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\begin{document}
\title{Orlicz convolutions in QHA with
applications to Orlicz Schatten properties for
Toeplitz operators}
\author{Joachim Toft}
\maketitle
For any (Weyl symbol) $a\in \mascS (\rr {2d})$, the Weyl operator $\op ^w(a)$
is the pseudo-differential operator, defined by
$$
\op ^w(a)f(x) = (2\pi )^{-d}\iint _{\rr {2d}}
a({\textstyle{\frac 12}}(x+y),\xi )f(y)e^{i\scal {x-y}\xi}\, dyd\xi ,
$$
when $f\in \mascS (\rr d)$. The definition extends to any $a\in \mascS '(\rr {2d})$,
and then $\op ^w(a)$ is continuous from $\mascS (\rr d)$ to $\mascS '(\rr d)$.
\par
Let $s_p^w$ be the set of all $a\in \mascS '(\rr {2d})$ such that
$\op ^w(a)$ belongs $\mascI _p$,
the set of all all Schatten-$p$ operators on $L^2(\rr d)$, $p\in (0,\infty ]$.
Then essential convolutions were established by R. Werner during the
80th.
In the language of Weyl symbols, these convolution results can
be formulated as
$$
s_p^w*s_q^w \subseteq L^r
\quad \text{and}\quad
s_p^w*L^q \subseteq s_r^w,
\quad
\frac 1p+\frac 1q=1+\frac 1r,
\ p,q,r\in [1,\infty].
$$
Here $*$ denotes the usual convolution.
\par
In the talk we first extend these convolution properties to Orlicz situations.
Thereafter we apply these results to deduce Orlicz Schatten properties for
Toeplitz operators.
\medspace
\emph{The project behind the talk is based on collaborations with Wolfram
Bauer and Robert Fulsche.}
\end{document}