fixed we now get
%%
\begin{align}
V_\Phi (e^{it \scal {D_\xi}{D_x}}a) &= \sum _{\mabfj ,\mabfk \in \Lambda _\ep}
c(\mabfj ,\mabfk )H_{\mabfj ,\mabfk},\label{Eq:STFTExp1}
\intertext{where}
H_{\mabfj ,\mabfk} &= V_\Phi (e^{it \scal {D_\xi}{D_x}}
(\Psi (\cdo -\mabfj )e^{i\scal \cdo {\rho (\mabfk)}}).
\notag
\end{align}
%%
We need to simplify $H_{\mabfj ,\mabfk}$. By the definitions and
straightforward computations, using Fourier's inversion formula
we get
$$
H_{\mabfj ,\mabfk}(X,\rho (Y))=e^{i(\scal X{\rho (\mabfk -Y)} +it\scal k\kappa}
 (V_\Phi \Psi _t )(X-\mabfj +t \mabfk ,\rho (Y-\mabfk )),
$$
where
$$
\Psi _t = e^{it \scal {D_\xi}{D_x}}\Psi
\quad \text{and}\quad
\mabfk =(k,\kappa )
$$
(see (1.13) in \cite{Toft6}, and its proof).

\par

Since $\Psi \in \Sigma _1(\rr {2d})$, it follows from \cite{CaTo,Tr} that
$\sets {\Psi _t}{t\in [0,1]}$ is a bounded set in $\Sigma _1(\rr {2d})$.
Hence, by \cite{ChuChuKim}, for every $r>0$ there is a constant $C_r$
which is independent of $t \in [0,1]$ such that
$$
|H_{\mabfj ,\mabfk}(X,\rho (Y))|
=
|(V_\Phi \Psi _t )(X-\mabfj +t \mabfk ,\rho (Y-\mabfk ))|
\le C_re^{-r(|X-\mabfj +t \mabfk |+|Y-\mabfk |)}
$$

\par

By using the latter estimate in \eqref{Eq:STFTExp1} we get
with $r>0$ large enough that
%%
\begin{multline*}
|V_\Phi b(X,\rho (Y))|
\lesssim
\sum _{\mabfj ,\mabfk \in \Lambda _\ep}|c(\mabfj ,\mabfk )|
\int _0^1e^{-r(|X-\mabfj +t \mabfk |}\, dt e^{-2r|Y-\mabfk |}
\omega _0(\rho (Y))
\\[1ex]
\lesssim
\sum _{\mabfj ,\mabfk \in \Lambda _\ep}|c(\mabfj ,\mabfk )|
\int _0^1e^{-r(|X-\mabfj +t \mabfk |}\, dt e^{-r|Y-\mabfk |}
\omega _0(\rho (\mabfk)).
\end{multline*}
%%
It now follows from Minkowski's inequality that
$$
\nm {V_\Phi b(\cdo ,\rho (Y))}{\ell ^p(\Lambda _\ep )}\omega _0(\rho (Y))
\lesssim
\sum _{\mabfk \in \Lambda _\ep} h(\mabfk )e^{-r|Y-\mabfk |}
\omega _0(\rho (\mabfk))
$$
where
$$
h(\mabfk )
=
\int _0^1
\left (
\sum _{\mabfm \in \Lambda _\ep}
\left (
\sum _{\mabfj \in \Lambda _\ep}
|c(\mabfj ,\mabfk )|e^{-r|\mabfm -\mabfj+t \mabfk |}
\right )^p
\right )^{1/p}\, dt .
$$
We have
%%
\begin{multline*}
h(\mabfk )
\lesssim
\int _0^1
\left (
\sum _{\mabfm \in \Lambda _\ep}
\left (
\sum _{\mabfj \in \Lambda _\ep}
|c(\mabfj ,\mabfk )|e^{-r|\mabfm -\mabfj |}
\right )^p
\right )^{1/p}\, dt
\\[1ex]
=
\nm {c(\cdo ,\mabfk )*e^{-r|\cdo |}}{\ell ^p}
\lesssim
\nm {c(\cdo ,\mabfk )}{\ell ^p},
\end{multline*}
%%
where the last step follows from Young's inequality. Here $*$
denotes the discrete convolution. If $c_0(\mabfk )
=\nm {c(\cdo ,\mabfk )}{\ell ^p}\omega _0(\rho (\mabfk ))$, then
we get from these estimates that
$$
\nm {V_\Phi b(\cdo ,\rho (\mabfn ))}{\ell ^p(\Lambda _\ep )}\omega _0(\rho (\mabfn ))
\lesssim
\sum _{\mabfk \in \Lambda _\ep} c_0(\mabfk )e^{-r|\mabfn -\mabfk |}
= (c_0*e^{-r|\cdo |})(\mabfn )
$$
By applying the $\ell ^q$ norm on the last inequality we get
%%
\begin{multline*}
\nm b{M^{p,q}_{(\omega )}}
\asymp
\nm c{\ell ^{p,q}_{(\omega )}}
\\[1ex]
\lesssim
\nm {c_0*e^{-r|\cdo |}}{\ell ^q}
\le
\nm {c_0}{\ell ^q}\nm {e^{-r|\cdo |}}{\ell ^r}
\asymp 
\nm {c_0}{\ell ^q} \asymp \nm a{M^{p,q}_{(\omega )}}.
\end{multline*}
%%
Hence we have proved
$$
\nm b{M^{p,q}_{(\omega )}}\lesssim \nm a{M^{p,q}_{(\omega )}},
$$
and the result follows.
%\vspace{2cm}
%
%
%
%
%Let $t$, $\Phi$, $\Phi _t$, $a$, $a_t$ and $Q_t$
%be the same as in Lemma \ref{LemmaProp1.7}, and let
%$b\in \Sigma _1'(\rr {2d})$ be such that
%$$
%\opp 0 (b) = \opBJ (a).
%$$
%
%\par
%
%Let
%$$
%F_t (x,\xi ,\eta ,y)\quad \text{and}\quad
%G_t (x,\xi ,\eta ,y)
%$$
%be the real and imaginary parts of
%$$
%e^{iQ_t (x-t y ,\xi -t \eta ,\eta ,y)}
%(V_{\Phi _t}a)(x-t y,\xi -t \eta ,\eta ,y)\omega (\eta ,y).
%$$
%Then
%%%
%\begin{align}
%\sup _{t \in [0,1]}
%\nm {F_t (\cdo ,\eta ,y ) }{L^{p}(\rr {2d})}&\le 
%(H_0*e^{-r|\cdo |})(\eta ,y)
%\nm {(V_{\Phi _t} a)(\cdo ,\eta ,y )}{L^p}\omega (\eta ,y)
%\label{FGTauEst1}
%\intertext{and}
%\sup _{t \in [0,1]}
%\nm {G_t (\cdo ,\eta ,y )}{L^{p}}
%&\le \nm {(V_{\Phi _t} a)(\cdo ,\eta ,y )}{L^p}\omega (\eta ,y).
%\label{FGTauEst2}
%\intertext{for every $r>0$, where}
%H_0(\eta ,y) &= \nm {(V_{\Phi } a)(\cdo ,\eta ,y )}{L^p(\rr {2d})} \omega ()
%\end{align}
%%%
%
%\par
%
%In fact, we have
%%%
%\begin{multline*}
%\sup _{t \in [0,1]}\nm {F_t (\cdo ,\eta ,y ) }{L^{p}}
%=
%\sup _{t \in [0,1]} \left ( \iint _{\rr {2d}}
%|F_t (x,\xi ,\eta ,y)|^p\, dxd\xi
%\right ) ^{1/p}
%\\[1ex]
%\le
%\sup _{t \in [0,1]} \left ( \iint _{\rr {2d}}
%|(V_{\Phi _t}a) (x-t y,\xi -t \eta ,\eta ,y)|^p\, dxd\xi
%\right ) ^{1/p}\omega (\eta ,y)
%\\[1ex]
%\le
%\left ( \iint _{\rr {2d}}
%|(V_{\Phi}a) (x,\xi ,\eta ,y)|^p\, dxd\xi
%\right ) ^{1/p} \omega (\eta ,y)
%=\nm {(V_\Phi a)(\cdo ,\eta ,y )}{L^p}\omega (\eta ,y).
%\end{multline*}
%%%
%In the same way we get
%$$
%\sup _{t \in [0,1]}\nm {G_t (\cdo ,\eta ,y ) }{L^{p}}\le
%\nm {(V_\Phi a)(\cdo ,\eta ,y )}{L^p}\omega (\eta ,y),
%$$
%and \eqref{FGTauEst} follows.
%
%\par
%
%By applying the $L^q$ quasi-norm
%in the $(\eta ,y)$-variables, Minkowski's inequality and \eqref{FGTauEst} give
%%%
%\begin{multline*}
%\nm b{M^{p,q}_{(\omega )}} \asymp
%\left ( \iint _{\rr {2d}} \left ( \iint _{\rr {2d}}
%|(V_{\Phi}b) (x,\xi ,\eta ,y)\omega (\eta ,y)|^p\, dxd\xi
%\right ) ^{q/p}\, dyd\eta \right )^{1/q}
%\\[1ex]
%=
%\left ( \iint _{\rr {2d}} \left ( \iint _{\rr {2d}}
%\left |
%\int _0^1 (F_t (x,\xi ,\eta ,y)+iG_t (x,\xi ,\eta ,y))\, dt 
%\right | ^p \, dxd\xi
%\right ) ^{q/p}\, dyd\eta \right )^{1/q}
%\\[1ex]
%\le
%\left ( \iint _{\rr {2d}} \left (\int _0^1 \left ( \iint _{\rr {2d}}
%\left |
%(F_t (x,\xi ,\eta ,y)+iG_t (x,\xi ,\eta ,y)) 
%\right | ^p \, dxd\xi \right )^{1/p} \, dt
%\right ) ^{q}\, dyd\eta \right )^{1/q}
%\\[1ex]
%\lesssim
%\left ( \iint _{\rr {2d}} \left (\int _0^1
%\nm {(V_\Phi a)(\cdo ,\eta ,y )}{L^p}\omega (\eta ,y) \, dt
%\right ) ^{q}\, dyd\eta \right )^{1/q}
%\\[1ex]
%\asymp
%\nm a{M^{p,q}_{(\omega )}},
%\end{multline*}
%%%
%and the result follows.
%
% Hoppades kunna visa satsen aeven foer p,q i (0,\infty ].
%
%
%By the mean-value theorem, we have
%$$
%V_\Phi b = \int _0^1 (F_t +iG_t )\, dt =F_{t _1}+iG_{t _2}, 
%$$
%for some 
%
%it follows from Lemma \ref
%
\end{proof}

\par

The next result follows from the previous one and the fact that
$M^{p,q}(\rr {2d})\subseteq s_{t,p}(\rr {2d})$ when $q\le \min (p,p')$.
Here $s_{t ,p}(\rr {2d})$
is the set of all symbols $a$ in $\mathscr S'(\rr {2d})$ such that
$\opp t (a)$ belongs to $\mathscr I_p$, the set of all Schatten-von
Neumann operators on $L^2(\rr d)$ of order $p$.

\par

\begin{thm}
Let $p,q \in [1,\infty ]$ be such that $q\le \min (p,p')$. If
$a\in M^{p,q}(\rr {2d})$, then $\opBJ (a)\in \mathscr I_p$.
\end{thm}

\par


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Born-Jordan operators with H{\"o}rmander symbols}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\par

In this section we show that any Born-Jordan operator with symbol in certain
H{\"o}rmander class is a pseudo-differential operator with symbol in the same
class. Furthermore, we prove sharp-G{\aa}rding's and Feffermann-Phong's inequalities
for
such operators.

\par

First we recall the definition of the involved symbol classes. Let $g$
be a Riemannian metric on the phase space $W=T^*\rr d\simeq \rr {2d}$, let
$m>0$ be a function in $L^\infty _{loc}(W)$, and let $N\ge 0$
be an integer. For any $a\in C^N(W)$ and $X=(x,\xi )\in W$, let $|a|_0^g(X) =
|a(X)|$, and
$$
|a|_k^g(X)\equiv \sup |a^{(k)}(X; Y_1,\dots ,Y_k)|, \quad k\ge 1.
$$
Here the supremum is taken over all $Y_1,\dots ,Y_k\in W$ such that
$g_X(Y_j)\le 1$ for every $j=1,\dots ,k$. We also set
$$
\nm a{m,N}^g \equiv \sum _{k=0}^N\sup _{X\in W} (|a|_k^g(X)/m(X)).
$$
Then $S_N(m,g)$ is the Banach space which consists of all $a\in C^N(W)$
such that $\nm a{m,N}^g$ is finite. Furthermore,
$$
S(m,g) = \bigcap _{N\ge 0} S_N(m,g),
$$
and equip this space by the topology, induced by the semi-norms
$\nm \cdo {m,N}^g$, $N\ge 0$.

\par

The dual metric $g^\sigma$ with respect to the (standard) symplectic form
$\sigma$ is defined by
$$
g^\sigma _X(Z) \equiv \sup |\sigma (Y,Z)|^2,\quad X,Z\in W
$$
where the supremum is taken over all $Y\in W$ such that $g_X(Y)\le 1$.
Furthermore, the Planck's function $h_g(X)$ is defined by
$$
h_g(X) \equiv \sup g_X(Y)^{1/2},\quad X\in W.
$$
where the supremum is taken over all $Y\in W$ such that
$g^\sigma _X(Y)\le 1$.

\par

As in \cite{BuTo,Toft4} we need some restrictions on $m$ and $g$. More precisely,
the metric $g$ on $W$ is called slowly varying if there are constants
$c,C>0$ such that
$$
C^{-1}g_X\le g_Y\le Cg_X,\quad \text{when}\quad g_X(Y)<c,
$$
and $m$ is called $g$-continuous if there are constants $c,C>0$ such that
$$
C^{-1}m(X)\le m(Y)\le Cm(X),\quad \text{when}\quad g_X(Y)<c,
$$
The metric $g$ is called $\sigma$-temperate if there are constants $C,N>0$
such that
$$
g_X(Z)\le Cg_Y(Z)(1+g_X(X-Y))^N,\quad \text{for all}\quad X,Y,Z\in W,
$$
and $m$ is called $(\sigma ,g)$-temperate if there are constants $C,N>0$
such that
$$
m(X)\le Cm(Y)(1+g_X(X-Y))^N,\quad \text{for all}\quad X,Y\in W.
$$
Furthermore, it is most of the times assumed that the uncertainty principle
holds, i.{\,}e. $h_g(X)\le 1$ for every $X\in W$.

\par

The metric $g$ on $W$ is called \emph{strongly feasible} if $g$ is slowly varying,
$\sigma$-temperate and $h_g\le 1$. The weight $m$ is called
\emph{$g$-feasible} if $m$ is $g$-continuous and $(\sigma ,g)$-temperate.

\par

Finally, the metric $g$ on $W$ is called \emph{split}, if
$$
g_X(z,\zeta ) = g_X(z,-\zeta ),\quad \text{when} \quad X,(z,\zeta )\in W.
$$

\par

We have now the following.

\par

\begin{thm}\label{BornJordanHormWeyl}
Let $g$ be strongly feasible and split on $W$, $m$ be $g$-feasible,
$N> 0$ be an integer, and let $a\in S(m,g)$. Then
$$
\opBJ (a) =\op ^w(b)
$$
for some $b\in S(m,g)$. Furthermore,
%%
\begin{equation}\label{bassexp}
b-\sum _{2j<N} \frac {(-1)^j\scal {D_\xi }{D_x}^{2j}a}{4^j(2j+1)!}\in S(h_g^{N}m,g)
\end{equation}
%%
\end{thm}

\par

For the proof we need the following proposition. We omit the proof, since
the result is essentially a restatement of
Proposition 18.5.10 (?) in \cite{Horm}.

\par

\begin{prop}\label{Calculitransfer}
Let $g$ be strongly feasible and split on $W$, $m$ be $g$-feasible, $I\subset \mathbf R$
be bounded, $N> 0$ be an integer, $a\in S(m,g)$, and let $a_t \in \mathscr S'(W)$,
be such that
%%
\begin{equation}\label{CalculitransferForm}
\op ^w(a)=\op _t (a_t),\qquad t \in I.
\end{equation}
%%
Then $\{ a_t \} _{t \in I}$ is a bounded subset of $S(m,g)$, and for some bounded
subset $\Omega$ of $S(h_g^{N}m,g)$ it holds
%%
\begin{equation}\label{atexpansions}
a_t  -\sum _{k<N} \left ( t -\frac 12 \right )^k
\frac {i^k\scal {D_\xi }{D_x} ^ka}{k!}\in \Omega .
\end{equation}
%%
\end{prop}

\par

\begin{proof}[Proof of Theorem \ref{BornJordanHormWeyl}]
Let $a_t $ be the same as in Proposition \ref{Calculitransfer}. Then
$$
b=\int _0^1 a_t \, dt  ,
$$
and Proposition \ref{Calculitransfer} shows that $b\in S(m,g)$.
Furthermore, by \eqref{atexpansions} we get
$$
b-  \sum _{k<N} \left (\int _0^1\left ( t -\frac 12 \right ) ^k\, dt \cdot
\frac {i^k\scal {D_\xi }{D_x} ^ka}{k!} \right ) \in S(h_g^{N}m,g),
$$
which gives \eqref{bassexp}. The proof is complete.
\end{proof}

\par

Theorem \ref{BornJordanHormWeyl} shows that several continuity
properties in the Weyl calculus in Chapter XVIII in \cite{Horm} carry
over to Born-Jordan operators. For example, the following results are
immediate consequences of Propositions 18.6.2 and 18.6.3
in \cite{Horm}, and Theorem \ref{BornJordanHormWeyl}.

\par

\begin{prop}\label{BJSchwartzcont}
Let $g$ be strongly feasible and split on $W$, $m$ be $g$-feasible,
and let $a\in S(m,g)$. Then $\opBJ (a)$
is continuous on $\mascS (\rr d)$ and is uniquely extendable to
a continuous operator on $\mascS '(\rr d)$.
\end{prop}

\par

\begin{prop}\label{BJL2cont}
Let $g$ be strongly feasible and split on $W$, %$m$ be $g$-feasible,
and let $a\in S(1,g)$. Then $\opBJ (a)$
is continuous on $L^2 (\rr d)$.
\end{prop}

\par

\begin{rem}
In \cite{BoCh}, Bony and Chemin introduced a broad family of Sobolev
type spaces, where each space $H(m_0,g)$ is a Hilbert space and depends on
the choice of the strongly feasible metric $g$ and the $g$-feasible weight
$m_0$. By Th{\'e}or{\`e}me 4.5 in \cite{BoCh}, the fact that $H(1,g)=L^2$ and
Theorem \ref{BornJordanHormWeyl}, it follows that Proposition
\ref{BJL2cont} can be extended in the following way.

\par

Let $g$ be strongly feasible and split on $W$, $m$ and $m_0$ be $g$-feasible,
and let $a\in S(1,g)$. Then $\opBJ (a)$
is continuous from $H(m_0,g)$ to $H(m_0/m,g)$.
\end{rem}

\par

We also have the following slight extension of the previous result. Here
and in what follows we let $L^\infty _0(\rr d)$ be the set of all $f\in L^\infty (\rr d)$
vanishing at infinity, i.{\,}e. $f$ should satisfy
$$
\lim _{R\to \infty} \,  \underset {|x|\ge R} {\operatorname {ess\, sup}}\,  |f(x)| =0. 
$$

\par

\begin{prop}\label{HormSchattenBJ}
Let $p\in [1,\infty ]$, $t\in\mathbf R$, $g$ be strongly
feasible and split, $m$ be $g$-continuous and
$(\sigma,g)$-temperate, and let $a\in S(m,g)$. Then the following is
true:
\begin{enumerate}
\item if $h_g^{k/2}m\in L^p(W)$ for some $k\ge 0$, and $a\in L^p(W)$,
then $\opBJ (a)\in \mascI _p$;

\vspace{0.1cm}

\item if $h_g^{k/2}m\in L^\infty _0(W)$ for some $k\ge 0$, and $a\in
L^\infty _0(W)$, then $\opBJ (a)$ is compact on $L^2(\rr d)$.
\end{enumerate}
\end{prop}

\par

\begin{cor}\label{HormSchattenBJcor}
Let $p\in [1,\infty ]$, $t\in\mathbf R$, $g$ be strongly
feasible and split, $m$ be $g$-continuous and
$(\sigma,g)$-temperate, and let $a\in S(m,g)$. Then the following is
true:
\begin{enumerate}
\item if $m\in L^p(W)$, then $\opBJ (a)\in \mascI _p$;

\vspace{0.1cm}

\item if $m\in L^\infty _0(W)$, then $\opBJ (a)$ is compact on
$L^2(\rr d)$.
\end{enumerate}
\end{cor}

\par

For the proof of Proposition \ref{HormSchattenBJ} we need the following result
which is a slight extension of \cite[Proposition 2.10]{BuTo}. The proof is omitted,
since the result follows by the arguments in the proof of \cite[Proposition 2.10]{BuTo}.

\begin{lemma}\label{intersectionlemma}
Let $p$ and $g$ be the same as in Proposition \ref{HormSchattenBJ}
and let $m$ be $g$-continuous, $(\sigma,g)$-temperate and satisfies
$h_g^{N/2}m \in L^p(W)$ ($h_g^{N/2}m \in L^\infty _0(W)$) for some
$N\ge 0$. Also let $t \in [0,1]$, and let $a,a_t \in \mascS '(W)$
be related as in
\eqref{CalculitransferForm}. Then the map $a\mapsto a_t $ on
$\mathscr S'(W)$ restricts to a continuous isomorphism on $S(m,g)\cap
L^p(W)$ ($S(m,g)\cap L^\infty _0(W)$), which is uniformly bounded with
respect to $t \in [0,1]$.
\end{lemma}

\par

\begin{proof}[Proof of Proposition \ref{HormSchattenBJ}]
Again let $b$ be chosen such that $\op ^w(b)= \opBJ (a)$. Then
Lemma \ref{intersectionlemma} shows that $b\in L^p(W)\cap S(m,g)$
when $a\in L^p(W)\cap S(m,g)$, and that $b\in L^\infty _0(W)\cap S(m,g)$
when $b\in L^\infty _0(W)\cap S(m,g)$. The result now follows from
Theorem 2.9 in \cite{BuTo}.
\end{proof}

\par

Finally we also have the following Feffermann-Phong's inequality inequality
for Born-Jordan operators, which in particular shows that Sharp-G{\aa}rding's
inequality is also true for such operators.

\par

\begin{prop}\label{GaardingIneq}
Let $g$ be strongly feasible and split on $W$, %$m$ be $g$-feasible,
and let $0\le a\in S(h_g^{-2},g)$. Then $\opBJ (a) \ge -C$
for some constant $C\ge 0$.
\end{prop}

\par

\begin{proof}
By letting $b$ be defined by $\op ^w(b)=\op _{BJ}(a)$ choosing $N=2$ in
Theorem \ref{BornJordanHormWeyl}, it follows that $b=a+c$, where
$c\in S(1,g)$. The result now follows from the facts that $\op ^w(a)$ is lower
bounded on $L^2$ by the Feffermann-Phong's
inequality for Weyl operators (cf. \cite[Theorem 18.6.?]{Horm}),
and the fact that $\op ^w(c)$ is bounded on $L^2$ in view of
\cite[Proposition 18.6.3]{Horm}.
\end{proof}

\par

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{Weyl H.Weyl Gruppentheorie und Quantenmechanik, Verlag von S.Hirzel, Leipzig, 1928.