We study the link between $\Psi$dos
and Wick operators via the Bargmann transform.
We deduce a formula for the symbol of the Wick operator
in terms of the short-time Fourier transform of the Weyl symbol.
This gives characterizations of Wick symbols of
$\Psi$dos of Shubin type and of infinite order,
and results on composition.
We prove a series expansion of Wick operators in

which leads to a sharp Gårding inequality and transition of
hypoellipticity between Wick and and Shubin symbols.
Finally we show continuity results for anti-Wick operators, and
estimates for the Wick symbols of anti-Wick operators.

§ INTRODUCTION

In the paper we investigate conjugation with the Bargmann
transformation of pseudo-differential and Toeplitz
operators on $\rr d$ with isotropic symbols, and we
explore relations between Wick and anti-Wick operators.
Particularly we consider
Shubin operators and operators of infinite order.
This gives rise to analytic type pseudo-differential operators on $\cc d$
that are called Wick or Berezin operators
because of the fundamental contributions
by F. Berezin [6, 7], which in turns goes back to
some ideas in [31] by G. C. Wick.

Let $a$ be a suitable locally bounded function on $\cc {2d}$ such that
$z\mapsto a(z,w)$ is analytic, $z,w\in \cc d$. Then the Wick operator
$\op _{\mathfrak V}(a)$ with symbol $a$ is the operator which takes an
appropriate entire function $F$ on $\cc d$ into the entire function

\begin{equation}\label{Eq:AnalPseudoIntro}
\op _{\mathfrak V}(a)F (z) = \pi ^{-d}
\int _{\cc d} a(z,w)F(w)e^{(z-w,w)}\, d\lambda (w),
\end{equation}

where $d\lambda$ is the Lebesgue measure
and $(\cdo ,\cdo )$ is the scalar product on $\cc d$.
(See [18] and Section <ref>
for notation.) Wick operators appear naturally in
several problems in analysis and its applications, e. g.
in quantum mechanics.
For example, the harmonic oscillator, the creation
and annihilation operators take the simple forms

F\mapsto \scal z{\nabla _z}F +cF,
\quad
F\mapsto z_jF
\quad \text{and}\quad
F\mapsto \partial _{z_j}F,

respectively, for some constant $c$, in the Wick formulation (see [4]).

An advantage of the Wick calculus compared to corresponding
operators on functions and distributions defined on $\rr d$
is that in almost all situations, the involved functions are entire,
which admits the use of the powerful techniques of complex analysis.
(A more general approach is studied in [28], where the
Wick calculus is formulated in terms of spaces of
formal power series expansions instead of spaces of entire functions.)
The possible lack of analyticity of $a(z,w)$
in (<ref>) with respect to the $w$ variable
is removable in the sense
that for any Wick symbol $a$, there is a unique $a_0$ such that
$(z,w)\mapsto a_0(z,\overline w)$ is entire, and
$\op _{\mathfrak V}(a)=\op _{\mathfrak V}(a_0)$. Consequently it is no restriction
to assume that $a(z,w)$ in (<ref>) is analytic in $z$ and
conjugate analytic in $w$, which we do in the introduction henceforth.
Any linear and continuous operator from the Schwartz space,
a Fourier invariant Gelfand-Shilov space or Pilipović space, to
the corresponding distribution spaces, respectively, is in a unique way transformed
into a Wick operator by the Bargmann transform (see [28]).

Several operators in quantum mechanics are so-called Shubin operators, i. e.
pseudo-differential operators

\op (\fka )f(x) = (2\pi )^{-\frac d2}\int _{\rr d}\fka (x,\xi )
\widehat f(\xi )e^{i\scal x\xi}\, d\xi ,
\qquad f\in \mascS (\rr d),

where the symbol $\fka$ belongs to the Shubin class
$\Sh _\rho ^{(\omega )}(\rr {2d})$, the set of all $\fka \in C^\infty (\rr {2d})$
such that

|\partial _x^\alpha \partial _\xi ^\beta \fka (x,\xi )|
\lesssim
\omega (x,\xi )(1+|x|+|\xi |)^{-\rho |\alpha +\beta |},\quad \alpha ,\beta \in \nn d.

Here $\omega$ is a suitable weight function on $\rr {2d}$ and $0 \le \rho \le 1$.
Partial differential operators with polynomial coefficients,
e. g. the creation and annihilation operators or the harmonic oscillator
mentioned above, are examples of Shubin operators.
In Section <ref> we prove that the Bargmann image of Shubin
operators with symbols in
$\Sh _\rho ^{(\omega )}(\rr {2d})$ is the set of all
Wick operators in (<ref>) such that
$a$ belongs to $\wideparen \maclA _{\Sh ,\rho}^{(\omega )}(\cc {2d})$.
This means that $\cc {2d} \ni (z,w)\mapsto a(z,\overline w)$
is an entire function that satisfies

\begin{equation}\label{Eq:AnalShubinIntro}
|\partial _z^\beta \overline \partial _w^\gamma a(z,w)|
\lesssim
e^{\frac 12|z-w|^2}\omega (\sqrt 2 \overline z)
\eabs {z+w}^{-\rho |\beta +\gamma |} \eabs {z-w}^{-N}
\end{equation}

for every $N\ge 0$.

An important subclass of Wick operators are the
anti-Wick operators, which are
Wick operators where the symbol
$a(z,w)$ does not depend on $z$. That is, for an appropriate
measurable function $a_0$ on $\cc d$, its anti-Wick operator is given by

\begin{equation}\tag*{(\ref{Eq:AnalPseudoIntro})$'$}
\op _{\mathfrak V}^{\aw}(a_0)F (z) = \pi ^{-d}
\int _{\cc d} a_0(w)F(w)e^{(z-w,w)}\, d\lambda (w).
\end{equation}

Again $F$ is a suitable entire function on $\cc d$. The anti-Wick
operators can also be described as the Bargmann image of
Toeplitz operators on $\rr d$. (See e. g. [21, 25, 29] for the definition
of Toeplitz operators.)

A feature of Toeplitz operators and anti-Wick operators, useful for energy
estimates in quantum mechanics and
time-frequency analysis, is that non-negative symbols give rise to
non-negative operators. (Cf. e. g. [21, 19, 20].)
An operator $T=\op _{\mathfrak V}(a)$

with $a$ satisfying (<ref>) for every $N\ge 0$, is called
positive (non-negative), if there is a constant
$C> 0$ ($C\ge 0$) such that

(TF,F)_{A^2} \ge C\nm F{A^2}^2,

for every analytic polynomial $F$ on $\cc d$, where $(\cdo ,\cdo )_{A^2}$
is the scalar product induced by the Hilbert norm

\nm F{A^2} =\pi ^{-\frac d2}
\left ( \int _{\cc d} |F(z)|^2e^{-|z|^2}\, d\lambda (z) \right )^{\frac 12}.

The implication from non-negative symbols to non-negative operators
is not relevant for Wick operators in (<ref>)
when $a(z,w)$ is not constant with respect to $z$, since
the analyticity of the map $z\mapsto a(z,w)$ implies that
$a(z,w)$ is non-real almost everywhere. For such symbols it is
instead natural to check whether positivity of the map
$w\mapsto a(w,w)$ leads to positive operators (see e. g.
[6, 7, 13]). By choosing

d=1,\quad a(z,w) = 1-2z\overline w +2z^2\overline w^2
\quad \text{and}\quad

we obtain

a(w,w)=(1-|w|^2)^2+|w|^4> 0
\quad \text{but}\quad
(\op _{\mathfrak V}(a)F,F)_{A^2} =-1<0.

$\op _{\mathfrak V}(a)$
may fail to be a non-negative operator even though $a(w,w)$ is positive.

On the other hand, for certain conditions on $a$, we deduce in Section <ref>
a weaker positivity result for Wick operators, which is equivalent to the
sharp Gårding inequality in isotropic pseudo-differential calculus on $\rr d$
(see Theorem 18.6.7 and the proof of Theorem 18.6.8 in [18]). That is
for $a\in \wideparen \maclA _{\Sh ,\rho}^{(\omega )}(\cc {2d})$
$\omega (z)=\eabs z^{2\rho}$ and $\rho > 0$ we prove

\begin{align}
\repart  (\op _{\mathfrak V}(a)F,F)_{A^2}
&\ge -C\nm F{A^2}^2
\label{Eq:SharpGardingIntro}
\intertext{and}
|\impart  (\op _{\mathfrak V}(a)F,F)_{A^2}|
&\le C\nm F{A^2}^2,
\quad \text{when}\quad
a(w,w) \ge 0
\label{Eq:SharpGardingIntroImPart}
\end{align}

(cf. Theorem <ref>).
In particular we obtain energy estimates also for
Wick operators with symbols that are non-negative on the diagonal.

The latter result is obtained by approximating Wick operators by
anti-Wick operators, using for the Wick operator (<ref>) with
$a\in \wideparen \maclA _{\Sh ,\rho}^{(\omega )}(\cc {2d})$
the remarkable identity

\begin{equation}\label{Eq:WickToAntiWickIntro}
\op _{\mathfrak V}(a)
\sum _{|\alpha |<N} \frac {(-1)^{|\alpha |}}{\alpha !}
\op _{\mathfrak V}^{\aw}(b_\alpha ) + \op _{\mathfrak V}(c_N)
\quad \text{where} \quad
b_\alpha (w)
\partial _z^\alpha \overline \partial _{w}^\alpha a(w,w),
\end{equation}

for some $c_N\in \maclA _{\Sh ,\rho}^{(\omega _N)}(\cc {2d})$
with $\omega _N(z)= \omega (z)\eabs z^{-2N\rho}$.
Here we again assume $\rho > 0$.
The decay conditions on $b_\alpha$ and $c_N$ are, respectively,

\begin{align}
|\partial _w^\beta \overline \partial _w^\gamma b_\alpha (w)|
& \lesssim
\omega (\sqrt 2 \overline w)\eabs {w}^{-\rho |2\alpha +\beta +\gamma |},
\qquad
\alpha, \beta, \gamma  \in \nn d,
\label{Eq:WickAntiWickbalphaIntro}
\intertext{and}
|\partial _z^\beta \overline \partial _w^\gamma c_N(z,w)|
& \lesssim
e^{\frac 12|z-w|^2}\omega (\sqrt 2 \overline z)\eabs z^{-2N\rho}
\eabs {z+w}^{-\rho |\beta +\gamma |} \eabs {z-w}^{-N}.
\end{align}

Consequently, many Wick operators can essentially be expressed
as linear combinations of anti-Wick operators. The expansion
(<ref>) is deduced in Section <ref> using
Taylor expansion and integration by parts, see Proposition
<ref> and Remark <ref>.

The conditions on $b_\alpha$ are the same as the conditions on $a$
(<ref>), restricted to the diagonal $z=w$, and with improved decay.
On the diagonal, the growth term $e^{\frac 12|z-w|^2}$ disappears, which dominates in
(<ref>) when $|z-w|\gtrsim |z|$ or $|z-w|\gtrsim |w|$.
The right-hand side of (<ref>)
becomes as large as possible when
$\alpha =\beta =\gamma =0$, that is $b_0$
is the dominating term in the sum (<ref>).

The conditions on $c_N$ are the same as the estimates (<ref>)
again with improved decay due to the factor $\eabs z^{-2N\rho}$.

For polynomial symbols, (<ref>)
agree with the integral formula <cit.> due to Berezin
which carry over Wick operators into anti-Wick operators. For
the general case, (<ref>) is analogous to the
approximation technique of pseudo-differential operators on $\rr d$
in terms of Toeplitz operators given in <cit.>
and its proof, by Shubin.

The anti-Wick symbols in (<ref>)
$b_\alpha (w)=\partial _z^\alpha \overline \partial _w^\alpha a (w,w)$ extend to have the property that
$\partial _z^\alpha \overline \partial _w^\alpha a (z,w)$ is entire in
$z$ and conjugate entire in $w$.
Note that restriction to the diagonal also appears in the positivity condition
(<ref>) on Wick symbols.

The sharp Gårding inequality (<ref>) is
reached by using the fact that $\op _{\mathfrak V}^{\aw}(b_0)$
is non-negative,
and that
if $T$ is either $\op ^{\aw}_{\mathfrak V}(b_\alpha )$
or $\op _{\mathfrak V}(c_N)$ for $\alpha \neq 0$, then
$\nm {TF}{A^2}\lesssim \nm F{A^2}$ when $F\in A(\cc d)$ is a polynomial.

In Section <ref> we deduce links concerning ellipticity, hypoellipticity
(in Shubin's sense) and weak ellipticity between Shubin and Wick symbols.
The notion of hypoelliptic symbol resembles hypoelliptic symbols
in Shubin's sense (see [25]).
More specifically, we say that the symbol $\fka \in \Sh _\rho ^{(\omega )}(\rr {2d})$ is
hypoelliptic of order $\rho _0\ge 0$, whenever there is an $R>0$ such that

|\fka (x,\xi )|\gtrsim \omega (x,\xi )\eabs {(x,\xi )}^{-\rho _0}
\quad \text{and}\quad |\partial ^\alpha \fka (x,\xi )|
\lesssim |\fka (x,\xi )|\eabs {(x,\xi )}^{-\rho |\alpha |}

when  $|(x,\xi )|\ge R$.

A linear operator $T$ from $\mascS '(\rr d)$ to $\mascS '(\rr d)$
is called globally hypoelliptic if

Tf=g,\ f\in \mascS '(\rr d),\ g\in \mascS (\rr d)
\quad \Rightarrow \quad
f\in \mascS (\rr d).

(See e. g. [12].)
It can be proved that a pseudo-differential operator with hypoelliptic
symbol in Shubin's sense is globally hypoelliptic as operator (see e. g.

We show, similarly to our investigations of
the sharp Gårding
inequality and for expansion (<ref>), that ellipticity, hypoellipticity
and to some degree weak ellipticity for the Shubin symbol $\fka$
can be characterized by certain conditions for the corresponding Wick symbol
$a(z,w)$ along the diagonal $z=w$. For example, let
$\fka$ be a polynomial on $\rr d$ with principal symbol $\fka _p$, and let $a(z,w)$
be a polynomial in $z, \overline w \in \cc d$ with principal part
$a_p$. Then $\fka$ is elliptic means that
$\fka _p(x,\xi )\neq 0$ when $(x,\xi )\neq (0,0)$, and
$a$ is elliptic means that $a_p(z,z)\neq 0$ when $z\neq 0$.
For such $\fka$ we prove

\fka \ \text{is elliptic}
\quad \Leftrightarrow \quad
a\ \text{is elliptic},

when $a(z,w)$ is the Wick symbol corresponding to $\fka$
(which must be a polynomial in $z$ and $\overline w$).

Our investigations include the Bargmann transform of certain
operators of infinite order, i. e. pseudo-differential operators
with ultra-differentiable symbols that are permitted to grow faster
than polynomially at infinity together with their derivatives. Particularly
we consider Wick operators of infinite order, i. e. the Bargmann
images $\op _{\mathfrak V}(a)$ of operators $\op (\fka )$ of infinite order
in [1],
and characterize their images under the  Bargmann transform
(see Theorem <ref>).

Then we deduce in Subsections
<ref> and <ref> continuity results for anti-Wick operators
which holds for the symbols $b_\alpha$ in (<ref>) when
$\op _{\mathfrak V}(a)$ is the Bargmann image of an
operator of infinite order.

In fact, in Subsection <ref>
we show that $\op _{\mathfrak V}^{\aw}(b_\alpha )$ possess several
other continuity properties than what is valid for $\op _{\mathfrak V}(a)$
in the expansion (<ref>)
(see Propositions <ref> and <ref>). In Subsection
<ref> we deduce estimates of the Wick symbol $b_{\alpha}^{\aw}$
to the anti-Wick operator $\op _{\mathfrak V}^{\aw}(b_\alpha )$, i. e. the unique element
$b_{\alpha}^{\aw}\in \wideparen A(\cc {2d})$ such that $\op _{\mathfrak V}(b_{\alpha}^{\aw})
=\op _{\mathfrak V}^{\aw}(b_\alpha )$. We show that usually, $b_{\alpha}^{\aw}$
satisfies stronger conditions than $a$ when $\op _{\mathfrak V}(a)$
is a Wick operator of infinite order (see Theorems <ref>,
<ref> and <ref>).

The paper is organized as follows. In Section <ref> we
recall useful properties for weight functions, Gelfand-Shilov spaces, the Bargmann
transform, pseudo-differential operators, Wick and anti-Wick operators.
Thereafter we characterize in Section <ref> Shubin operators
and operators of infinite order in terms of appropriate classes of
Wick operators on the Bargmann side.
These considerations are based on a formula for the Wick symbol expressed
in terms of a short-time Fourier transform of the Weyl symbol, and
admits characterization of the Wick symbols corresponding to
Shubin Weyl symbols and symbols for operators of infinite order
(see Proposition <ref>).

In Section <ref> we also study composition and show for example
that the well-known closure under composition of Shubin operators
and operators of infinite orders have simple and natural proofs on the Wick symbol side.

In Section <ref> we deduce series expansions of Wick operators
in terms of anti-Wick operators, and between Wick symbols and symbols
to corresponding Shubin operators.
We also consider anti-Wick operators, and
show continuity results for them. We show
that the upper bounds for the Wick symbols of anti-Wick operators
are stricter than for general Wick symbols.

In Section <ref> we discuss lower bounds for Wick operators
and deduce the sharp Gårding's inequality.
Section <ref> concerns ellipticity, hypoellipticity and weak ellipticity.

Finally we observe in Section <ref> that a polynomial bound of a Wick symbol implies
that the symbol is a polynomial. For pseudo-differential operators this
corresponds to partial differential operators with polynomial coefficients.
This gives a characterization of such operators as those having
polynomially bounded Wick symbols.

§ PRELIMINARIES

In this section we recall some facts on function and distribution
spaces as well as on pseudo-differential operators, Wick and anti-Wick operators.
Subsection <ref> concerns weight functions and
Subsection <ref> treats Gelfand-Shilov spaces.
In Subsection <ref> we introduce the Bargmann transform and
topological spaces of entire functions on $\cc d$, and
in Subsection <ref> we recall the definitions
and some facts on pseudo-differential operators on $\rr d$ as well
as Wick and anti-Wick operators on $\cc d$.
Subsection <ref> defines certain symbol classes for pseudo-differential
operators on $\rr d$.

§.§ Weight functions

A weight on $\rr d$ is a positive function $\omega \in  L^\infty _{loc}(\rr d)$
such that $1/\omega \in  L^\infty _{loc}(\rr d)$. The weight $\omega$
is called moderate if there is a positive locally bounded function
$v$ such that

\begin{equation}\label{eq:2}
\omega(x+y)\le C\omega(x)v(y),\quad x,y\in\rr{d},
\end{equation}

for some constant $C\ge 1$. If $\omega$ and $v$ are weights such
that (<ref>) holds, then $\omega$ is also called $v$-moderate.
The set of all moderate weights on $\rr d$ is denoted by $\mascP _E(\rr d)$.
The set $\mascP (\rr d)$ consists of weights that are $v$-moderate for
a polynomially bounded weight, that is a weight of the form $v(x) = {\eabs x}^s$
where $\eabs x = (1+|x|^2)^{\frac 12}$ and $s \ge 0$.
The bracket notation is also used for complex arguments as
$\eabs z=(1+|z|^2)^{\frac 12}$ when $z\in \cc d$.
If $s\in \mathbf R$ then $x\mapsto \eabs x^s$ belongs to $\mascP (\rr d)$,
due to Peetre's inequality

\begin{equation}\label{eq:Peetre}
\eabs{x+y}^s \leqs 2^{|s|} \eabs{x}^s\eabs{y}^{|s|}\qquad x,y\in\rr{d},
\qquad s \in \ro.
\end{equation}

The weight $v$ is called submultiplicative
if it is even and (<ref>)
holds for $\omega =v$.
If (<ref>) holds and $v$ is submultiplicative then

\begin{equation}\label{eq:2Next}
\begin{gathered}
\frac {\omega (x)}{v(y)} \lesssim \omega(x+y) \lesssim \omega(x)v(y),
\\[1ex]
\quad
v(x+y) \lesssim v(x)v(y)
\quad \text{and}\quad v(x)=v(-x),
\quad x,y\in\rr{d}.
\end{gathered}
\end{equation}

The notation
$A(\theta )\lesssim B(\theta )$, $\theta \in \Omega$,
means that there is a constant $c>0$ such that $A(\theta )\le cB(\theta )$
for all $\theta \in \Omega$.

If $\omega$ is a moderate weight then by [29] there is a
submultiplicative weight
$v$ such that (<ref>) and (<ref>)
hold. If $v$ is submultiplicative then

\begin{equation}\label{Eq:CondSubWeights}
1\lesssim v(x) \lesssim e^{r|x|}
\end{equation}

for some constant $r>0$ (cf. [16]). In particular, if $\omega$ is moderate, then

\begin{equation}\label{Eq:ModWeightProp}
\omega (x+y)\lesssim \omega (x)e^{r|y|}
\quad \text{and}\quad
e^{-r|x|}\lesssim \omega (x)\lesssim e^{r|x|},\quad
x,y\in \rr d
\end{equation}

for some $r>0$.
If not otherwise specified the symbol $v$ always denote a submultiplicative weight.

§.§ Gelfand-Shilov spaces

Let $s ,\sigma > 0$. The
Gelfand-Shilov space $\maclS _s^\sigma (\rr d)$ ($\Sigma _s^\sigma (\rr d)$) of
Roumieu (Beurling) type consists of all $f\in C^\infty (\rr d)$
such that

\begin{equation}\label{gfseminorm}
\nm f{\maclS _{s,h}^\sigma}\equiv \sup \frac {|x^\alpha \partial ^\beta
f(x)|}{h^{|\alpha  + \beta |}\alpha !^s \, \beta !^\sigma}
\end{equation}

is finite for some (every) $h>0$. The supremum refers to all
$\alpha ,\beta \in \mathbf N^d$ and $x\in \rr d$.
The seminorms
$\nm \cdo {\maclS _{s,h}^\sigma}$ induce an inductive limit topology for the
space $\maclS _s^\sigma (\rr d)$ and a projective limit topology for
$\Sigma _s^\sigma (\rr d)$. The latter space is a Fréchet space under this topology.
The space $\maclS _s^\sigma (\rr d)\neq \{ 0\}$ ($\Sigma _s^\sigma (\rr d)\neq \{0\}$),
if and only if $s+\sigma \ge 1$ ($s+\sigma \ge 1$ and $(s,\sigma) \neq (\frac 12, \frac 12)$).
We write $\maclS _s (\rr d) = \maclS _s^s (\rr d)$ and $\Sigma _s (\rr d) = \Sigma _s^s (\rr d)$.

The Gelfand-Shilov distribution spaces $(\maclS _s^\sigma )'(\rr d)$
and $(\Sigma _s^\sigma )'(\rr d)$ are the dual spaces of $\maclS _s^\sigma (\rr d)$
and $\Sigma _s^\sigma (\rr d)$, respectively.

The embeddings

\begin{multline}\label{GSembeddings}
\maclS _{s_1}^{\sigma _1} (\rr d) \hookrightarrow \Sigma _{s_2}^{\sigma _2}(\rr d)
\hookrightarrow
\maclS _{s_2}^{\sigma _2} (\rr d)
\hookrightarrow
\mascS (\rr d)
\\[1ex]
\hookrightarrow \mascS '(\rr d) %\hspace{2cm} \phantom k
\hookrightarrow  (\maclS _{s_2}^{\sigma _2})'(\rr d)
\hookrightarrow  (\Sigma _{s_2}^{\sigma _2})'(\rr d)
\hookrightarrow (\maclS _{s_1}^{\sigma _1}) '(\rr d),
\\[1ex]
s_1+\sigma _1 \ge 1,\ s_1<s_2,\ \sigma _1<\sigma _2,
\end{multline}

are dense.
For topological spaces $A$ and $B$, $A\hookrightarrow B$ means that the inclusion $A\subseteq B$ is continuous.

The spaces $\maclS _s$ and $\Sigma_s$, and their duals spaces, admit
characterizations in terms of  coefficients with respect to expansions with respect to
the Hermite functions

h_\alpha (x) = \pi ^{-\frac d4}(-1)^{|\alpha |}
(2^{|\alpha |}\alpha !)^{-\frac 12}e^{\frac {|x|^2}2}
(\partial ^\alpha e^{-|x|^2}),\quad \alpha \in \nn d.

The set of Hermite functions on $\rr d$ is an orthonormal basis for
$L^2(\rr d)$.
We use $\maclH _0(\rr d)$ to denote the space of finite linear combinations
of Hermite functions.
Then $\maclH _0(\rr d)$ is dense in the Schwartz space $\mascS (\rr d)$,
as well as in $\mascS '(\rr d)$, with respect to its weak$^*$ topology.
The same conclusion is true for $\Sigma _s(\rr d)$ when $s>\frac 12$,
$\maclS _s(\rr d)$ when $s\ge \frac 12$ and their distribution dual
spaces $\Sigma _s'(\rr d)$ and $\maclS _s'(\rr d)$.
An $f$ in any of these spaces possess an expansion of the form

\begin{equation}\label{Eq:HermiteExpansions}
f=\sum _{\alpha \in \nn d}c(f,\alpha )h_\alpha ,
\quad
c (f,\alpha )=(f,h_\alpha ), \quad \alpha \in \nn d.
\end{equation}

Here $(\cdo , \cdo )$ denotes the unique extensions of the $L^2$ form,
which is linear in the first variable and conjugate linear in the second variable,
from $\maclH _0(\rr d)\times \maclH _0(\rr d)$ to
$\maclS _s'(\rr d)\times \maclS _s(\rr d)$ or
$\Sigma _s'(\rr d)\times \Sigma _s(\rr d)$.
We recall that (cf. <cit.>)

\begin{equation}
\begin{alignedat}{5}
f&\in \mascS (\rr d) & \quad &\Leftrightarrow &\quad
|c(f,\alpha )| &\lesssim \eabs \alpha ^{-N} &
\ &\text{for every}& \ N&\ge 0 ,
\\[1ex]
f&\in \mascS '(\rr d) & \quad &\Leftrightarrow &\quad
|c(f,\alpha )| &\lesssim \eabs \alpha ^{N} &
\ &\text{for some}& \ N&\ge 0.
\end{alignedat}
\end{equation}

The topology on $\mascS (\rr d)$ is equivalent to the Fréchet space
topology defined by the sequence space seminorms

\begin{equation*}
\mascS (\rr d) \ni f \mapsto \sum_{\alpha \in \nn d}
\eabs \alpha ^{2N} |c(f,\alpha )|^2, \quad N \ge 0.
\end{equation*}

For $f \in \mascS '(\rr d)$ the sum in (<ref>)
converges in the weak$^*$ topology.

The Hermite functions are eigenfunctions to the harmonic
oscillator $H=H_d\equiv |x|^2-\Delta$ and to the Fourier transform
$\mathscr F$, given by

\mathscr Ff (\xi )= \widehat f(\xi ) \equiv (2\pi )^{-\frac d2}\int _{\rr
{d}} f(x)e^{-i\scal  x\xi }\, dx, \quad \xi \in \rr d,

when $f\in L^1(\rr d)$. Here $\scal \cdo \cdo$ denotes the
scalar product on $\rr d$. In fact

H_dh_\alpha = (2|\alpha |+d)h_\alpha, \quad \alpha \in \nn d.

The Fourier transform $\mathscr F$ extends
uniquely to homeomorphisms on $\mathscr S'(\rr d)$,
from $(\maclS _s^\sigma )'(\rr d)$ to $(\maclS ^s_\sigma )'(\rr d)$ and
from $(\Sigma _s^\sigma )'(\rr d)$ to $(\Sigma ^s_\sigma )'(\rr d)$.
It also restricts to
homeomorphisms on $\mathscr S(\rr d)$, from
$\maclS _s^\sigma (\rr d)$ to $\maclS ^s_\sigma (\rr d)$, from
$\Sigma _s^\sigma (\rr d)$ to $\Sigma ^s_\sigma (\rr d)$,
and to a unitary operator on $L^2(\rr d)$. Similar facts hold true
when the Fourier transform is replaced by a partial
Fourier transform.

Let $\phi \in \mascS  (\rr d) \setminus 0$ be
fixed. We use the transform

\begin{equation}\label{eq:cTdef}
\begin{aligned}
\cT_\phi  f(x,\xi) &= (2\pi)^{-\frac d2}e^{i \langle x, \xi \rangle}
(f,e^{i\scal \cdo \xi}\phi (\cdo -x))
\\[1ex]
e^{i \langle x, \xi \rangle}\mascF (f\cdot \overline {\phi (\cdo -x)})(\xi )
\mascF (f(\cdo +x)\overline \phi )(\xi ), \quad x, \xi \in \rr d,
\end{aligned}
\end{equation}

where $f \in \mascS '(\rr d)$ and $\phi  \in \mascS (\rr d) \setminus 0$
(cf. [9]). If $f ,\phi \in \mascS (\rr d)$ then

\begin{align*}
\cT _\phi f(x,\xi )
(2\pi )^{-\frac d2}e^{i \langle x, \xi \rangle}
\int _{\rr d} f(y)\overline {\phi
(y-x)}e^{-i\scal y\xi}\, dy
\\[1ex]
(2\pi )^{-\frac d2}
\int _{\rr d} f(y+x)\overline {\phi
(y)}e^{-i\scal y\xi}\, dy, \quad x,\xi \in \rr d.
\end{align*}

We notice that the short-time Fourier transform $V_{\phi}f$
of $f$ is given by

\begin{equation}\label{Eq:LinkSTFTandTmap}
V_{\phi}f(x,\xi) = e^{-i \langle x, \xi \rangle} \cT_\phi  f(x,\xi).
\end{equation}

Thus by <cit.> it follows that the definition of the map
$(f,\phi)\mapsto \cT _{\phi} f$ from $\mascS (\rr d) \times \mascS (\rr d)$
to $\mascS(\rr {2d})$ is uniquely extendable to a continuous map from
$\maclS _s'(\rr d)\times \maclS_s'(\rr d)$
to $\maclS_s'(\rr {2d})$, and restricts to a continuous map
from $\maclS _s (\rr d)\times \maclS _s (\rr d)$
to $\maclS _s(\rr {2d})$.
The same conclusion holds with $\Sigma _s$ in place of
$\maclS_s$, at each place.

The adjoint $\cT _\phi ^*$ is given by

(\cT _\phi^* F, g)_{L^2(\rr d)}
(F, \cT _\phi g)_{L^2(\rr {2d})}

for $F \in \maclS _s'(\rr {2d})$ and $g \in \maclS _s(\rr d)$,
and similarly with $\Sigma _s$ or with $\mascS$ in place of
$\maclS _s$ at each occurrence.
When $F$ is a polynomially bounded measurable function we

\begin{equation}\label{Eq:Moyalsformula}
\cT _\phi^* F(y) = (2\pi)^{-\frac d2} \iint _{\rr {2d}}
F(x,\xi) \, e^{i\scal {y-x}\xi}\phi (y-x)
\, d x d \xi ,
\end{equation}

where the integral is defined weakly so that
$(\cT _\phi^* F, g)_{L^2(\rr d)}
(F, \cT _\phi g)_{L^2(\rr {2d})}$
for $g \in \cS(\rr d)$.
The identity (<ref>) is called Moyal's formula.

We have

\begin{equation}\label{eq:reproducing}
(\cT_\psi ^*\circ \cT _\phi )f
=(\psi ,\phi ) f, \qquad f \in \maclS _s'(\rr d),\ \phi ,\psi \in \maclS _s(\rr d),
\end{equation}

and similarly with $\Sigma _s$ or with $\mascS$ in place of
$\maclS _s$ at each occurrence.

Two important features of $\cT _\phi$ which distinguish
it from the short-time Fourier transform are the differential identities

\begin{align}
\label{eq:diffident}
\partial_x^\alpha \cT _\phi f (x,\xi) & = \cT _\phi (\partial^\alpha f) (x,\xi),
\qquad \alpha \in \nn d
\intertext{and}
\label{eq:diffidentstar}
D_{\xi}^\beta \cT _\phi f (x,\xi) & = \cT_{g_\beta} f(x,\xi), \qquad \beta \in \nn d, \qquad
\phi _\beta (x) = (-x)^\beta \phi (x).
\end{align}

By (<ref>) it follows that
characterizations of Gelfand-Shilov spaces and their distribution spaces
in terms of estimates of their short-time Fourier transforms
carry over to estimates on $\cT _\phi$
in place of $V_\phi$. For example we have
the following (see e. g. [17, 26] for the proof of (1)
and [30] for the proof of (2)).
See also [11] for related results.

Let $s,\sigma >0$,
$\phi \in \maclS _s^\sigma (\rr d)\setminus 0$
($\phi \in \Sigma _s^\sigma (\rr d)\setminus 0$) and let
$f\in (\maclS _s^\sigma )'(\rr d)$ ($f\in (\Sigma _s^\sigma )'(\rr d)$).
Then the following is true:

* $f\in \maclS _s^\sigma (\rr d)$ ($f\in \Sigma _s^\sigma (\rr d)$) if and only if

\begin{equation}\label{stftexpest2}
|\cT _\phi f(x,\xi )| \lesssim  e^{-r (|x|^{\frac 1s}+|\xi |^{\frac 1\sigma})}, \quad x,\xi \in \rr d,
\end{equation}

for some (every) $r > 0$.

* $f\in (\maclS _s^\sigma )'(\rr d)$ ($f\in (\Sigma _s^\sigma )'(\rr d)$) if and only if

\begin{equation}\label{stftexpest2Dist}
|\cT _\phi f(x,\xi )| \lesssim  e^{r(|x|^{\frac 1s}+|\xi |^{\frac 1\sigma})}, \quad
x,\xi \in \rr d,
\end{equation}

for every (some) $r > 0$.

§.§ The Bargmann transform and spaces of analytic

If $\Omega \subseteq \cc d$ is open then $A(\Omega)$
consists of all (complex-valued) analytic functions on $\Omega$.
Complex derivatives are denoted, with $z = x+iy \in \Omega$,

\begin{equation*}
\partial_{z_j} = \frac{1}{2} \left( \partial_{x_j} - i \partial_{y_j} \right), \quad
\overline{\partial}_{z_j} = \frac{1}{2} \left( \partial_{x_j} + i \partial_{y_j} \right)
\end{equation*}

for $1 \le j \le d$, which admits the Cauchy-Riemann equations to be written as
$\overline{\partial}_{z_j} f = 0$, $1 \le j \le d$.

The Bargmann kernel is defined by

\mathfrak A_d(z,y)=\pi ^{-\frac d4} \exp \Big ( -\frac 12(\scal
zz+|y|^2)+2^{1/2}\scal zy\Big ), \quad z \in \cc d, \quad y \in \rr d,

\scal zw = \sum _{j=1}^dz_jw_j\quad \text{and} \quad
(z,w)= \scal z{\overline w}

z=(z_1,\dots ,z_d) \in \cc d\quad  \text{and} \quad w=(w_1,\dots ,w_d)\in \cc d.

Sometimes $\scal \cdo \cdo$ denotes the duality between a test function
space and its dual. The context precludes confusion between its double use.
The Bargmann transform $\mathfrak V_df$ of $f\in \maclS _{1/2}'(\rr d)$
is the entire function

\begin{equation}\label{bargdistrform}
\mathfrak V_d f (z) =\scal f{\mathfrak A_d(z,\cdo )},
\quad z \in \cc d.
\end{equation}

The right-hand side is a well defined element in $A(\cc d)$,
since $y\mapsto \mathfrak A_d(z,y)$ belongs to
$\maclS _{1/2} (\rr d)$ for $z \in \cc d$ fixed, and
$\mathfrak A_d(\cdo,y)$ is entire for all $y \in \rr d$.
Let $p\in [1,\infty ]$ and $\omega \in \mascP _E(\rr d)$. Then
$L^p_{(\omega )}(\rr d)$ consists of all $f\in L^1_{loc}(\rr d)$ such
that $\nm f{L^p_{(\omega )}}\equiv \nm {f\cdot \omega }{L^p}$
is finite. If $f\in L^p_{(\omega )}(\rr d)$, then

\begin{multline}
\mathfrak V_d f (z) =\int_{\rr d} \mathfrak A_d(z,y)f(y)\, dy
\\[1ex]
\pi ^{-\frac d4}\int _{\rr d}\exp \Big ( -\frac 12(\scal
z z+|y|^2)+2^{1/2}\scal zy \Big )f(y)\, dy,\quad z \in \cc d.
\end{multline}

(Cf. [4, 5, 29, 30].)

For  $p\in (0,\infty]$, $\omega \in \mascP _E(\cc d)$ and
$\omega _0(z)=\omega (\sqrt 2\overline z)$, let
$A^p_{(\omega )}(\cc d)$ be the set of all $F\in A(\cc d)$
such that

\begin{equation*}
\nm F{A^p_{(\omega )}}
\equiv \pi ^{-\frac dp}
\nm {F\cdot e^{-\frac 12|\cdo |^2}\cdot \omega _0}{L^p},
\end{equation*}

and set $A^p=A^p_{(\omega )}$ when $\omega =1$.
It was proved by Bargmann [4] that

\begin{equation}\label{Eq:L2A2Isometry}
\mathfrak V_d : L^2(\rr d) \to A^2(\cc d)
\end{equation}

is bijective and isometric. The space $A^2(\cc d)$ is the Hilbert space
of entire functions with scalar product

\begin{equation*}
(F,G)_{A^2}\equiv  \int _{\cc d} F(z)\overline {G(z)}\, d\mu (z),\quad F,G\in A^2(\cc d),
\end{equation*}

where $d\mu (z)=\pi ^{-d} e^{-|z|^2}\, d\lambda (z)$ and $d\lambda (z)$ is
the Lebesgue measure on $\cc d$. The space $A^2(\cc d)$ is known as the
Fock space in quantum mechanics (see[13]).

In [4] it was proved that the Bargmann transform maps the Hermite functions
to monomials as

\begin{equation}\label{Eq:BargmannHermiteMap}
\mathfrak V_dh_\alpha = e_\alpha ,\qquad e_\alpha (z)= \frac {z^\alpha}{\alpha !^{\frac 12}},
\quad z\in \cc d,\quad \alpha \in \nn d.
\end{equation}

The orthonormal basis
$\{ h_\alpha \}_{\alpha \in \nn d} \subseteq L^2(\rr d)$
is thus mapped to the orthonormal basis
$\{ e_\alpha \} _{\alpha \in \nn d}\subseteq A^2(\cc d)$.
Bargmann also proved that there is a reproducing formula for
$A^2(\cc d)$. Let $\Pi _A$ be the operator from $L^2(d\mu )$
to $A(\cc d)$, given by

\begin{equation}\label{eq:projection}
\Pi _A F (z)= \int _{\cc d} F(w)e^{(z,w)}\, d\mu (w),
\quad z \in \cc d.
\end{equation}

Then $\Pi _A$ is the orthogonal projection from
$L^2(d\mu)$ to
$A^2(\cc d)$ (cf. [4]).

When we discuss extensions and restrictions of the
Bargmann transform to Gelfand-Shilov spaces and their
distribution spaces, we use

\begin{equation}\label{Eq:GevreyQuasinormMIxed}
|z|_{s,\sigma} = |\operatorname{Re} z|^{\frac 1s}
|\operatorname{Im} z|^{\frac 1\sigma},
\qquad
z\in \cc d,
\end{equation}

and consider the seminorms

when $F\in A(\cc d)$, $r>0$ and $s,\sigma \ge  \frac 12$. Then
$\maclA _{0,s}^\sigma (\cc d)$ for
$s, \sigma > \frac 12$,
$\maclA _{\mascS} (\cc d)$ and
$(\maclA _s^\sigma )' (\cc d)$ for
$s,\sigma \ge \frac 12$
are the sets of all
$F\in A(\cc d)$ such that

\begin{equation*}
\nm F{\maclA _{\maclS _{s;r}^\sigma}}<\infty,
\quad
\nm F{\maclA _{\mascS ;r}}<\infty
\quad \text{and}\quad
\nm F{\maclA _{\maclS _{s;r}^\sigma}'}<\infty,
\end{equation*}

respectively, for every $r>0$.
The spaces are equipped with the
projective limit topology with respect to $r>0$,
defined by each class of seminorms, respectively.

In the same way we
let $\maclA _{s}^\sigma (\cc d)$ for
$s,\sigma \ge \frac 12$,
$\maclA _{\mascS}' (\cc d)$ and
$(\maclA _{0,s}^\sigma )' (\cc d)$ for $s,\sigma >\frac 12$
be the sets of all
$F\in A(\cc d)$ such that

\begin{equation*}
\nm F{\maclA _{\maclS _{s;r}^\sigma}}<\infty,
\quad
\nm F{\maclA _{\mascS ;r}'}<\infty
\quad \text{and}\quad
\nm F{\maclA _{\maclS _{s;r}^\sigma}'}<\infty ,
\end{equation*}

respectively, for some $r>0$.
Their topologies are the inductive limit
topologies with respect to $r>0$,
defined by each class of seminorms, respectively.
We also set

\maclA _{0,s} =\maclA _{0,s}^s
\quad \text{and}\quad
\maclA _{s} =\maclA _{s}^s.

𝔙_d      :
(d)    →_(d),
𝔙_d      :
'(d)    →_'(d),

𝔙_d      :
_s^σ(d)    →_s^σ(d),
𝔙_d      :
→(_s^σ)' (d)
s,σ   ≥1/2
𝔙_d      :
𝔙_d      :
→(_0,s^σ)' (d),
s,σ   > 1/2

are homeomorphisms [30].

From these homeomorphisms, the fact that the map
(<ref>) is a homeomorphism and
duality properties for Gelfand-Shilov spaces, it follows that
$(\cdo ,\cdo )_{A^2}$ on $\maclA _{1/2}(\cc d)\times
\maclA _{1/2}(\cc d)$ is uniquely extendable to
a continuous sesqui-linear form on
$(\maclA _s^\sigma )'(\cc d)\times \maclA _s^\sigma (\cc d)$.
The dual of $\maclA _s^\sigma (\cc d)$
can be identified with $(\maclA _s^\sigma )'(\cc d)$ through
this form. Similar facts hold for $\maclA _{0,s}^\sigma$
in place of $\maclA _s^\sigma$ at each occurrence.
(Cf. e. g. [29, 30].)

Finally let $\maclA _{\flat _1;r}(\cc d)$ and $\maclA _{\flat _\infty ;r}(\cc d)$
for $r > 0$ be the Banach spaces which consist
of all $F\in A(\cc d)$ such that

\nm F{\maclA _{\flat _1;r}}\equiv \nm {F\cdot e^{-r|\cdo |}}{L^\infty}
\quad \text{respectively}\quad
\nm F{\maclA _{\flat _\infty ;r}}\equiv \nm {F\cdot e^{-r|\cdo |^2}}{L^\infty}

is finite, and let $\maclA _{\flat _1}(\cc d)$ be the inductive limit
of $\maclA _{\flat _1;r}(\cc d)$ with respect to $r>0$. Also let
$\maclA _{0,\flat _\infty}(\cc d)$ and $\maclA _{0,\flat _\infty}'(\cc d)$
be the projective respectively inductive limit topologies of
$\maclA _{\flat _\infty ;r}(\cc d)$ with respect to $r>0$.

It is evident that
$\maclA _{\flat _1}(\cc d)$ is densely embedded in $\maclA _s^\sigma (\cc d)$
for every $s,\sigma \ge \frac 12$, as well as in $\maclA _{0,s}^\sigma (\cc d)$
for every $s,\sigma > \frac 12$. The form
$(\cdo ,\cdo )_{A^2}$ on $\maclA _{\flat _1}(\cc d)\times
\maclA _{\flat _1}(\cc d)$ is uniquely extendable to
a continuous sesqui-linear form on
$A(\cc d)\times \maclA _{\flat _1} (\cc d)$ and the dual
of $\maclA _{\flat _1} (\cc d)$ can be identified with $A(\cc d)$.
The Fréchet space topology of $A(\cc d)$
can be defined by the seminorms

F\mapsto  \sup _{|z|\le N}|F(z)|,\qquad N=1,2,\dots .

(Cf. [30].)

The spaces $\maclA _{\flat _1}(\cc d)$ and $\maclA _{0,\flat _\infty}(\cc d)$
are examples of Bargmann images of special Pilipović spaces, a family
of Fourier invariant topological vector spaces which are smaller than any Fourier
invariant Gelfand-Shilov space, and which were introduced
and investigated in [30]. For any $\sigma >0$, the Bargmann image
of the Pilipović spaces $\maclH _{\flat _\sigma}(\rr d)$ and
$\maclH _{0,\flat _\sigma}(\rr d)$ are given by

\begin{align*}
\maclA _{\flat _\sigma}(\cc d)
& \equiv
\sets {F\in A(\cc d)}{|F(z)|\lesssim e^{r|z|^{\frac {2\sigma}{\sigma +1}}}
\ \text {for some}\ r>0 }
\intertext{respectively}
\maclA _{0,\flat _\sigma}(\cc d)
& \equiv
\sets {F\in A(\cc d)}{|F(z)|\lesssim e^{r|z|^{\frac {2\sigma}{\sigma +1}}}
\ \text {for every}\ r>0 }.
\end{align*}

If $\sigma >1$, then the (strong) duals of $\maclA _{\flat _\sigma}(\cc d)$ and
$\maclA _{0,\flat _\sigma}(\cc d)$ are given by

\begin{align*}
\maclA _{\flat _\sigma}'(\cc d)
& \equiv
\sets {F\in A(\cc d)}{|F(z)|\lesssim e^{r|z|^{\frac {2\sigma}{\sigma -1}}}
\ \text {for every}\ r>0 }
\intertext{respectively}
\maclA _{0,\flat _\sigma}'(\cc d)
& \equiv
\sets {F\in A(\cc d)}{|F(z)|\lesssim e^{r|z|^{\frac {2\sigma}{\sigma -1}}}
\ \text {for some}\ r>0 }
\end{align*}

through a unique extension of the $A^2$ scalar product on
$\maclA _{\flat _1}(\cc d)\times \maclA _{\flat _1}(\cc d)$.
In particular, if $\sigma$ tends to $\infty$, it follows that some of
these conditions tend to

\begin{align*}
\maclA _{0,\flat _\infty}(\cc d)
& \equiv
\sets {F\in A(\cc d)}{|F(z)|\lesssim e^{r|z|^2}
\ \text {for every}\ r>0 }
\intertext{respectively}
\maclA _{0,\flat _\infty}'(\cc d)
& \equiv
\sets {F\in A(\cc d)}{|F(z)|\lesssim e^{r|z|^2}
\ \text {for some}\ r>0 }.
\end{align*}

Note that in [30, 28], the set $\maclA _{0,\flat _\infty}(\cc d)$
is denoted by $\maclA _{0,\frac 12}(\cc d)$, and its dual
$\maclA _{0,\flat _\infty}'(\cc d)$ is denoted by $\maclA _{0,\frac 12}'(\cc d)$.

At many places it will be crucial to use the Gaussian window

\begin{equation}\label{Eq:phidef}
\phi (x)=\pi ^{-\frac d4}e^{-\frac 12|x|^2}, \quad x\in \rr d,
\end{equation}

in the transform $\cT_\phi$.
For this $\phi$ the relationship between
the Bargmann transform and $\cT _\phi$ is

\begin{equation}\label{bargstft1}
\mathfrak V_d = U_{\mathfrak V}\circ \cT _\phi ,\quad \text{and}\quad
U_{\mathfrak V}^{-1} \circ \mathfrak V_d =  \cT _\phi ,
\end{equation}

where $U_{\mathfrak V}$ is the linear, continuous and bijective operator on
$\mathscr D'(\rr {2d})\simeq \mathscr D'(\cc d)$, given by

\begin{equation}\label{UVdef}
U_{\mathfrak V} F (x+i\xi ) = (2\pi )^{\frac d2} e^{\frac 12(|x|^2+|\xi |^2)}e^{i\scal x\xi}
F(\sqrt 2\, x,-\sqrt 2\, \xi ), \quad x,\xi \in \rr d,
\end{equation}

cf. [29] in combination with (<ref>).

In analytic operator theory we need subspaces of

\wideparen A(\cc {2d})
\equiv
\Sets {\Theta K}
{K\in A(\cc {2d})},

where the semi-conjugation operator is

\begin{equation}\label{Eq:ThetaOpDef}
(\Theta K) (z,w) = K(z,\overline w),\qquad z,w\in \cc d.
\end{equation}

If $T$ is a linear and continuous operator
from $\maclS _{1/2}(\rr d)$ to $\maclS _{1/2}'(\rr d)$,
then there is a unique $K\in \wideparen A(\cc {2d})$
such that $\Theta K\in \maclA _{1/2}'(\cc {2d})$
and $\mathfrak V_d \circ T \circ \mathfrak V_d^{-1}$
is given by

\begin{equation}\label{Eq:CompIntOp}
F(z)\mapsto \int _{\cc d}K(z,w)F(w)\, d\mu (w).
\end{equation}

(See e. g. [28].) For these reasons we let

\wideparen \maclA _{0,s}(\cc {2d}),
\quad
\wideparen \maclA _s(\cc {2d}),
\quad
\wideparen \maclA _{\mascS}(\cc {2d}),
\quad
\wideparen \maclA _{\mascS}'(\cc {2d}),
\quad
\wideparen \maclA _s'(\cc {2d})
\quad \text{and}\quad
\wideparen \maclA _{0,s}'(\cc {2d})

be the images of

\maclA _{0,s}(\cc {2d}),
\quad
\maclA _s(\cc {2d}),
\quad
\maclA _{\mascS}(\cc {2d}),
\quad
\maclA _{\mascS}'(\cc {2d}),
\quad
\maclA _s'(\cc {2d})
\quad \text{and}\quad
\maclA _{0,s}'(\cc {2d})

respectively, under the map $\Theta$. We also let $\wideparen A^p(\cc {2d})$
and $\wideparen \maclA _{\flat _1}(\cc {2d})$
be the images of $A^p(\cc {2d})$ and $\maclA _{\flat _1}(\cc {2d})$,
respectively, under the map $\Theta$.
The topologies of the former spaces are
inherited from the corresponding latter spaces.

The semi-conjugated
Bargmann (SCB) transform is defined as

\mathfrak V_{\Theta ,d}
\Theta \circ \mathfrak V_{2d}.

All properties of the Bargmann transform
carry over naturally to analogous properties for the SCB transform.

§.§ Pseudo-differential operators

Let $A$ be a real $d\times d$ matrix.
The pseudo-differential operator
$\op _A(\fka)$ with symbol
$\fka \in \maclS _{1/2} (\rr {2d})$ is the linear and continuous operator on
$\maclS _{1/2} (\rr d)$ given by

\begin{equation}\label{e0.5}
\op _A(\fka )f (x)
(2\pi  ) ^{-d}\iint \fka (x-A(x-y),\xi )
f(y)e^{i\scal {x-y}\xi }\,
dyd\xi, \quad x\in \rr d.
\end{equation}

For $\fka \in \maclS _{1/2}'(\rr {2d})$ the
pseudo-differential operator $\op _A(\fka )$
is defined as the continuous
operator from $\maclS _{1/2}(\rr d)$ to $\maclS _{1/2}'(\rr d)$ with
distribution kernel

\begin{equation}\label{atkernel}
K_{\fka ,A}(x,y)
(2\pi )^{-\frac d2} \mascF _2^{-1}\fka (x-A(x-y),x-y), \quad
x,y \in \rr d,
\end{equation}

where $\mascF _2F$ is the partial Fourier
transform of $F(x,y)\in
\maclS _{1/2}'(\rr {2d})$ with respect to the $y$
variable. This definition makes sense since the

\begin{equation}\label{homeoF2tmap}
\mascF _2\quad \text{and}\quad F(x,y)\mapsto F(x,x-y)
\end{equation}

are homeomorphisms on $\maclS _{1/2}'(\rr {2d})$.
The map $a\mapsto K_{\fka ,A}$ is
hence a homeomorphism on $\maclS _{1/2}'(\rr {2d})$.

If $A$ and $B$ are real $d\times d$ matrices and
$\fka \in \maclS _{1/2}'(\rr {2d})$,
then there is a unique $\fkb \in \maclS _{1/2}'(\rr {2d})$ such that
$\op _A(\fka )= \op _B(\fkb )$, and that $\fkb$ can be obtained by

\begin{equation}\label{Eq:RealPseudoCalculiTransfer}
\op _A(\fka )= \op _B(\fkb )
\quad \Leftrightarrow \quad
e^{i\scal {AD_\xi }{D_x}}\fka (x,\xi )
e^{i\scal {BD_\xi }{D_x}}\fkb (x,\xi )
\end{equation}

(see [18, 10]).

By Fourier's inversion formula, (<ref>) and the
kernel theorem
<cit.>, <cit.> for
operators from
Gelfand-Shilov spaces to their duals,
it follows that the map $\fka \mapsto \op _A(\fka )$
is bijective from $\maclS _{1/2}'(\rr {2d})$
to the set of all linear and continuous operators from
$\maclS _{1/2}(\rr d)$ to $\maclS _{1/2}'(\rr {2d})$.

If $A=0$ then
$\op _A(\fka ) = \op _0(\fka ) = \op (\fka ) = \fka (x,D)$
is the Kohn-Nirenberg or standard representation.
If $A=\frac 12 I_d$
where $I_d$ is the $d\times d$ identity matrix
then $\op _A(\fka ) = \op ^w(\fka )$ is the Weyl quantization.
In this paper we use mainly the Weyl quantization and we put

\begin{equation*}
K_{\fka }^w = K_{\fka ,I_d/2} \ .
\end{equation*}

The Weyl product $\fka  \wpr \fkb$ of two Weyl symbols $\fka ,\fkb
\in \maclS _{1/2}(\rr {2d})$ is defined
as the product of symbols corresponding to operator composition. Thus

\begin{equation*}
\op ^w(\fka \wpr \fkb ) = \op ^w(\fka ) \circ \op ^w(\fkb )
\end{equation*}

and the Weyl product can be extended to larger spaces as long as
composition is well defined.

Next we recall the definition of Wick operators. Suppose that
$a\in \wideparen A(\cc {2d})$ satisfies

\begin{equation}\label{Eq:AntiWickL1Cond}
w\mapsto a(z,w)e^{r|w|-|w|^2} \in L^1(\cc d)
\end{equation}

locally uniformly with respect to $z\in \cc d$ for every $r>0$. Then the
analytic pseudo-differential operator,
or Wick operator $\op _{\mathfrak V}(a)$ with symbol $a$
and acting on $F\in \maclA _{\flat _1}(\cc d)$, is defined by

\begin{align}
\op _{\mathfrak V}(a)F(z) &= \int _{\cc d} a(z,w)F(w)e^{(z,w)}\, d\mu (w),
\quad z \in \cc d.
\label{Eq:AnalPseudo}
\end{align}

(Cf. e. g. [6, 13, 28, 29, 30].)
The condition (<ref>) and
$F\in \maclA _{\flat _1}(\cc d)$ imply that the integrand on
the right-hand side of (<ref>) is well defined.
The locally uniform
condition (<ref>) with respect to $z\in \cc d$
implies that $\op _{\mathfrak V}(a)F \in A(\cc d)$.

In [28] several extensions and restrictions of
$\op _{\mathfrak V}(a)$ are given.
The following result follows
from <cit.>. Here
$\mathcal L(\maclA _{\flat _1}(\cc d),A(\cc d))$
is the space of all linear and continuous operators from
$\maclA _{\flat _1}(\cc d)$ to $A(\cc d)$.

The map $a\mapsto \op _{\mathfrak V}(a)$
from $\wideparen \maclA _{\flat _1}(\cc {2d})$ to
$\mathcal L(\maclA _{\flat _1}(\cc d),A(\cc d))$ is uniquely
extendable to a bijective map from
$\wideparen A(\cc {2d})$ to
$\mathcal L(\maclA _{\flat _1}(\cc d),A(\cc d))$.

Let $L_A(\cc {2d})$ be the set of all $a \in L^1_{\rm loc}(\cc {2d})$ such that
$z\mapsto a(z,w)$ is entire for almost every $w\in \cc d$ and

\begin{equation}\label{Eq:LACond}
w \mapsto
\sup _{\alpha \in \nn d}
\left |
\frac { \partial _z^\alpha a(z,w)\cdot e^{r|w|-|w|^2}}{h^{|\alpha |}\alpha !}
\right |
\in L^1(\cc d)
\end{equation}

for every $h,r>0$ and $z\in \cc d$.
If $a\in \wideparen A(\cc {2d})$ satisfies (<ref>) then
$a\in L_A(\cc {2d})$ as a consequence of Cauchy's integral formula.
Thus $L_A(\cc {2d})$ is a relaxation of the former condition.

If $a\in L_A(\cc {2d})$ then $\op _{\mathfrak V}(a):
\maclA _{\flat _1}(\cc d) \to \maclA _{\flat _1}'(\cc d)=A(\cc d)$
is continuous.
Hence the following result is a straight-forward consequence of Proposition
<ref> and the fact that
$\wideparen {\maclA} _{\flat _1}'(\cc {2d})=\wideparen A(\cc {2d})$.

Let $a\in L_A(\cc {2d})$. Then there is a unique $a_0\in \wideparen A(\cc {2d})$
such that $\op _{\mathfrak V}(a) = \op _{\mathfrak V}(a_0)$ as mappings
from $\maclA _{\flat _1}(\cc d)$ to $\maclA _{\flat _1}'(\cc d)$. It holds

\begin{multline}\label{Eq:AntiWickAnalPseudoRel}
\op _{\mathfrak V}(a) = \op _{\mathfrak V}(a_0)
\\[1ex]
\text{where}\quad
a_0(z,w) = \pi ^{-d} \int _{\cc d}a(z,w_1)e^{-(z-w_1,w-w_1)}\, d\lambda (w_1).
\end{multline}

The operator $\Pi _A$ defined in (<ref>) is the orthogonal
projection from $L^2(d\mu)$ to $A^2(\cc d)$ which is uniquely extendable
to a continuous
map from

\begin{equation}\label{Eq:AntiWickL1SymbClass}
L_{0,A}(\cc d) \equiv
\sets {a_0 \in L^1_{\rm loc}(\cc d)}{w \mapsto a_0(w)e^{r|w|-|w|^2} \in L^1(\cc d)
\ \text{for every}\ r>0}
\end{equation}

to $A(\cc d)$ (see e. g. [29]). Hence, if $F,G\in \maclA _{\flat _1}(\cc d)$ and $a_0$ is
given by (<ref>) then

\begin{multline*}
(\op _{\mathfrak V}(a)F,G)_{A^2}
((\op _{\mathfrak V}(a)\circ \Pi _A)F,G)_{A^2}
\\[1ex]
= \left (
\int _{\cc d} \left ( \int _{\cc d}
a(\cdo ,w_1)e^{(\cdo ,w_1)}e^{(w_1,w)}\, d\mu (w_1)
\right )
F(w)\, d\mu (w),G \right )_{\!\! A^2}
\\[1ex]
\left (
\int _{\cc d} a_0(\cdo ,w)e^{(\cdo ,w)} F(w)\, d\mu (w),G
\right )_{\!\! A^2}
(\op _{\mathfrak V}(a_0)F,G)_{A^2},
\end{multline*}

and thus $\op _{\mathfrak V}(a) = \op _{\mathfrak V}(a_0)$ follows. The assertion
now follows from Proposition <ref> and the fact that $a_0$ in
the integral formula of (<ref>) defines an element in
$\wideparen A(\cc {2d})$.

We will also consider anti-Wick operators [13, 6, 7]
defined by

\begin{equation}\label{Eq:AntiWick}
\op _{\mathfrak V}^{\aw}(a_0)F(z) = \int _{\cc d} a_0(w)F(w)e^{(z,w)}
\, d\mu (w),\quad z \in \cc d,
\end{equation}

when $a_0\in L_{0,A}(\cc d)$ and $F$ belongs to $\maclA _0(\cc d)$, the space of
analytic polynomials on $\cc d$.
Then $a_0\in L_{0,A}(\cc d)$ if and only if $a(z,w)\equiv a_0(w)$ belongs to
$L_A(\cc {2d})$, and then $\op _{\mathfrak V}^{\aw}(a_0) = \op _{\mathfrak V}(a)$.
Consequently, all results for Wick operators with symbols in
$L_A(\cc {2d})$ hold for anti-Wick operators. In particular, if
$a_0\in L_{0,A}(\cc d)$, then $\op _{\mathfrak V}^{\aw}(a_0):
\maclA _{\flat _1}(\cc d) \to A(\cc d)$ is continuous.
We denote
the Wick symbol of the anti-Wick operator $\op _{\mathfrak V}^{\aw}(a_0)$
by $a_0^{\aw}$. Then (<ref>) takes the form

\begin{multline}\tag*{(\ref{Eq:AntiWickAnalPseudoRel})$'$}
\op _{\mathfrak V}^{\aw}(a_0) = \op _{\mathfrak V}(a_0^{\aw})
\\[1ex]
\text{where}\quad
a_0^{\aw} (z,w) = \pi ^{-d} \int _{\cc d}a_0(w_1)e^{-(z-w_1,w-w_1)}\, d\lambda (w_1).
\end{multline}

Pseudo-differential operators on $\rr d$ may be transferred to
Wick operators on $\cc d$ by means of the Bargmann transform.

Let $\fka \in \maclS _{1/2}'(\rr {2d})$.

* the Bargmann assignment $\mathsf S_{\mathfrak V}\fka$ of $\fka$ is
the unique element $a\in \wideparen A (\cc {2d})$ which fulfills

\begin{equation}\label{Eq:BargmannAssignment}
\op _{\mathfrak V}(a) = \mathfrak V_d \circ \op^w (\fka )\circ \mathfrak V_d^*
\quad \Leftrightarrow \quad a = \mathsf S_{\mathfrak V}\fka  \text ;
\end{equation}

* the Bargmann kernel assignment $K_{\mathfrak V,\fka}$ of $\fka$ is
the unique element $K \in \wideparen A (\cc {2d})$, which is the kernel of
the map $\mathfrak V_d \circ \op^w (\fka )\circ \mathfrak V_d^*$ with respect to
the sesquilinear $A^2$ form.

By the definitions we have

\begin{equation}\label{SVandSVK}
K_{\mathfrak V,\fka}(z,w) = e^{(z,w)}\mathsf S_{\mathfrak V}\fka (z,w).
\end{equation}

The creation and annihilation operators

2^{-\frac 12}(x_j-\partial _{x_j})
\quad \text{and}\quad
2^{-\frac 12}(x_j+\partial _{x_j}),

are transfered to the operators

\begin{equation}\label{Eq:LinearWickOps}
F\mapsto z_jF
\quad \text{and}\quad
F\mapsto \partial _{z_j}F,
\end{equation}

by the Bargmann transform (see [4]).
The Wick symbols of the operators in (<ref>)
are $z_j$ and $\overline w_j$, respectively [6, 29]. By combining these identities
with the fact that the Weyl symbol of $i^{-1}\partial _{x_j}$ equals $\xi _j$
we get

\begin{equation}\label{Eq:BargmannAssignBasicMaps}
\begin{gathered}
\mathsf S_{\mathfrak V}(2^{-\frac 12}(x_j-i\xi _j)) = z_j,
\qquad
\mathsf S_{\mathfrak V}(2^{-\frac 12}(x_j+i\xi _j)) = \overline w_j,
\\[1ex]
\mathsf S_{\mathfrak V}(x_j) = 2^{-\frac 12}(z_j+\overline w_j)
\qquad \text{and}\qquad
\mathsf S_{\mathfrak V}(\xi _j) = 2^{-\frac 12}i(z_j-\overline w_j).
\end{gathered}
\end{equation}

We need to compare $K_{\fka}^w$ and $K_{\mathfrak V,\fka}$.
On the one hand we have for $f,g \in \mascS (\rr d)$

\begin{equation*}
(\op^w (\fka ) f, g)_{L^2(\rr d)}
= (K_{\fka}^w, g \otimes \overline f)_{L^2(\rr {2d})}
= (\mathfrak V_{2d} K_{\fka}^w, \mathfrak V_{2d} (g \otimes \overline f)_{A^2(\cc {2d})}
\end{equation*}

and on the other hand

\begin{align*}
(\op^w (\fka ) f, g)_{L^2(\rr d)}
& = (\op _{\mathfrak V}(a) \mathfrak V_d f, \mathfrak V_d g)_{A^2(\cc d)}
\\[1ex]
& = ( K_{\mathfrak V,\fka}, \mathfrak V_d g \otimes \overline{\mathfrak V_d f})_{A^2 (\cc {2d})}
\\[1ex]
& = ( \Theta K_{\mathfrak V,\fka},
\Theta (\mathfrak V_d g \otimes \overline{\mathfrak V_d f} ))_{A^2 (\cc {2d})}.
\end{align*}

\begin{align*}
\Theta (\mathfrak V_d g \otimes \overline{\mathfrak V_d f} )(z,w)
= \mathfrak V_d g(z) \overline{\mathfrak V_d f(\overline w)}
= \mathfrak V_{2d} (g \otimes \overline f) (z,w)
\end{align*}

we obtain

\begin{equation}\label{eq:kernelreal2complex}
K_{\mathfrak V,\fka} = \Theta \mathfrak V_{2d} K_{\fka}^w
= \mathfrak V_{\Theta,d} K_{\fka}^w.
\end{equation}

§.§ Symbol classes for pseudo-differential
operators on $\rr d$

In order to define a generalized family of Shubin symbol classes
[25], we
need to add a restriction of the involved weights. Let $\rho \in [0,1]$, and let
$\mascP _{\Sh ,\rho }(\rr d)$
be the set of all $\omega \in \mascP (\rr d)\cap C^\infty (\rr d)$ such that
for every multi-index $\alpha \in \nn d$,

|\partial ^\alpha \omega (x)|\lesssim \omega (x)\eabs x^{-\rho |\alpha |},
\quad x\in \rr d.

For $\omega \in \mascP _{\Sh ,\rho }(\rr {d})$ the Shubin symbol class
$\Sh _\rho ^{(\omega )}(\rr {d})$ is the set of all $f \in C^\infty (\rr {d})$
such that for every $\alpha \in \nn {2d}$,

|\partial ^\alpha f(x)|\lesssim \omega (x) \eabs x^{-\rho |\alpha |},
\qquad x\in \rr {d}.

Let $\rho \in [0,1]$, $\omega \in \mascP _{\Sh ,\rho }(\rr {2d})$ and
$A$ be a real $d\times d$ matrix.
Then it follows from [25] or <cit.>
that $e^{i\scal {AD_\xi }{D_x}}$ is a homeomorphism on
$\Sh _\rho ^{(\omega )}(\rr {2d})$, which implies that the set

\sets {\op _A(\fka )}{\fka \in \Sh _\rho ^{(\omega )}(\rr {2d})}

is independent of the choice of $A$, in view of
(<ref>). If $B$ is another
real $d\times d$ matrix and
$\fka ,\fkb \in \Sh _\rho ^{(\omega )}(\rr {2d})$ satisfy
(<ref>), then it follows from
<cit.> that

\begin{gather}
\fka - \fkb \in \Sh _\rho ^{(\omega _\rho)}(\rr {2d}),
\quad \text{where}\quad
\omega _\rho (x,\xi )= \omega (x,\xi )\eabs {(x,\xi )}^{-2\rho}.
\label{Eq:DifferencesShubinSymbols}
\intertext{In particular}
|\fka (x,\xi ) - \fkb (x,\xi ) |
\lesssim \omega (x,\xi )\eabs {(x,\xi )}^{-2\rho}.
\label{Eq:DifferencesShubinSymbols2}
\end{gather}

We also need the symbol classes defined in <cit.>
with symbols satisfying estimates of the form

\begin{equation}\label{eq:symbolgevrey}
|\partial _x^\alpha \partial _\xi^\beta  \fka (x,\xi )|
\lesssim h^{|\alpha +\beta |} \alpha !^\sigma \beta !^s
e^{r (|x|^{\frac1{s}}+|\xi |^{\frac1{\sigma}})}, \qquad x,\xi  \in \rr d.
\end{equation}

(See also [10] for the restricted case when $s=\sigma$.)

Let $s,\sigma >0$. Then

* $\Gamma ^{\sigma ,s;0}_{s,\sigma}(\rr {2d})$ consists of all
$\fka \in C^\infty (\rr d)$ such that
(<ref>) holds for every $h > 0$ and some $r > 0$;

* $\Gamma ^{\sigma ,s}_{s,\sigma ;0}(\rr {2d})$ consists of all
$\fka \in C^\infty (\rr d)$ such that
(<ref>) holds for some $h > 0$ and every $r > 0$;

* $\Gamma ^{\sigma ,s}_{s,\sigma}(\rr {2d})$ consists of all
$\fka \in C^\infty (\rr d)$ such that
(<ref>) holds for some $h > 0$ and some $r > 0$.

The symbol classes $\Sh _\rho ^{(\omega )}(\rr {2d})$ have isotropic
behaviour with respect to phase space $T^* \rr d\simeq \rr {2d}$,
and the same holds for the symbol classes in Definition
<ref> when $\sigma = s$.
See also [10] for the restricted case when $s=\sigma$,
and [2] for a bilinear extension.
Important classes similar to those given by Definition
<ref> are considered in [23].

Pseudo-differential operators with symbols in the classes in
Definition <ref> are examples of so called
operators of infinite order.
These operators are continuous on appropriate Gelfand-Shilov (distribution)
spaces [1, 10].
The next result characterizes the symbol classes in
Definition <ref> by means of estimates of form

\begin{equation}\label{Eq:STFTgevrey}
|\cT_\psi \fka (x,\xi , \eta ,y)|
\lesssim
e^{r_1(|x|^{\frac1{s}}+|\xi|^{\frac1{\sigma}}) - r_2(|\eta |^{\frac1{\sigma}} + |y|^{\frac1{s}}) },
\quad x,\xi ,y, \eta \in \rr d.
\end{equation}

We omit the proof since the result is a special case of
We refer to
<cit.> for the definition of the Gelfand-Shilov spaces
$\maclS _{s,\sigma}^{\sigma ,s}(\rr {2d})$,
$\Sigma _{s,\sigma}^{\sigma ,s}(\rr {2d})$ and their distribution spaces.

Let $s,\sigma >0$ and let $\fka \in C^\infty (\rr {2d})$.
Then the following is true:

* if $\psi \in \maclS _{s,\sigma}^{\sigma ,s} (\rr {2d})\setminus 0$, then
$\fka  \in \Gamma ^{\sigma ,s}_{s,\sigma ;0}(\rr {2d})$
if and only if (<ref>) holds for every
$r_1 > 0$ and some $r_2 > 0$;

* if $\psi \in \Sigma _{s,\sigma}^{\sigma ,s} (\rr {2d})\setminus 0$, then
$\fka  \in \Gamma ^{\sigma ,s;0}_{s,\sigma }(\rr {2d})$ if and only if
(<ref>) holds for some $r_1 > 0$ and all $r_2 > 0$;

* if $\psi \in \Sigma _{s,\sigma}^{\sigma ,s} (\rr {2d})\setminus 0$, then
$\fka \in \Gamma ^{\sigma ,s}_{s,\sigma}(\rr {2d})$
if and only if (<ref>) holds for some $r_1 > 0$ and some $r_2 > 0$.

§.§ Elliptic, weakly elliptic and hypoelliptic elements in $\Sh ^{(\omega )}_\rho (\rr d)$

Let $\rho \ge 0$ and $\omega \in \mascP _{\Sh ,\rho}(\rr d)$. Then
$f\in \Sh ^{(\omega )}_\rho (\rr d)$ is called
weakly elliptic of order $\rho _0\ge 0$,
(in $\Sh ^{(\omega )}_\rho (\rr d)$),

or $\rho _0$-weakly elliptic,
if there is an $R>0$ such that

|f(x)| \gtrsim \eabs x^{-\rho _0}\omega (x),\qquad |x|\ge R.

A weakly elliptic function of order $0$ is called elliptic.

Let $A$ and $B$ be real $d\times d$ matrices, $\rho >0$, $\rho _0\in [0,2\rho )$,
$\omega \in \mascP _{\Sh ,\rho}(\rr {2d})$ and suppose that $\fka ,\fkb \in
\Sh _\rho ^{(\omega )}(\rr {2d})$ satisfy (<ref>).
It follows from (<ref>)
that $\fka$ is weakly elliptic of order $\rho _0$, if and only if
$\fkb$ is weakly elliptic of order $\rho _0$. In particular,
$\fka$ is elliptic, if and only if $\fkb$ is elliptic.

Next we define Shubin hypoelliptic symbols (cf. Definitions 5.1 and 25.1 in [25]).

Let $\rho >0$, $\rho _0 \ge 0$, $\omega _0\in \mascP _{\Sh ,\rho}(\rr {d})$
and $f\in \Sh _\rho ^{(\omega _0)}(\rr {d})$. Then
$f$ is called hypoelliptic (in Shubin's sense
in $\Sh _\rho ^{(\omega _0)}(\rr {d})$) of order $\rho _0$,
if there is an $R>0$
such that for every $\alpha \in \nn d$, it holds

|∂^αf(x)|    ≲|f(x)|x^-ρ|α|,
|x|   ≥R,
|f(x)|    ≳ω_0(x)x^-ρ_0,
|x|    ≥R .

Elliptic and hypoelliptic symbols are important since they give rise to parametrices.
For $\rho$, $\omega$ as above and $\fka \in \Sh ^{(\omega )}_\rho (\rr {2d})$ elliptic,
there is an elliptic symbol $\fkb \in \Sh ^{(1/\omega )}_\rho (\rr {2d})$
such that

\op _A(\fka )\circ \op _A(\fkb ) =I+\op _A(\fkc _1)
\quad \text{and}\quad
\op _A(\fkb )\circ \op _A(\fka ) =I+\op _A(\fkc _2)

for some $\fkc _1,\fkc _2\in \mascS (\rr {2d})$.
An operator $\op (\fkc )$ with $c\in \mascS (\rr {2d})$
is regularizing in the sense that $\op (\fkc )$ is continuous from $\mascS '(\rr d)$
to $\mascS (\rr d)$. (Cf. e. g. [8, 25].)

§ REFORMULATION OF PSEUDO-DIFFERENTIAL CALCULUS
USING THE BARGMANN TRANSFORM

In this section we characterize the Bargmann assignment of
pseudo-differential operator symbols from Subsection <ref>,
using estimates of complex derivatives.
In Subsection <ref>
we show how pseudo-differential operators on $\rr d$ with Shubin symbols
are transformed to Wick operators by the Bargmann transform.
In Subsection <ref>
we deduce similar links between pseudo-differential operators
of infinite order, given in the second part of Subsection <ref>,
and suitable classes of Wick operators.
Subsection <ref> treats composition formulae for symbols
of Wick operators, which leads to algebraic
properties for operators in Subsection <ref> and <ref>.
As an application we obtain short proofs of composition results for
pseudo-differential operators on $\rr d$ from Subsection <ref>.

§.§ Wick symbols of Shubin pseudo-differential

The following proposition is essential in the characterization of Shubin
type pseudo-differential operators on $\rr d$ by means of the corresponding
Wick symbols. The Shubin classes can be characterized using the
transform $\cT_\phi$ by means of estimates of the form

\begin{align}
|\partial_x^\alpha \partial_\xi^\beta \cT _\phi f(x,\xi)|
\omega(x) \eabs{x}^{- \rho |\alpha|} \eabs{\xi}^{-N},
\label{eq:Gineq1}
\\[1ex]
|\partial_x^\alpha\cT _\phi f(x,\xi)|
\omega(x) \eabs{x}^{- \rho |\alpha|} \eabs{\xi}^{-N}
\label{eq:Gineq2}
\intertext{and}
|\cT _\phi f(x,\xi)|
\omega(x) \eabs{\xi}^{-N}.
\label{eq:Gineq3}
\end{align}

The proof of the following result is similar to the proof of

Let $0 \le \rho \leqs 1$, let $\omega \in \mascP _{\Sh ,\rho }(\rr {d})$,
and suppose $f\in \cS '(\rr d)$ and $\phi \in\cS (\rr d) \setminus 0$.
The following conditions are equivalent:

* $f \in \Sh _\rho^{(\omega )}(\rr d)$,

* (<ref>) holds true for any $N \geqs 0$ and
$\alpha, \beta \in \nn d$,

* (<ref>) holds true for any $N \geqs 0$ and
$\alpha \in \nn d$,

and the following conditions are equivalent:

(1)$'$ $f \in \Sh _0^{(\omega )}(\rr d)$,

(2)$'$ (<ref>) holds true for any $N \geqs 0$.

First we prove that (1) implies (2).
Suppose $f \in \Sh _\rho^{(\omega )}(\rr d)$ and let
$\alpha,\beta, \gamma \in \nn d$ be arbitrary. We will show

\begin{equation*}
|\xi^\gamma \partial_x^\alpha \partial_\xi^\beta \cT _\phi f(x,\xi)|
\lesssim
\omega(x) \eabs{x}^{-\rho|\alpha|}.
\end{equation*}

By (<ref>), (<ref>) and integration by parts we get

\begin{multline*}
|\xi^\gamma \partial_x^\alpha \partial_\xi^\beta \cT _\phi f(x,\xi)|
\left
| \xi^\gamma \cT_{\phi _\beta}(\partial^\alpha f)(x,\xi)
\right |
\\[1ex]
(2 \pi)^{-\frac d2} \left| \int_{\rr d} \left((i\partial_{y})^\gamma e^{- i \la \xi,y \ra }\right)
\overline{\phi _\beta(y)} \, \partial^\alpha f(x+y)\, dy \right |
\\[1ex]
\lesssim
\int_{\rr{d}} \left| \partial_{y}^\gamma \left[\overline{\phi _\beta(y)} \,
\partial^\alpha f(x+y)\right] \right|\, dy
\\[1ex]
\int_{\rr{d}} \left| \sum_{\kappa \leqs \gamma} \binom{\gamma}{\kappa}
\partial^{\gamma-\kappa}
\overline{\phi _\beta(y)} \, \partial^{\alpha+\kappa} f(x+y)\right|\, dy
\\[1ex]
\lesssim
\sum_{\kappa \leqs \gamma} \binom{\gamma}{\kappa}  \int_{\rr d} \left |
\partial^{\gamma-\kappa}
\phi _\beta(y) \right| \, \omega(x+y) \eabs{x+y}^{-\rho|\alpha+\kappa|}\, dy.
\end{multline*}

Since $\omega$ is polynomially moderate, Peetre's inequality
(<ref>) and the fact that $\phi \in \cS$ give

\begin{multline*}
|\xi^\gamma \partial_x^\alpha \partial_\xi^\beta \cT _\phi f(x,\xi)|
\\[1ex]
\lesssim
\omega(x) \eabs{x}^{-\rho |\alpha |} \sum_{\kappa \leqs \gamma}
\binom{\gamma}{\kappa}
\int_{\rr d} \left| \partial^{\gamma-\kappa} \phi _\beta(y) \right| \, \omega (y)
\, \eabs{y}^{|m|+\rho |\alpha+\kappa|}\, dy
\\[1ex]
\asymp \omega(x) \eabs{x}^{- \rho |\alpha|}.
\end{multline*}

Thus $f\in \Sh _\rho^{(\omega )}(\rr d)$ implies
implies (<ref>), and as a special case (<ref>),
and $f \in \Sh _0^{(\omega )}(\rr d)$ implies (<ref>).
We have proved that (1) implies (2) which in turn implies (3),
and that (1)$'$ implies (2)$'$.

Conversely, suppose (3), that is $f \in \cS'(\rr d)$ and (<ref>)
holds for all $ N \ge 0$ and all $\alpha \in \nn d$,
which is a weaker assumption than (2).
We obtain from (<ref>)

\begin{align*}
& = \|\phi \|_{L^2}^{-2} \, \cT _\phi^* \cT _\phi f(x)
\\[1ex]
& = \|\phi \|_{L^2}^{-2} \, (2 \pi)^{-\frac d2} \iint _{\rr {2d}}
\cT _\phi f(y,\xi) \, e^{i \la \xi,x-y \ra} \, \phi (x-y) \, dyd\xi
\end{align*}

which is an absolutely convergent integral due to (<ref>)
and the fact that $\phi \in\cS(\rr d)$.  We may differentiate under the
integral, so integration by parts, (<ref>) and Peetre's
inequality give for some $N_0 \ge 0$, any $\alpha \in \nn d$ and any $x \in \rr d$

\begin{multline*}
\left |
\partial^\alpha f(x)
\right |
\|\phi \|_{L^2}^{-2} \, (2 \pi)^{-\frac d2}  \left| \iint _{\rr {2d}} \cT _\phi f(y,\xi)
\, \partial_y^\alpha
\left ( e^{i \la \xi,x-y \ra} \, \phi (x-y) \right ) \, dyd\xi
\right |
\\[1ex]
\|\phi \|_{L^2}^{-2} \, (2 \pi)^{-\frac d2}  \left| \iint _{\rr {2d}} \partial_y^\alpha
\cT _\phi f(y,\xi) \, e^{i \la \xi,x-y \ra} \, \phi (x-y) \, dyd\xi \right |
\\[1ex]
\|\phi \|_{L^2}^{-2} \, (2 \pi)^{-\frac d2}  \left| \iint _{\rr {2d}} \partial_y^\alpha
\cT _\phi f(x-y,\xi) \, e^{i \la \xi,y \ra} \, \phi (y) \, dyd\xi \right |
\\[1ex]
\lesssim
\iint _{\rr {2d}}  \omega(x-y) \eabs{x-y}^{- \rho |\alpha|} \,
\eabs{\xi}^{-d-1} \, |\phi (y)| \, dyd\xi
\\[1ex]
\lesssim
\omega (x) \eabs {x}^{- \rho |\alpha|} \iint _{\rr {2d}}  \eabs{\xi}^{-d-1}
\eabs{y}^{N_0+\rho |\alpha|} \, |\phi (y)| \, dyd\xi
\\[1ex]
\asymp
\omega(x) \eabs{x}^{- \rho |\alpha|}.
\end{multline*}

Thus $f \in \Sh _\rho^{(\omega )}(\rr d)$ and we have proved the
equivalence of (1), (2) and (3).

It remains to show that (2)$'$ implies (1)$'$, that is
(<ref>) for all $N \ge 0$ implies
$f \in \Sh _0^{(\omega )}(\rr d)$.
We have for some $N_0 \geqs 0$, any $\alpha \in \nn d$, $x \in \rr d$ and $N \geqs 0$,

\begin{align*}
\left |
\partial^\alpha f(x)
\right |
& =
\|\phi \|_{L^2}^{-2} \, (2 \pi)^{-\frac d2}  \left| \iint _{\rr {2d}} \cT _\phi f(y,\xi) \, \partial _x^\alpha
\left ( e^{i \la \xi,x-y \ra} \, \phi (x-y) \right ) \, dyd\xi
\right |
\\[1ex]
& \lesssim
\sum_{\beta \leqs \alpha}
\binom{\alpha}{\beta}
\iint _{\rr {2d}} \left|  \cT _\phi f(y,\xi) \right| \eabs{\xi}^{|\beta|}
\left| \partial^{\alpha-\beta} \phi (x-y) \right| \, dyd\xi
\\[1ex]
& \lesssim
\sum_{\beta \leqs \alpha}
\binom{\alpha}{\beta}
\iint _{\rr {2d}} \omega(y) \eabs{\xi}^{|\alpha|-N}
\left| \partial^{\alpha-\beta} \phi (x-y) \right| \, dyd\xi
\\[1ex]
& \lesssim
\omega (x) \sum_{\beta \leqs \alpha}
\binom{\alpha}{\beta}
\iint _{\rr {2d}} \eabs{\xi}^{|\alpha|-N}
\eabs{x-y}^{N_0} \left| \partial^{\alpha-\beta} \phi (x-y) \right| \, dyd\xi
\\[1ex]
& \lesssim
\omega(x)
\end{align*}

provided $N$ is sufficiently large, since $\phi \in \cS$.
This shows that $f \in \Sh _0^{(\omega )}(\rr d)$.

We may now characterize the Shubin classes $\Sh _\rho^{(\omega )}(\rr {2d})$
by estimates on their Bargmann (kernel) assignments of the forms

\begin{align}
\big |
( \partial_z  + \overline \partial _w)^\alpha (\partial _z- \overline \partial _w)
^\beta & {\mathsf S}_{\mathfrak V} \fka (z,w)
\big |
\notag
\\[1ex]
e^{\frac1{2} |z-w|^2}\omega (\sqrt 2\, \overline z)
\eabs{z+w}^{- \rho |\alpha + \beta|} \eabs{z-w}^{-N} ,
\label{eq:characShubinBargmann}
\\[1ex]
\left| \partial_z^\alpha \overline \partial _w
^\beta \mathsf S_{\mathfrak V} \fka (z,w) \right |
e^{ \frac1{2} |z-w|^2}\omega (\sqrt 2\, \overline z)\eabs{z+w}^{- \rho |\alpha + \beta|}
\eabs{z-w}^{-N} ,
\label{eq:characShubinBargmann2}
\\[1ex]
\left | \mathsf S_{\mathfrak V} \fka (z,w) \right|
e^{ \frac1{2} |z-w|^2} \omega (\sqrt 2\, \overline z)
\eabs{z-w}^{-N}
\label{eq:characShubinBargmann0}
\intertext{and}
\left | K_{\mathfrak V,\fka} (z,w) \right|
\omega (\sqrt 2\, \overline z)
\eabs{z-w}^{-N} e^{ \frac1{2} \left( |z|^2 + |w|^2 \right)}.
\label{eq:characShubinBargmannkernel0}
\end{align}

Let $0 \le \rho \leqs 1$, $\omega \in \mascP _{\Sh ,\rho}(\rr {2d})$ and
$\fka \in \mascS '(\rr {2d})$.
The following conditions are equivalent:

* $\fka \in \Sh ^{(\omega )}_\rho (\rr {2d})$,

* (<ref>) holds true for every $N\ge 0$, $z,w\in \cc d$ and $\alpha,\beta \in \nn d$,

* (<ref>) holds true for every $N\ge 0$, $z,w\in \cc d$ and $\alpha,\beta \in \nn d$,

and the following conditions are equivalent:

(1)$'$ $\fka \in \Sh ^{(\omega )}_0 (\rr {2d})$,

(2)$'$ (<ref>) holds true for any $N \in \no$
and $z,w\in \cc d$,

(3)$'$  (<ref>) holds true for any $N \in \no$
and $z,w\in \cc d$.

For the proof we need the following proposition of independent interest. Here
we recall that $\mathsf S_{\mathfrak V}$ is bijective from $\maclS _{1/2}'(\rr {2d})$
to the set

\begin{equation}\label{Eq:SBGelfandShilovLargeDist}
\sets{a\in \wideparen A(\cc {2d})}
{|a(z,w)|\lesssim e^{(\frac 12+r)|z-w|^2}\ \text{for every}\ r>0 }.
\end{equation}

Let $\psi (x,\xi ) = (\frac 2\pi )^{\frac d2} e^{-(|x|^2+|\xi |^2)}$, $x,\xi \in \rr d$,
$\fka \in \maclS _{1/2}'(\rr {2d})$ and $a$ belongs to the set in
(<ref>). Then

\begin{align}
\mathsf S_{\mathfrak V} \fka (z,w)
(2 \pi)^{\frac d2} e^{\frac1{2} |z-w|^2}
\cT_\psi \fka \left( \frac{x+y}{\sqrt{2}}, - \frac{\xi+\eta}{\sqrt{2}},
\sqrt{2}(\eta-\xi),  \sqrt{2} (y-x) \right),
\label{eq:BargmannsymbolSTFT}
\intertext{and}
(\mathsf S_{\mathfrak V}^{-1}a)(x,-\xi )
%2^d\iint _{\cc {2d}}a \left( w_1+\frac z{\sqrt 2},w_2+\frac z{\sqrt 2} \right)
%e^{2i\impart (w_1,w_2)}\, d\mu (w_1)d\mu (w_2),
\left( \frac{2}{\pi} \right)^d \int _{\cc {d}}
a \left( \frac z{\sqrt 2} - w, \frac z{\sqrt 2} + w \right)
e^{- 2 |w|^2}\, d\lambda (w),
\label{eq:InvBargmannsymbolSTFT}
\end{align}

with $z=x+i\xi $, $w=y+i\eta$ and $x,y,\xi ,\eta \in \rr d$.

$\phi (x,y) = \pi^{-\frac d2} e^{-\frac1{2}(|x|^2+|y|^2)}$ for $x,y \in \rr {d}$,
and let $K_{\fka}^w$ be the kernel of $\op^w (\fka )$. Then
$\psi =\mascF _2(\phi \circ \kappa)$, where $\kappa (x,y)=(x+y/2,x-y/2)$.
By (<ref>) (or <cit.>)
and <cit.>
we have

\begin{align*}
& \mathfrak V _{\Theta ,d} K_{\fka}^w (z,w)
= \mathfrak V _{2d} K_{\fka}^w (z,\overline w)
= \mathfrak V _{2d} K_{\fka}^w ( (x,y) + i (\xi,-\eta))
\\[1ex]
& = (2 \pi)^d e^{\frac1{2} \left( |z|^2 + |w|^2 \right) + i \left( \langle x, \xi \rangle
- \langle y, \eta \rangle \right)}
\cT_\phi K_{\fka}^w \left( \sqrt{2} (x,y), - \sqrt{2} (\xi,-\eta) \right)
\\[1ex]
& = (2 \pi)^{\frac d2} e^{\frac1{2} \left( |z|^2 + |w|^2 \right) + i \left( \langle y, \xi \rangle -
\langle x, \eta \rangle \right)}
\cT_\psi \fka \left( \frac{x+y}{\sqrt{2}}, - \frac{\xi+\eta}{\sqrt{2}}, \sqrt{2}(\eta-\xi),
\sqrt{2} (y-x) \right),
\\[1ex]
& = (2 \pi)^{\frac d2} e^{\frac1{2} \left( |z|^2 + |w|^2 \right) + i \, \impart (z,w)}
\cT_\psi \fka \left( \frac{x+y}{\sqrt{2}}, - \frac{\xi+\eta}{\sqrt{2}}, \sqrt{2}(\eta-\xi),
\sqrt{2} (y-x) \right).
\end{align*}

Together with the identity

|z|^2 + |w|^2 + 2i \, \impart (z,w) = |z-w|^2 +2(z,w)

this gives

\begin{multline}
\mathfrak V _{\Theta ,d} K_{\fka}^w (z,w)
\\[1ex]
= (2 \pi)^{\frac d2} e^{\frac1{2}|z-w|^2 + (z,w)}
\cT_\psi \fka \left( \frac{x+y}{\sqrt{2}}, - \frac{\xi+\eta}{\sqrt{2}}, \sqrt{2}(\eta-\xi),
\sqrt{2} (y-x) \right ).
\end{multline}

A combination of this identity with (<ref>) and (<ref>)
gives (<ref>).

In order to prove (<ref>), we use
Moyal's formula (<ref>), (<ref>) and the fact that $\nm \psi {L^2}=1$.
This implies that the inverse of $\cT _\psi$ is given by

\begin{multline*}
(\cT _\psi ^{-1}F) (x,\xi ) = (\cT _\psi ^*F) (x,\xi )
\\[1ex]
(2\pi )^{-d}\iiiint _{\rr {4d}}
F(x_1,\xi _1,\eta _1,y_1)\psi (x-x_1,\xi -\xi _1)
e^{i(\scal {x-x_1}{\eta _1}+\scal {y_1}{\xi -\xi _1}}\, dx_1d\xi _1d\eta _1dy_1.
\end{multline*}

G(z,w) = F(x,\xi ,\eta ,y),\qquad z=x+i\xi ,\ w=y+i\eta,

we obtain

\begin{equation}\label{Eq:TInvOp1}
\cT _\psi ^*F(x,\xi )
2^d(2\pi )^{-\frac {3d}2}\iint _{\cc {2d}} G(w_1,w_2)e^{-|z-w_1|^2}
e^{i\impart \scal {z-w_1}{w_2}}\, d\lambda (w_1)d\lambda (w_2).
\end{equation}

If $\fka = \cT _\psi ^*F$ and $a=\mathsf S_{\mathfrak V} \fka$,
then (<ref>) shows that

(2\pi )^{\frac d2}e^{\frac 12|z-w|^2}G \left( \frac {\overline z+\overline w}{\sqrt 2},\sqrt 2(w-z) \right)

which gives

(2\pi )^{-\frac d2}e^{-\frac 14|w|^2}
a \left( \frac {2\overline z -w}{2\sqrt 2},\frac {2\overline z +w}{2\sqrt 2} \right).

Inserting this into (<ref>) we get

\begin{multline*}
\cT _\psi ^*F (x,-\xi )
\\[1ex]
\frac 1{2^d\pi ^{2d}}\iint _{\cc {2d}}
a \left( \frac {2\overline w_1 -w_2}{2\sqrt 2},\frac {2\overline w_1 +w_2}{2\sqrt 2} \right)
e^{-|\overline z-w_1|^2}e^{-\frac 14|w_2|^2}
e^{i\impart \scal {\overline z-w_1}{w_2}}\, d\lambda (w_1)d\lambda (w_2),
\end{multline*}

and by taking

\frac {2\overline w_1 -w_2}{2\sqrt 2}-\frac z{\sqrt 2}
\quad \text{and}\quad
\frac {2\overline w_1 +w_2}{2\sqrt 2}-\frac z{\sqrt 2}

as new variables of integration, we obtain using (<ref>)

\begin{align*}
& \cT _\psi ^*F (x,-\xi )
\\[1ex]
& =
\frac {2^d}{\pi ^{2d}}\iint _{\cc {2d}}
a \left( w_1+\frac z{\sqrt 2},w_2+\frac z{\sqrt 2} \right)
e^{2i\impart ({w_1},{w_2})}\, d\lambda (w_1)d\lambda (w_2)
\\[1ex]
& =
2^d \iint _{\cc {2d}}
a \left( w_1+\frac z{\sqrt 2},w_2+\frac z{\sqrt 2} \right)
e^{2i\impart ({w_1},{w_2})}\, d\mu (w_1)d\mu (w_2)
\\[1ex]
& =
2^d \int _{\cc {d}} \left( \int _{\cc {d}}
a \left( w_1+\frac z{\sqrt 2},w_2+\frac z{\sqrt 2} \right)
e^{ (w_1,w_2)} \, e^{ (-w_2,w_1)} \, d\mu (w_1) \right) d\mu (w_2)
\\[1ex]
& =
2^d \int _{\cc {d}}
a \left( -w_2+\frac z{\sqrt 2},w_2+\frac z{\sqrt 2} \right)
e^{ -|w_2|^2}  \, d\mu (w_2)
\\[1ex]
& =
\left( \frac{2}{\pi} \right)^d \int _{\cc {d}}
a \left(\frac z{\sqrt 2} - w,\frac z{\sqrt 2} + w \right)
e^{ -2 |w|^2}  \, d\lambda (w).
\end{align*}

Combining Propositions <ref> and <ref>,
writing $z + w = 2 z + w - z$, we obtain that
$\fka \in \Sh _\rho^{(\omega )}(\rr {2d})$ if and only if for all $\alpha,\beta \in \nn d$
and $N \in \no$ we have

\begin{multline*}
\left | (\partial_x + \partial_y)^\alpha (\partial_\xi + \partial_\eta)^\beta
\left( e^{ - \frac1{2} |z-w|^2} \mathsf S_{\mathfrak V} \fka (z,w) \right)\right |
\\[1ex]
\lesssim
\omega \left( \frac{\overline{z+w}}{\sqrt{2}} \right)
\eabs{z+w}^{- \rho |\alpha + \beta|} \eabs{z-w}^{-N}
\\[1ex]
\lesssim
\omega (\sqrt 2\, \overline z)\eabs{z+w}^{- \rho |\alpha + \beta|} \eabs{z-w}^{-N+k}
\end{multline*}

for some $k \in \no$ that can be absorbed into $N$.

Note that multi-index powers of the differential operators $\partial_x + \partial_y$ and
$\partial_\xi + \partial_\eta$ acting on the factor $e^{ - \frac1{2} |z-w|^2} = e^{- \frac1{2}
\left( |x-y|^2 + |\xi-\eta|^2 \right)}$ are zero.
Thus we obtain the equivalent condition

\begin{multline*}
\left| (\partial_x + \partial_y)^\alpha (\partial_\xi + \partial_\eta)^\beta
\mathsf S_{\mathfrak V} \fka (z,w) \right |
\\[1ex]
\lesssim
\omega (\sqrt 2\, \overline z)\eabs{z+w}^{- \rho |\alpha + \beta|}
\eabs{z-w}^{-N} e^{ \frac1{2} |z-w|^2} .
\end{multline*}

Using the (conjugate) analyticity of $\mathsf S_{\mathfrak V} \fka (z,w)$
with respect to $z \in \cc d$ ($w \in \cc d$)
we can formulate this as (<ref>).
We have now shown the equivalence between (1) and (2).

The equivalence between (2) and (3)

follows from the binomial formulae

\begin{align*}
(\partial _z+ t\overline \partial _w)^\alpha
\sum _{\gamma \le \alpha}
{\alpha \choose \gamma}t^{|\gamma|}\partial _z^{\alpha -\gamma}
\overline \partial _w^{\gamma},\qquad t\in \{ -1,1\} ,
\\[1ex]
\partial _z^\alpha
2^{-|\alpha |}
\sum _{\gamma \le \alpha}
{\alpha \choose \gamma}
(\partial _z+  \overline \partial _w)^{\alpha -\gamma}
(\partial _z-  \overline \partial _w)^{\gamma}
\intertext{and}
\overline \partial _w^\beta
2^{-|\beta |}
\sum _{\gamma \le \beta}
{\beta \choose \gamma} (-1)^{|\gamma |}
(\partial _z+  \overline \partial _w)^{\beta -\gamma}
(\partial _z-  \overline \partial _w)^{\gamma}.
\end{align*}

It remains to consider the case $\rho = 0$.
We obtain from

Propositions <ref> and <ref>
that $\fka \in \Sh _0^{(\omega )}(\rr {2d})$ if and only if for all $N \in \no$ we have

\begin{equation*}
\left | \mathsf S_{\mathfrak V} \fka (z,w) \right|
\lesssim
\omega (\sqrt 2\, \overline z)
\eabs{z-w}^{-N} e^{ \frac1{2} |z-w|^2}, \quad z, \zeta \in \cc d.
\end{equation*}

This shows the equivalence between (1)$'$ and (2)$'$.

Finally the equivalence of (2)$'$ and (3)$'$

is an immediate consequence of
(<ref>) and

|e^{(|z|^2+|w|^2)/2} e^{-(z,w)}| = e^{(|z|^2-2 \, \repart (z,w)+|w|^2)/2}= e^{|z-w|^2/2}.
\qquad \qedhere

Let $\wideparen \maclA _{\Sh ,\rho}^{(\omega )}(\cc {2d})$,
be the set of all $a\in \wideparen A(\cc {2d})$ such that

\begin{equation}\label{Eq:ShubinWickEstimates}
\left| \partial_z^\alpha \overline \partial _w
^\beta a(z,w) \right |
\le C
e^{ \frac1{2} |z-w|^2}\omega (\sqrt 2\, \overline z)\eabs{z+w}^{- \rho |\alpha + \beta|}
\eabs{z-w}^{-N} ,\quad N\ge 0.
\end{equation}

The smallest constant $C\ge 0$ defines a semi-norm parameterized by $\alpha$,
$\beta$ and $N$,
and we equip $\wideparen \maclA _{\Sh ,\rho}^{(\omega )}(\cc {2d})$ with
the Fréchet space topology defined by these semi-norms.
The following result is an immediate consequence of Theorem
<ref> and its proof.

Let $0 \le \rho \leqs 1$ and $\omega \in \mascP _{\Sh ,\rho}(\rr {2d})$. Then
${\mathsf S}_{\mathfrak V}$ is a homeomorphism from
$\Sh ^{(\omega )}_\rho (\rr {2d})$ to $\wideparen
\maclA _{\Sh ,\rho}^{(\omega )}(\cc {2d})$.

§.§ Wick operators corresponding to Gevrey
type pseudo-differential operators

Using (<ref>) and (<ref>)
we obtain the following theorem expressed with estimates of the form

\begin{equation}\label{Eq:BargmannSTFT}
\lesssim \exp \left( \frac1{2} |z-w|^2  + r_1 |z+w|_{s,\sigma} - r_2 |z-w|_{s,\sigma}  \right)
\end{equation}

(cf. Definition <ref>). The verification is left for the reader.

The following is true:

* if $s,\sigma \ge \frac 12$, then $\mathsf S_{\mathfrak V}$ is homeomorphic
from $\Gamma ^{\sigma ,s}_{s,\sigma ;0}(\rr {2d})$ to the set of all
$a\in \wideparen A(\cc {2d})$ such that (<ref>) holds for all
$r_1 > 0$ and some $r_2 > 0$;

* if $s,\sigma > \frac 12$, then $\mathsf S_{\mathfrak V}$ is homeomorphic
from $\Gamma ^{\sigma ,s ;0}_{s,\sigma}(\rr {2d})$ to the set of all
$a\in \wideparen A(\cc {2d})$ such that
(<ref>) holds for some $r_1 > 0$ and every $r_2 > 0$;

* if $s,\sigma >\frac 12$, then  $\mathsf S_{\mathfrak V}$ is homeomorphic
from $\Gamma ^{\sigma ,s}_{s,\sigma}(\rr {2d})$ to the set of all
$a\in \wideparen A(\cc {2d})$ such that
(<ref>) holds for some $r_1 > 0$ and some $r_2 > 0$.

The restrictions on $s$ and $\sigma$ in Theorem
<ref> are needed since
we must choose $\psi$ in (<ref>) as
the Gauss function in Proposition <ref>.
According to the proof of Theorem <ref>
this is necessary for the use of
the formula (<ref>)
that relates $\cT _\phi K_{\fka}^w$ and the Bargmann transform
$\mathfrak V _{2d} K_{\fka}^w$.
For this $\psi$ we have $\psi \in \maclS _s^\sigma (\rr d)$
($\psi \in \Sigma _s^\sigma (\rr d)$),
if and only if $s,\sigma \ge \frac 12$ ($s,\sigma > \frac 12$).

Theorem <ref> can
be combined
with continuity results in [1] to deduce
continuity of Wick operators
acting on the Bargmann images
of $\Sigma _s^\sigma (\rr d)$, $\maclS _s^\sigma (\rr d)$,
$(\maclS _s^\sigma )'(\rr d)$
and $(\Sigma _s^\sigma )'(\rr d)$, respectively.
The following result follows by a straight-forward combination of
Theorems 3.8, 3.15 and 3.16 in [1],
(<ref>) and Theorem

Let $a\in \wideparen A(\cc {2d})$. Then the following
is true:

* if $s,\sigma \ge \frac 12$ and
(<ref>) holds for all
$r_1 > 0$ and some $r_2 > 0$, then
$\op _{\mathfrak V}(a)$ is continuous
on $\maclA _s^\sigma (\cc d)$ and on
$(\maclA _s^\sigma )'(\cc d)$;

* if $s,\sigma > \frac 12$ and (<ref>)
holds for
some $r_1 > 0$ and all $r_2 > 0$, then
$\op _{\mathfrak V}(a)$ is continuous
on $\maclA _{0,s}^\sigma (\cc d)$ and on $(\maclA _{0,s}^\sigma )'(\cc d)$;

* if $s,\sigma >\frac 12$ and (<ref>) holds for some
$r_1 > 0$ and some $r_2 > 0$, then $\op _{\mathfrak V}(a)$ is continuous
from $\maclA _{0,s}^\sigma (\cc d)$ to $\maclA _{s}^\sigma (\cc d)$, and from
$(\maclA _{s}^\sigma )'(\cc d)$ to $(\maclA _{0,s}^\sigma )'(\cc d)$.

§.§ Composition of Wick operators

Let $a_1,a_2 \in \wideparen A(\cc {2d})$. If composition is well defined then
the complex twisted product $a_1 \wpr _{\mathfrak V}a_2$ is defined by

\op _{\mathfrak V}(a_1)\circ \op _{\mathfrak V}(a_2) =
\op _{\mathfrak V}(a_1 \wpr _{\mathfrak V}a_2).

By straight-forward computations it follows that
the product $\wpr _{\mathfrak V}$ is given by

\begin{equation}\label{Eq:DefCompTwistProd}
a_1 \wpr _{\mathfrak V} a_2 (z,w)
= \pi ^{-d}\int_{\cc d} a_1(z,u) a_2 (u,w) e^{-(z-u,w-u)}\, d \lambda (u),
\quad z,w \in \cc d,
\end{equation}

provided the integral is well defined. Inserting derivatives,
(<ref>) takes the form

\begin{multline}\tag*{(\ref{Eq:DefCompTwistProd})$'$}
(\partial _z^{\alpha _1} \overline \partial _w^{\beta _1} a_1) \wpr _{\mathfrak V}
(\partial _z^{\alpha _2} \overline \partial _w^{\beta _2}a_2) (z,w)
\\[1ex]
= \pi ^{-d}\int_{\cc d} (\partial _z^{\alpha _1} \overline \partial _u^{\beta _1} a_1)(z,u)
(\partial _u^{\alpha _2} \overline \partial _w^{\beta _2} a_2)(u,w) e^{-(z-u,w-u)}
\, d \lambda (u), \quad z,w \in \cc d.
\end{multline}

The following lemma is a product rule
for the complex twisted product.

Let $a_1,a_2\in \wideparen A(\cc {2d})$ and suppose
the integral in (<ref>)$'$ is well defined for all $z,w\in \cc d$
and all $\alpha _1,\alpha _2,\beta _1,\beta _2\in \nn d$ such that

|\alpha _1+\alpha _2+\beta _1+\beta _2|\le 1.

Suppose also that the integrand in (<ref>) is zero at infinity.

∂_z_j(a_1 _𝔙a_2)
(∂_z_ja_1) _𝔙a_2
a_1 _𝔙(∂_z_ja_2),
j    = 1,…,d

∂_w_j(a_1 _𝔙a_2)
(∂_w_ja_1) _𝔙a_2
a_1 _𝔙(∂_w_ja_2),
j    = 1,…,d.

a_1(z,u) a_2 (u,w) e^{(z,u-w) + (u,w)}

\pi ^d(a_1 \wpr _{\mathfrak V}a_2)(z,w)
= \int_{\cc d}F_{a_1,a_2}(z,w,u)e^{-|u|^2}\, d\lambda (u).

This gives

\pi ^d \partial _{z_j}(a_1 \wpr _{\mathfrak V}a_2) (z,w) = b_1(z,w)

\begin{align*}
\int_{\cc d} F_{\partial _{z_j}a_1,a_2}(z,w,u)e^{-|u|^2}\, d\lambda (u),
\\[1ex]
\int_{\cc d} F_{a_1,a_2}(z,w,u)\overline u_je^{-|u|^2}\, d\lambda (u)
\intertext{and}
\overline w_j\int_{\cc d} F_{a_1,a_2}(z,w,u)e^{-|u|^2}\, d\lambda (u)
\\[1ex]
\overline w_j \pi^d (a_1 \wpr _{\mathfrak V}a_2)(z,w).
\end{align*}

The conjugate analyticity of $u \mapsto a_1(z,u)$ and $u \mapsto e^{(z,u-w)}$ implies
$\partial _{u_j}a_1(z,u) = \partial _{u_j}e^{(z,u-w)} = 0$ which gives

\begin{multline*}
\partial _{u_j}F_{a_1,a_2}(z,w,u)
\\[1ex]
\left( a_1(z,u)  \partial _{u_j}a_2 (u,w)
\overline w_ja_1(z,u) a_2 (u,w) \right) e^{(z,u-w) + (u,w)}
\\[1ex]
F_{a_1,\partial _{z_j}a_2}(z,w,u) + \overline w_jF_{a_1,a_2}(z,w,u).
\end{multline*}

Consider $b_2(z,w)$. Integration by parts gives

\begin{multline*}
\int_{\cc d} F_{a_1,a_2}(z,w,u)\overline u_je^{-|u|^2}\, d\lambda (u)
\\[1ex]
= - \int_{\cc d} F_{a_1,a_2}(z,w,u) \, \partial _{u_j}e^{-|u|^2} \, d\lambda (u)
\\[1ex]
\int_{\cc d} \partial _{u_j}F_{a_1,a_2}(z,w,u) e^{-|u|^2}\, d\lambda (u)
\\[1ex]
\int_{\cc d} F_{a_1,\partial _{z_j}a_2}(z,w,u)
e^{-|u|^2}\, d\lambda (u)
\overline w_j \int_{\cc d} F_{a_1,a_2}(z,w,u)
e^{-|u|^2}\, d\lambda (u)
\\[1ex]
\int_{\cc d} F_{a_1,\partial _{z_j}a_2}(z,w,u)
e^{-|u|^2}\, d\lambda (u)
\end{multline*}

A combination of these identities now gives

\begin{multline*}
\pi ^d \partial _{z_j} (a_1 \wpr _{\mathfrak V}a_2) (z,w)
\\[1ex]
\int_{\cc d} (
F_{\partial _{z_j}a_1,a_2}(z,w,u)
F_{a_1,\partial _{z_j}a_2}(z,w,u)
)e^{-|u|^2}\, d\lambda (u)
\\[1ex]
=\pi ^d(\partial _{z_j}a_1) \wpr _{\mathfrak V}a_2(z,w)
\pi ^d a_1 \wpr _{\mathfrak V}(\partial _{z_j}a_2) (z,w),
\end{multline*}

and (<ref>) follows.

The assertion (<ref>) is proved by similar arguments.

The characterization

in Theorem <ref> (3)
can be applied
to prove the following composition result,
which is a generalization of <cit.> to include
the case when $\rho = 0$.

Let $0 \le \rho \leqs 1$ and $\omega_j \in \mascP _{\Sh ,\rho}(\rr {2d})$ for $j = 1,2$.
If $\fka _j \in \Sh_\rho^{(\omega _j)}(\rr {2d})$ for $j = 1,2$, then $\fka _1 \wpr \fka _2
\in \Sh _\rho^{(\omega _1\omega _2)}(\rr {2d})$.

If $\fka _0 = \fka _1 \wpr \fka _2$ and $a_j = \mathsf S_{\mathfrak V} \fka _j$, $j=0,1,2$, then
$a_0 = a_1 \wpr _{\mathfrak V} a_2$.
From Lemma <ref> and (<ref>)
we obtain for $\alpha, \beta \in \nn d$,

\begin{align*}
& \partial_z  ^\alpha \overline \partial _w
^\beta a_0 (z,w)
\\[1ex]
& = \sum_{\gamma \leqs \alpha} \sum_{\kappa \leqs \beta}
\binom{\alpha}{\gamma} \binom{\beta}{\kappa}
\left( (\partial _z^{\alpha - \gamma} \overline \partial _w^{\beta-\kappa}
a_1) \wpr _{\mathfrak V}
(\partial _z^{\gamma} \overline \partial _w^{\kappa}a_2) \right)(z,w)
\\[1ex]
& = \pi^{-d} \sum_{\gamma \leqs \alpha} \sum_{\kappa \leqs \beta}
\binom{\alpha}{\gamma} \binom{\beta}{\kappa}
\int_{\cc d} \partial _z^{\alpha-\gamma} {\overline \partial} _u^{\beta-\kappa} a_1(z,u)
\partial _u^{\gamma} \overline \partial _w^{\kappa} a_2 (u,w) e^{(z,u-w) + (u,w)} d \mu (u).
\end{align*}

Since $\omega _2 \in \mascP (\rr {2d})\simeq \mascP (\cc {d})$ is moderate,
Theorem <ref> gives for some $N_0 \ge 0$ and any $N_1, N_2 \geqs 0$

\begin{align*}
|\partial _u^{\alpha-\gamma} \overline \partial _w^{\beta-\kappa} a_1(z,u)|
& \lesssim
\omega _1(\sqrt 2\, \overline z)\eabs {z+u}^{-\rho |\alpha+\beta-\gamma -\kappa |}
\eabs {z-u}^{-N_1} e^{\frac1{2} | z - u |^2}
\intertext{and}
|\partial _u^\gamma \overline \partial _w^\kappa a_2(u,w)|
& \lesssim
\omega _2(\sqrt 2\, \overline z)\eabs {z-u}^{N_0}\eabs {u+w}^{-\rho |\gamma +\kappa |}
\eabs {u-w}^{-N_2} e^{\frac1{2} | u - w |^2}.
\end{align*}

This gives

\begin{multline}\label{Eq:Derb0Est}
\left | \partial _z^\alpha \overline \partial _w
^\beta a_0 (z,w) \right|
\\[1ex]
\lesssim
\omega _1(\sqrt 2\, \overline z)\omega _2(\sqrt 2\, \overline z) e^{\frac 12|z-w|^2}
\int_{\cc d}
F(z,w,u)e^{\Phi (z,w,u)}\, d \lambda (u)
\end{multline}

where for any $N_1 \geqs 0$

\begin{align*}
\eabs{z+u}^{-\rho|\alpha+\beta-\gamma-\kappa|} \eabs{z-u}^{N_0-N_1}
\eabs{u+w}^{-\rho|\gamma+\kappa|} \eabs{u-w}^{-N_2}
\intertext{and}
\Phi (z,w,u)
- \frac1{2} |z-w|^2 + \frac1{2} |z-u|^2 + \frac1{2} |u-w|^2 - |u|^2
\\[1ex]
& \qquad \qquad \qquad \qquad \qquad \qquad \qquad
+ \repart (z,u-w) + \repart (u,w) = 0.
\end{align*}

By Peetre's inequality and the facts that $\gamma \le \alpha$ and $\kappa \le \beta$
we get

\begin{multline*}
\eabs {z+u}^{\rho |\gamma +\kappa|}\eabs{u+w}^{-\rho|\gamma+\kappa|}
\lesssim
\eabs {z-w}^{\rho |\gamma +\kappa|}
\\[1ex]
\lesssim
\eabs {z-u}^{\rho |\gamma +\kappa|}\eabs {u-w}^{\rho |\gamma +\kappa|}
\\[1ex]
\le
\eabs {z-u}^{\rho |\alpha +\beta|}\eabs {u-w}^{\rho |\alpha +\beta|}
\end{multline*}

\begin{equation*}
\eabs {z+u}^{-\rho |\alpha +\beta |}
\lesssim
\eabs {z+w}^{-\rho |\alpha +\beta |}\eabs {u-w}^{\rho |\alpha +\beta |}
\end{equation*}

\begin{equation}\label{Eq:FuncFEst}
\le
\eabs{z+w}^{-\rho |\alpha +\beta|}
\eabs{z-u}^{\rho |\alpha +\beta| +N_0-N_1}
\eabs{u-w}^{2\rho |\alpha +\beta |-N_2}.
\end{equation}

Hence a combination of (<ref>) and (<ref>)
gives for any $N \geqs 0$

\begin{align*}
& (\omega _1(\sqrt 2\, \overline z)\omega _2(\sqrt 2\, \overline z))^{-1}
\eabs{z+w}^{\rho|\alpha+\beta|}
\left| \partial _z^\alpha \overline \partial _w
^\beta a_0 (z,w) \right|
\\[1ex]
& \lesssim
e^{\frac1{2} |z-w|^2}
\int_{\cc d} \eabs{z-u}^{\rho|\alpha+\beta| + N_0 - N_1} \eabs{u-w}
^{2\rho|\alpha+\beta| - N_2}\, d \lambda (u)
\\[1ex]
& \lesssim
\eabs{z-w}^{-N} e^{\frac1{2} |z-w|^2}
\int_{\cc d} \eabs{z-u}^{\rho|\alpha+\beta| + N_0+N - N_1} \eabs{u-w}
^{2\rho|\alpha+\beta| + N - N_2}\, d \lambda (u).
\end{align*}

By letting

N_1 \geqs \rho|\alpha+\beta| + N_0 + N
\quad \text{and}\quad
N_2 > 2\rho|\alpha+\beta| + N  + 2 d

we obtain

\begin{equation*}
\left| \partial _z^\alpha \overline \partial _w^\beta
a_0 (z,w) \right|
\lesssim
\omega _1(\sqrt 2\, \overline z)\omega _2(\sqrt 2\, \overline z)
\eabs{z+w}^{-\rho|\alpha+\beta|}
\eabs{z-w}^{-N} e^{\frac1{2} |z-w|^2}.
\end{equation*}

According to
Theorem <ref> (3)

this estimate
implies that $\fka _0 \in \Sh _\rho^{(\omega _1\omega _2)}(\rr {2d})$.

Eq. (<ref>)
combined with Theorem <ref>
can be used to show composition results for pseudo-differential
operators with symbols in $\Gamma ^{\sigma ,s}_{s,\sigma ;0}(\rr {2d})$.
In fact we may use an argument similar to the proof of Proposition
<ref>, but simpler since derivatives can be avoided.
We obtain

_1 _2 ∈Γ^σ,s_s,σ;0(2d)
when  _1,_2 ∈Γ^σ,s_s,σ;0(2d),

and similarly with $\Gamma ^{\sigma ,s;0}_{s,\sigma}$

in place of
$\Gamma ^{\sigma ,s}_{s,\sigma ;0}$, provided $\sigma >\frac 12$.
Thereby we regain parts of <cit.> for certain
restrictions on $s$ and $\sigma$.

§ RELATIONS AND ESTIMATES FOR WICK AND
ANTI-WICK OPERATORS

In this section we first show how to approximate a Wick operator by means of
a sum of anti-Wick operators.
Then we prove continuity results for anti-Wick operators with symbols having
exponential type bounds.  Finally we deduce estimates for the Wick symbol
of these anti-Wick operators.

§.§ Expansion of Shubin type Wick operators with respect to
anti-Wick operators

The first result can be stated for semi-conjugate analytic symbols on $\cc {2d}$.

Suppose $s\ge \frac 12$, $a \in \wideparen \maclA _s'(\cc {2d})$,
let $N\ge 0$ be an integer, and let

\begin{align*}
a_\alpha (w) &= \partial _z^\alpha \overline \partial _w^\alpha a (w,w), \quad \alpha \in \nn d,
\intertext{and}
b_\alpha (z,w) &= |\alpha |
\int _0^1 (1-t)^{|\alpha |-1} \partial _z^\alpha \overline \partial _w^\alpha a
(w+t(z-w),w)\, dt, \quad \alpha \in \nn d\setminus 0.
\end{align*}

\begin{equation}\label{Eq:WickToAntiWick}
\op _{\mathfrak V}(a)
\sum _{|\alpha |\le N}
\frac {(-1)^{|\alpha |}\op _{\mathfrak V}^{\aw}(a_\alpha )}{\alpha !}
\sum _{|\alpha | = N+1}\frac {(-1)^{|\alpha |}\op _{\mathfrak V}(b_\alpha )}{\alpha !}.
\end{equation}

Taylor expansion gives

\begin{align*}
a(z,w) & = \sum _{|\alpha |\le N} \frac {(-1)^{|\alpha |} c_{\alpha}(z,w)} {\alpha !}
\sum _{|\alpha | = N+1}\frac {(-1)^{|\alpha |}c_{0,\alpha}(z,w)} {\alpha !},
\intertext{where}
c_{\alpha} (z,w) &= (-1)^{|\alpha |} (z-w)^\alpha \partial _z^\alpha a (w,w)
\intertext{and}
c_{0,\alpha} (z,w) &= (-1)^{|\alpha |} |\alpha |(z-w)^\alpha
\int _0^1 (1-t)^{|\alpha |-1} \partial _z^\alpha a (w+t(z-w),w)\, dt.
\end{align*}

\op _{\mathfrak V}(a)
\sum _{|\alpha |\le N}\frac {(-1)^{|\alpha |}\op _{\mathfrak V}(c_{\alpha} )}{\alpha !}
\sum _{|\alpha | = N+1}\frac {(-1)^{|\alpha |}\op _{\mathfrak V}(c_{0,\alpha} )}{\alpha !},

and the result follows if we prove

\begin{equation}\label{Eq:WickAntiWickExpIdent}
\op _{\mathfrak V}(c_{\alpha} )
\op _{\mathfrak V}^{\aw}(a_\alpha )
\quad \text{and}\quad
\op _{\mathfrak V}(c_{0,\alpha} )
\op _{\mathfrak V}(b_\alpha ) .
\end{equation}

It follows from (<ref>) that

\op _{\mathfrak V}(b_\alpha ) = \op _{\mathfrak V}(c_{1,\alpha} )
\quad \text{and}\quad
\op _{\mathfrak V}(c_{0,\alpha} ) = \op _{\mathfrak V}(c_{2,\alpha} )

\begin{align}
(-1)^{|\alpha |} \pi ^{-d} |\alpha |
\int _0^1 (1-t)^{|\alpha |-1} h_{j,\alpha}(a;t,z,w)\, dt, \label{eq:cjdef}
\intertext{$j=1,2$, with}
(-1)^{|\alpha |} \int _{\cc d}
\partial _z^\alpha \overline \partial _w^\alpha a
e^{-(z-w_1,w-w_1)}\, d\lambda (w_1) \label{eq:h1def}
\intertext{and}
\int _{\cc d}(z-w_1)^\alpha
\partial _z^\alpha a
e^{-(z-w_1,w-w_1)}\, d\lambda (w_1). \nonumber
\end{align}

(z-w_1)^\alpha e^{-(z-w_1,w-w_1)}
\overline \partial _{w_1}^\alpha  e^{-(z-w_1,w-w_1)}

integration by parts yields

\begin{multline*}
\int _{\cc d}
\partial _z^\alpha a
\overline \partial _{w_1}^\alpha e^{-(z-w_1,w-w_1)} \, d\lambda (w_1)
\\[1ex]
(-1)^{|\alpha |}
\int _{\cc d}
\partial _z^\alpha \overline \partial _{w}^\alpha a
e^{-(z-w_1,w-w_1)}\, d\lambda (w_1)
\end{multline*}

and the second equality in (<ref>)
follows. The first equality in (<ref>)
follows by similar arguments. The details are left for the reader.

Proposition <ref> and its proof
show that

\begin{equation}\tag*{(\ref{Eq:WickToAntiWick})$'$}
\op _{\mathfrak V}(a)
\sum _{|\alpha |\le N}
\frac {(-1)^{|\alpha |}\op _{\mathfrak V}^{\aw}(a_{\alpha})}{\alpha !}
\sum _{|\alpha | = N+1}\frac {(-1)^{|\alpha |}\op _{\mathfrak V}(c_{1,\alpha})}{\alpha !}
\end{equation}

where $c_{1,\alpha}$ is defined by (<ref>) and (<ref>).

In the following result we estimate $a_\alpha$ in Proposition <ref>
and $c_{1,\alpha}$ in (<ref>)
when $a =\mathsf S_{\mathfrak V} \fka$ satisfies
(<ref>) for every $N\ge 0$ and $\alpha,\beta \in \nn d$.
By Theorem <ref> this means
that $\op _{\mathfrak V}(a)$
is the Bargmann transform of a Shubin type operator.

Let $0 \le \rho \le 1$, $\omega \in \mascP _{\Sh ,\rho}(\rr {2d})$,
$a \in \wideparen \maclA _{\Sh ,\rho}^{(\omega )}(\cc {2d})$,
and let $a_\alpha$ and $b_\alpha$
be as in Proposition <ref> for $\alpha \in \nn d$.
Then  $\op _{\mathfrak V}(b_\alpha)
= \op _{\mathfrak V}(c_{1,\alpha})$ for
a unique $c_{1,\alpha}\in \wideparen A(\cc {2d})$,

\begin{align}
|\partial _w^\beta \overline \partial _w^\gamma a_\alpha (w)|
\omega (\sqrt 2\overline {w}) \eabs {w}^{-\rho (2 |\alpha| +|\beta +\gamma |)},
\quad
\alpha ,\beta ,\gamma \in \nn d,
\label{Eq:ShubinWickExpEst1}
\intertext{and}
|\partial _z^\beta \overline \partial _w^\gamma  c_{1,\alpha} (z,w)|
e^{\frac{1}{2} |z-w|^2}
\omega (\sqrt 2\overline {z}) \eabs {z+w}^{-\rho (2 |\alpha| + |\beta +\gamma|)}
\eabs {z-w}^{-N},
\quad
\alpha ,\beta ,\gamma \in \nn d.
\label{Eq:ShubinWickExpEst2}
\end{align}

The Wick symbol $c_{1,\alpha}$  in Proposition <ref> is
uniquely defined and given by (<ref>) in view of
Proposition <ref>, when $h_{1,\alpha}$ is
defined by (<ref>).
The conditions in Proposition <ref>
imply that $c_{1,\alpha} \in \wideparen \maclA _{\Sh ,\rho}^{(\omega _\alpha )} (\cc {2d})$
where $\omega _\alpha =\eabs \cdo ^{-2\rho |\alpha |}\cdot \omega$.

The estimate (<ref>) is an immediate consequence of

\begin{equation*}
\partial _w^\beta \overline \partial _w^\gamma a_\alpha (w)
= \partial _w^{\alpha+\beta} \overline \partial _w^{\alpha +\gamma} a (w,w)
\end{equation*}

and (<ref>).

In order to prove (<ref>) we first note that
the uniqueness assertion for $c_{1,\alpha}$ is a consequence of
Remark <ref>.
Let $h_{1,\alpha}(a;z,w)$
be the same as in the proof of Proposition <ref>.
Integration by parts gives

\partial _z^\beta \overline \partial _w^\gamma h_{1,\alpha}(a;t,z,w)
h_{1,\alpha}(\partial _z^\beta \overline \partial _w^\gamma a;t,z,w),

which reduce the problem to prove that (<ref>)
holds for $\beta =\gamma =0$.

The assumption $a \in \wideparen \maclA _{\Sh ,\rho}^{(\omega )}(\cc {2d})$
combined with $\omega$
and $\eabs \cdo ^{-|\alpha |}$ being moderate imply

| \partial_z^\alpha \overline \partial_w^\beta a(z,w) |
\lesssim
e^{\frac1{2} |z-w|^2}
\omega (\sqrt 2\, \overline w)\eabs{w}^{- \rho |\alpha +\beta |}
\eabs{z-w}^{-N}

for every $N\ge 0$. This gives

\begin{multline*}
e^{\repart (z,w)} |h_{1,\alpha} (a;t,z,w)|
\\[1ex]
\lesssim
\int _{\cc d}
\omega (\sqrt 2\overline {w_1})
e^{\frac{t^2}{2} |z-w_1|^2}\eabs {w_1}^{-2\rho |\alpha |}\eabs {t(z-w_1)}^{-N}
%|(\partial _z^\alpha a_0)(w_1+t(z-w_1),w_1)|
e^{\repart (z+w-w_1,w_1)}\, d\lambda (w_1),
\end{multline*}

that is

\begin{multline}\label{Eq:LongEsthalpha}
e^{-\frac 14|z-w|^2} |h_{1,\alpha} (a;t,z,w)|
\\[1ex]
\lesssim
\int _{\cc d}
\omega (\sqrt 2\overline {w_1})
e^{\frac{t^2}{2} |z-w_1|^2}\eabs {w_1}^{-2\rho |\alpha |}\eabs {t(z-w_1)}^{-N}
e^{-|w_1-z_2|^2}\, d\lambda (w_1)
\\[1ex]
\int _{\cc d}
\omega (\sqrt 2(\overline {z_2+w_1}))
e^{\frac{t^2}{2} |z_1-w_1|^2}\eabs {z_2+w_1}^{-2\rho |\alpha |}\eabs {t(z_1-w_1)}^{-N}
e^{-|w_1|^2}\, d\lambda (w_1)
\end{multline}

for every $N\ge 0$, where $z_1=\frac 12(z-w)$ and $z_2=\frac 12(z+w)$.

If $t\in [0,\frac 12]$, then the last estimate together with the
moderateness of $\omega$ give

\begin{multline*}
e^{-|z_1|^2} |h_{1,\alpha} (a;t,z,w)|
\lesssim
\omega (\sqrt 2\overline {z_2})\eabs {z_2}^{-2\rho |\alpha |}
\int _{\cc d}
e^{\frac 18|w_1|^2}
e^{\frac{1}{8} |z_1-w_1|^2}
e^{-|w_1|^2}\, d\lambda (w_1)
\\[1ex]
\lesssim
\omega (\sqrt 2\overline {z_2})\eabs {z_2}^{-2\rho |\alpha |} e^{\frac{1}{4} |z_1|^2}\int _{\cc d}
e^{\frac{1}{4} |w_1|^2}
e^{-\frac 78|w_1|^2}\, d\lambda (w_1)
\\[1ex]
\lesssim
\omega (\sqrt 2\overline {z_2})\eabs {z_2}^{-2\rho |\alpha |} e^{\frac{1}{2} |z_1|^2}
\eabs {z_1}^{-N},
\end{multline*}

for every $N\ge 0$.
The moderateness of $\omega$ again gives

\begin{equation}\label{Eq:halphaEst}
|h_{1,\alpha} (a;t,z,w)|\lesssim e^{\frac{1}{2} |z-w|^2}
\omega (\sqrt 2\overline {z}) \eabs {z+w}^{-2\rho |\alpha |}\eabs {z-w}^{-N}
\end{equation}

or every $N\ge 0$, when $t\in [0,\frac 12]$.

Suppose instead $t\in [\frac 12,1]$. Then
$\eabs {t(z_1-w_1)}^{-N}\asymp \eabs {z_1-w_1}^{-N}$.
Moderateness again gives

\omega (\sqrt 2(\overline {z_2+w_1}))
\eabs {z_2+w_1}^{-2\rho |\alpha |}\eabs {z_1-w_1}^{-N_0}
\lesssim
\omega (\sqrt 2\overline z)
\eabs {z}^{-2\rho |\alpha |}

for some $N_0$. Hence (<ref>) gives

\begin{multline*}
e^{-|z_1|^2} \omega (\sqrt 2\overline z)^{-1}\eabs {z}^{2\rho |\alpha |}
| h_{1,\alpha} (a;t,z,w)|
\\[1ex]
\lesssim
\int _{\cc d}
e^{\frac{1}{2} |z_1-w_1|^2}\eabs {z_1-w_1}^{-N}
e^{-|w_1|^2}\, d\lambda (w_1)
\\[1ex]
e^{|z_1|^2}\int _{\cc d}
\eabs {z_1-w_1}^{-N}
e^{- \frac{1}{2} |w_1+z_1|^2}\, d\lambda (w_1)
\asymp
e^{|z_1|^2}\eabs {z_1}^{-N}
\end{multline*}

for every $N\ge 0$. This gives (<ref>) also for $t\in [\frac 12,1]$.

The result now follows by using (<ref>) when estimating
$|c_{1,\alpha}(z,w)|$ in (<ref>)
and evaluating the arising integral.

The next result, analogous to Proposition <ref>,
will be useful in Section <ref> when we discuss hypoellipticity
for Shubin operators in the Wick setting.

Let $\rho \ge 0$, $\omega \in \mascP _{\Sh ,\rho}(\cc d)$,
$\omega _t=\omega \cdot \eabs \cdo ^{-2\rho t}$ when $t\ge 0$,
$a\in \wideparen \maclA _{\Sh ,\rho}^{(\omega )}(\cc {2d})$,
$\fka = \mathsf S_{\mathfrak V}^{-1}a$ and $N\ge 0$ be an integer.

\begin{equation}\label{Eq:ExpansionWickTaylor}
\fka (x,-\xi ) = \sum _{|\alpha |\le N}
\frac {(-1)^{|\alpha |}
(\partial _z^\alpha \overline \partial _w^\alpha a)(2^{-\frac 12}z,2^{-\frac 12}z)}
{2^{|\alpha |}\alpha !} +c_N(z), \quad z = x + i \xi,
\end{equation}

\begin{equation}\label{Eq:EstExpansionWickTaylor}
\begin{alignedat}{2}
\partial _z^\alpha \overline \partial _w^\alpha a
\in
\wideparen \maclA _{\Sh ,\rho}^{(\omega _{|\alpha |})}(\cc {2d})
\quad &\text{and} &\quad
(x,\xi )\mapsto c_N(x-i\xi ) \in \Sh _{\rho}^{(\omega _{N+1})}(\rr {2d}).
\end{alignedat}
\end{equation}

The first claim in (<ref>) $\partial _z^\alpha \overline \partial _w^\alpha a \in
\wideparen \maclA _{\Sh ,\rho}^{(\omega _{|\alpha |})}(\cc {2d})$ is an immediate consequence
of the definition (<ref>) and Peetre's inequality.

By Taylor expanding the right-hand side of
we obtain

\begin{equation}\label{Eq:ExpansionWickTaylor1}
\fka (x,-\xi ) = \sum _{|\alpha +\beta |\le 2N+1}
\frac {(2/\pi)^d I_{\alpha ,\beta}\cdot
(\partial _z^\alpha \overline \partial _w^\beta a)(2^{-\frac 12}z,2^{-\frac 12}z)}
{\alpha !\beta !} +c_N(z),
\end{equation}

\begin{equation*}
I_{\alpha ,\beta}
\int _{\cc {d}}(-w)^\alpha \overline w^\beta e^{- 2 |w|^2}
\, d\lambda (w),
\end{equation*}

\begin{equation}\label{Eq:cNdef}
\sum _{|\alpha +\beta |= 2N+2}
\frac {(-1)^{|\beta |}}{\alpha !\beta !}
\int _0^1(1-\theta )^{2N+1}
H_{\alpha ,\beta } (z ,\theta )\, d\theta
\end{equation}

\begin{equation}\label{Eq:Hdef}
H_{\alpha ,\beta } (z ,\theta ) =
\left ( \frac 2\pi \right )^d \int _{\cc {d}}
(\partial _z^\alpha \overline \partial _w^\beta a)
\left ( \frac z{\sqrt 2}- \theta w,\frac z{\sqrt 2}+\theta w \right )
w^\alpha \overline w^\beta
e^{- 2 |w|^2 }\, d\lambda (w).
\end{equation}

The orthonormality of $\{ e_\alpha \} _{\alpha \in \nn d}\subseteq A^2(\cc d)$
(cf. (<ref>))
yields $I_{\alpha,\beta} = 0$ if $\alpha \neq \beta$ and

\begin{align*}
& = \int _{\cc {d}}(-w)^\alpha \overline w^\alpha e^{- 2 |w|^2} \, d\lambda (w)
\\[1ex]
& = (-1)^{|\alpha|} 2^{-d -|\alpha|} \alpha! \pi^d \int _{\cc {d}} |e_\alpha(w) |^2
\, d\mu (w)
\\[1ex]
& = (-1)^{|\alpha|} 2^{-d -|\alpha|} \alpha! \pi^d.
\end{align*}

Comparing (<ref>) with (<ref>)
we see that the sum in the latter formula has been proven correct.

It remains to study the remainder $c_N$.
We need to prove that $\fkc (x,\xi )=c_N(x-i\xi )$ belongs to
$\Sh _{\rho}^{(\omega _{N+1})}(\rr {2d})$. If

h_{\alpha ,\beta } (z ,w,\theta ) =
(\partial _z^\alpha \overline \partial _w^\beta a)
\left ( \frac z{\sqrt 2}- \theta w,\frac z{\sqrt 2}+\theta w \right )
w^\alpha \overline w^\beta e^{- 2 |w|^2 }

H_{\alpha ,\beta } (z ,\theta )
\left ( \frac 2\pi \right )^d \int _{\cc {d}}
h_{\alpha ,\beta } (z ,w,\theta )
\, d\lambda (w).

First we notice that

\begin{align*}
\partial _z^\alpha \overline \partial _z^\beta c_N(z)
\sum _{|\gamma +\delta |= 2N+2}
\frac {(-1)^{|\delta |}}{\gamma !\delta !}
\int _0^1(1-\theta )^{2N+1}
\partial _z^{\alpha}\overline \partial _z^{\beta}
H_{\gamma ,\delta } (z ,\theta )\, d\theta,
\\[1ex]
\partial _z^\alpha \overline \partial _z^\beta H_{\gamma ,\delta} (z,\theta )
\left ( \frac 2\pi \right )^d
\int _{\cc {d}} \partial _z^\alpha \overline \partial _z^\beta h_{\gamma ,\delta}
(z,w,\theta ) \, d\lambda (w)
\intertext{and}
\partial _z^\alpha \overline \partial _z^\beta h_{\gamma ,\delta}
(z,w,\theta )
2^{-\frac {|\alpha +\beta |}2}(\partial _z^{\alpha +\gamma}
\overline \partial _z^{\beta +\delta}
a)\left( \frac z{\sqrt 2} - \theta w,\frac z{\sqrt 2} + \theta w \right )w^\gamma
\overline w^\delta e^{- 2 |w|^2 }.
\end{align*}

From the definition (<ref>) this implies that
for every $M\ge 0$ and some $M_0\ge 0$ we have

\begin{multline*}%\tag*{(*)}
|\partial _z^\alpha \overline \partial _z^\beta h_{\gamma ,\delta}
(z,w,\theta )|
\\[1ex]
\lesssim
e^{-2(1-\theta ^2)|w|^2}\omega (\overline z-\sqrt 2\theta \overline w)
\eabs {z}^{-\rho (|\alpha +\beta |+2N+2)}\eabs {\theta w}^{-M-M_0}|w|^{2N+2}
\\[1ex]
\lesssim
e^{-2(1-\theta )|w|^2}\omega (\overline z)
\eabs {z}^{-\rho (|\alpha +\beta |+2N+2)}\eabs {\theta w}^{-M}|w|^{2N+2}.
\end{multline*}

This gives

|\partial _z^\alpha \overline \partial _z^\beta H_{\gamma ,\delta} (z,\theta )|
\lesssim
\omega (\overline z)
\eabs {z}^{-\rho (|\alpha +\beta |+2N+2)}\cdot J(\theta ),

J(\theta ) = \int _{\cc d} e^{-2(1-\theta )|w|^2}\eabs {\theta w}^{-M}|w|^{2N+2}
\, d\lambda (w).

For $\theta \in [0,\frac 12]$ we get

J(\theta ) = \int _{\cc d} e^{-|w|^2}|w|^{2N+2}\, d\lambda (w),

which is finite and independent of $\theta$. If instead $\theta \in [\frac 12,1]$,
and choosing $M>2d+2N+2$, then

J(\theta ) \le
\int _{\cc d} \eabs {\theta w}^{-M}|w|^{2N+2}\, d\lambda (w),

which is again finite and independent of $\theta$.

A combination of these estimates give

|\partial _z^\alpha \overline \partial _z^\beta H_{\gamma ,\delta} (z,\theta )|
\lesssim
\omega (\overline z)
\eabs {z}^{-\rho (|\alpha +\beta |+2N+2)},

which in turn implies

|\partial _z^\alpha \overline \partial _z^\beta c_N(z)|
\lesssim
\omega (\overline z)
\eabs {z}^{-\rho (|\alpha +\beta |+2N+2)}.

This means that $\fkc \in \Sh _{\rho}^{(\omega _{N+1})}(\rr {2d})$.

§.§ Continuity of anti-Wick operators with exponentially
bounded symbols

Next we consider anti-Wick symbols that satisfy
exponential bounds of the form

\begin{align}
|a_0 (w)|
e^{-r_0 |w|^{\frac 1s}},
\label{Eq:NonAWGSEst1}
\intertext{or}
|a_0 (w)|
e^{r_0 |w|^{\frac 1s}}.
\label{Eq:NonAWGSEst2}
\end{align}

In order to formulate our results we introduce new spaces of entire functions.
Let $s > \frac{1}{2}$, $t_0, r > 0$, and
let $\maclA _{s, t_0,r}(\cc d)$ be the Banach space of all $F \in A(\cc d)$ such that

\begin{equation*}
\nm F{\maclA _{s, t_0,r}} \equiv
\nm {F\cdot e^{- t_0 |\cdo |^2 + r |\cdo|^{\frac{1}{s}}}}{L^\infty}
< \infty.
\end{equation*}

\begin{equation*}
\maclA _{0,(s, t_0)}(\cc d) = \bigcap_{r > 0} \maclA _{s, t_0,r} (\cc d)
\quad \text{and}\quad
\maclA _{(s, t_0)}'(\cc d) = \bigcap_{r > 0} \maclA _{s, t_0,-r} (\cc d)
\end{equation*}

equipped with the projective limit topology.
Likewise we set

\begin{equation*}
\maclA _{(s, t_0)}(\cc d) = \bigcup_{r > 0} \maclA _{s, t_0,r} (\cc d)
\quad \text{and}\quad
\maclA _{0,(s, t_0)}'(\cc d) = \bigcup_{r > 0} \maclA _{s, t_0,-r} (\cc d)
\end{equation*}

equipped with the inductive limit topology.

Referring to Section <ref> it is clear that
the spaces $\maclA _{0,(s, t_0)}(\cc d)$, $\maclA _{(s, t_0)}(\cc d)$, $\maclA _{(s, t_0)}'(\cc d)$
and $\maclA _{0,(s, t_0)}'(\cc d)$ are generalizations of

\begin{align*}
\maclA _{0,(s, \frac{1}{2})}(\cc d) &= \mathfrak V_d ( \Sigma_s (\rr d) ) = \maclA _{0,s}(\cc d)
\\[1ex]
\maclA _{(s, \frac{1}{2})}(\cc d) &= \mathfrak V_d ( \maclS _s (\rr d) ) = \maclA _{s}(\cc d)
\\[1ex]
\maclA _{(s, \frac{1}{2})}'(\cc d) &= \mathfrak V_d ( \maclS _s '(\rr d) ) = \maclA _{s}'(\cc d)
\intertext{and}
\maclA _{0,(s, \frac{1}{2})}'(\cc d) &= \mathfrak V_d ( \Sigma_s' (\rr d) ) = \maclA _{0,s}'(\cc d),
\end{align*}

Let $a_0\in L^\infty _{loc}(\cc d)$, $0 < t_0 < 1$ and

\begin{equation}\label{eq:t0t1}
t_1 = \frac{1}{4(1-t_0)}.
\end{equation}

Then the following is true:

* if (<ref>) holds for some $r_0 > 0$ then

\begin{equation}\label{Eq:ContAW1}
\begin{alignedat}{2}
\op _{\mathfrak V}^{\aw}(a_0) &: & \maclA _{0,(s, t_0)}(\cc d) &\to \maclA _{0,(s, t_1)}(\cc d),
\\[1ex]
\op _{\mathfrak V}^{\aw}(a_0) &: & \maclA _{0,(s, t_0)}'(\cc d) &\to \maclA _{0,(s, t_1)}'(\cc d)
\end{alignedat}
\end{equation}

are continuous;

* if (<ref>) holds for every $r_0 > 0$ then

\begin{equation}\label{Eq:ContAW2}
\begin{alignedat}{2}
\op _{\mathfrak V}^{\aw}(a_0) &: &\maclA _{(s, t_0)}(\cc d) &\to \maclA _{(s, t_1)}(\cc d),
\\[1ex]
\op _{\mathfrak V}^{\aw}(a_0) &: &\maclA _{(s, t_0)}'(\cc d) &\to \maclA _{(s, t_1)}'(\cc d)
\end{alignedat}
\end{equation}

are continuous.

We only prove that the first map in (<ref>) is continuous. The
other continuity assertions follow by similar arguments and are left
for the reader.

Let $r_2 > 0$ be given, $r_1 > r_0$ and $F\in \maclA _{0,(s, t_0)}(\cc d)$.
We have for $z \in \cc d$

\begin{align*}
& |\op _{\mathfrak V}^{\aw}(a_0)F(z)| e^{- t_1 |z |^2 + r_2 |z|^{\frac{1}{s}}}
\\[1ex]
& \lesssim e^{- t_1 |z |^2 + r_2 |z|^{\frac{1}{s}}}
\int_{\cc d} |a_0 (w)| \, |F(w)| \, e^{\repart (z,w) - |w|^2}\,  d\lambda (w)
\\[1ex]
& \lesssim e^{- t_1 |z |^2 + r_2 |z|^{\frac{1}{s}}}  \nm F{\maclA _{s, t_0,r_1}}
\int_{\cc d} e^{r_0 |w|^{\frac{1}{s}}
+ t_0 |w|^2 - r_1 |w|^{\frac{1}{s}} + \repart (z,w) - |w|^2}\,  d\lambda (w)
\\[1ex]
& = e^{r_2 |z|^{\frac{1}{s}}}  \nm F{\maclA _{s, t_0,r_1}}
\int_{\cc d} e^{-(r_1-r_0) |w|^{\frac{1}{s}}  -(1- t_0) |w|^2 +
\repart (z,w) - t_1 |z |^2 }\,  d\lambda (w)
\\[1ex]
& = e^{r_2 |z|^{\frac{1}{s}}}  \nm F{\maclA _{s, t_0,r_1}}
\int_{\cc d} e^{-(r_1-r_0)
|w|^{\frac{1}{s}}  - \left |  \sqrt{1- t_0} w - \frac{1}{2 \sqrt{1- t_0} } z \right|^2 }
\,  d\lambda (w)
\\[1ex]
& = e^{r_2 |z|^{\frac{1}{s}}}  \nm F{\maclA _{s, t_0,r_1}}
\int_{\cc d} e^{-(r_1-r_0) \left| w + \frac{1}{2 (1- t_0) } z \right |
^{\frac{1}{s}}  - (1- t_0)\left|  w  \right|^2 }\,  d\lambda (w)
\\[1ex]
& \le
e^{(r_2 - c_1(r_1 - r_0)) |z|^{\frac{1}{s}}}
\nm F{\maclA _{s, t_0,r_1}}
\int_{\cc d} e^{ c_2 (r_1-r_0) | w |^{\frac{1}{s}} - (1- t_0)\left|  w  \right|^2 }\, d\lambda (w)
\\[1ex]
\nm F{\maclA _{s, t_0,r_1}}
e^{(r_2 - c_1(r_1 - r_0)) |z|^{\frac{1}{s}}}
\end{align*}

for some constants $c_1, c_2>0$. By choosing $r_1$ sufficiently large we

\begin{equation*}
\nm {\op _{\mathfrak V}^{\aw}(a_0) F} {\maclA _{s, t_1,r_2}}
\lesssim
\nm F{\maclA _{s, t_0,r_1}}.
\end{equation*}

The estimates and (<ref>) imply $\op _{\mathfrak V}^{\aw}(a_0)F \in A(\cc d)$.

Note that (<ref>) implies $t_1 > \frac{1}{4}$
and $t_0 \le t_1$ with equality if and only if $t_0 = \frac{1}{2}$.
Hence $\maclA _{0,(s, t_0)}(\cc d) \subseteq \maclA _{0,(s, t_1)}(\cc d)$,
and similarly for the other spaces.

The particular case $t_0 = \frac{1}{2}$ gives

Let $a_0\in L^\infty _{loc}(\cc d)$.
If (<ref>) holds for some (every) $r_0 > 0$
then $\op _{\mathfrak V}^{\aw}(a_0)$ is continuous on $\maclA _{0,s}(\cc d)$
(on $\maclA _{s}(\cc d)$).

With a technique similar to the proof of Proposition <ref>
one shows the following result.

Let $a_0\in L^\infty _{loc}(\cc d)$, $0 < t_0 < 1$ and suppose (<ref>) holds.
Then the following is true:

* if (<ref>) holds for all $r_0 > 0$ then

\begin{equation}\label{Eq:RegAW1}
\op _{\mathfrak V}^{\aw}(a_0): \maclA _{0,(s, t_0)}'(\cc d) \to \maclA _{0,(s, t_1)}(\cc d)
\end{equation}

is continuous;

* if (<ref>) holds for some $r_0 > 0$ then

\begin{equation}\label{Eq:RegAW2}
\op _{\mathfrak V}^{\aw}(a_0): \maclA _{(s, t_0)}'(\cc d) \to \maclA _{(s, t_1)}(\cc d)
\end{equation}

is continuous.

Again the particular case $t_0 = \frac{1}{2}$ gives

Let $a_0\in L^\infty _{loc}(\cc d)$. Then the following is true:

* if (<ref>) holds for all $r_0 > 0$

\begin{equation*}
\op _{\mathfrak V}^{\aw}(a_0): \maclA _{0,s}'(\cc d) \to \maclA _{0,s}(\cc d)
\end{equation*}

is continuous;

* if (<ref>) holds for some $r_0 > 0$

\begin{equation*}
\op _{\mathfrak V}^{\aw}(a_0): \maclA _{s}'(\cc d) \to \maclA _{s}(\cc d)
\end{equation*}

is continuous.

§.§ Estimates of Wick symbols of anti-Wick operators with
exponentially bounded symbols

For anti-Wick operators in <cit.> we have the following result.

If $a_0\in L^\infty _{loc}(\cc d)$ satisfies

\begin{equation}\label{Eq:ExpQuadratic}
|a_0(w)| \lesssim e^{r|w|^2}, \quad w \in \cc d, \quad \text{for some} \quad r < 1,
\end{equation}

then $a_0\in L_{0,A}(\cc d)$ and (<ref>)$'$ holds for some
$a_0^{\aw}\in \wideparen A(\cc {2d})$ with

\begin{equation*}
|a_0^{\aw}(z,w)|\lesssim e^{r_0 |z+w|^2-\operatorname{Re}(z,w)},
\qquad
\end{equation*}

The claim $a_0\in L_{0,A}(\cc d)$ is an immediate consequence of the assumption (<ref>)
and the definition (<ref>).
The integral in (<ref>)$'$ can be estimated as

\begin{multline*}
\left |
\int _{\cc d}a_0(w_1)e^{-(z-w_1,w-w_1)}\, d\lambda (w_1)
\right |
\\[1ex]
\lesssim
\int _{\cc d} e^{r|w_1|^2} \left | e^{-(z-w_1,w-w_1)} \right |
\, d\lambda (w_1)
\\[1ex]
\int _{\cc d} e^{-(1-r)|w_1|^2} e^{\operatorname{Re}(z+w,w_1)}
\, d\lambda (w_1)
\\[1ex]
e^{\frac 1{4(1-r)}|z+w|^2-\operatorname{Re}(z,w)}
\int _{\cc d} e^{-(1-r)|w_1-(z+w)/(2(1-r))|^2}
\, d\lambda (w_1)
\\[1ex]
\asymp e^{r_0|z+w|^2-\operatorname{Re}(z,w)} .\qedhere
\end{multline*}

The condition on $a_0^{\aw}$ in Theorem <ref>
implies that $a_0^{\aw}$ belongs to
$\wideparen {\maclA} _{0,\frac 12}^\prime (\cc {2d})$
(see [28]).
In particular it follows that $\op _{\mathfrak V}^{\aw}(a_0)=\op _{\mathfrak V}(a_0^{\aw})$
is continuous from $\maclA _{0,\frac 12}(\cc d)$ to $\maclA _{0,\frac 12}'(\cc d)$
(cf. <cit.> and
Remark <ref>).

The following result concerns exponentially moderate weight functions.

Let $a_0\in L_{0,A}(\cc d)$, $a_0^{\aw} \in \wideparen A(\cc {2d})$
is given by (<ref>)$'$
and $\omega \in \mascP _E(\cc d)$. If

\begin{equation*}
|a_0(w)|\lesssim  \omega (2w), \quad w \in \cc d,
\end{equation*}

\begin{equation*}
|a_0^{\aw}(z,w)|\lesssim e^{\frac 14|z-w|^2} \omega (z+w), \quad z,w \in \cc d.
\end{equation*}

Let $r\ge 0$ be chosen such that $\omega (z+w)\lesssim \omega (z)e^{r|w|}$,
$z,w\in \cc d$. From (<ref>)$'$ we get

\begin{multline*}
\lesssim
\int _{\cc d} \omega (2w_1)
e^{-\operatorname{Re}(z-w_1,w-w_1)}\, d\lambda (w_1)
\\[1ex]
\int _{\cc d}
\omega (2w_1)
e^{\operatorname{Re}(z+w,w_1)-|w_1|^2}\, d\lambda (w_1)
\\[1ex]
e^{-\operatorname{Re}(z,w)+\frac 14|z+w|^2}
\int _{\cc d}
\omega (2w_1)
e^{-|w_1-(z+w)/2|^2}\, d\lambda (w_1)
\\[1ex]
e^{\frac 14|z-w|^2}
\int _{\cc d}
\omega (2w_1+z+w)
e^{-|w_1|^2}\, d\lambda (w_1)
\\[1ex]
\lesssim
e^{\frac 14|z-w|^2}
\omega (z+w)\int _{\cc d}  e^{2r|w_1|-|w_1|^2}\, d\lambda (w_1)
\asymp
e^{\frac 14|z-w|^2}
\omega (z+w). \qedhere
\end{multline*}

The anti-Wick operators in Propositions <ref> and
<ref> can also be described as Wick operators
with symbols that have smaller growth bounds than
$\wideparen \maclA _s(\cc {2d})$ and its dual.
The following result extends Theorem <ref>
for weights of the form $e^{c |z|^{\frac1s}}$ with $c \in \ro$
from $s \ge 1$ to $s  \ge \frac 12$.

Let $s\ge \frac 12$ ($s> \frac 12$), $a_0\in L_{0,A}(\cc {d})$ and
let $a_0^{\aw}$ be given by (<ref>)$'$.
Then the following is true:

* if (<ref>) holds for
some (every) $r_0 > 0$  then

\begin{equation}\label{Eq:AWGSEst1}
|a_0^{\aw} (z,w)|
\lesssim
e^{\frac 14|z-w|^2-r |z+w|^{\frac 1s}}
\end{equation}

for some (every) $r>0$;

* if (<ref>) holds for every (some) $r_0>0$ then

\begin{equation}\label{Eq:AWGSEst2}
|a_0^{\aw} (z,w)|
\lesssim
e^{\frac 14|z-w|^2+r |z+w|^{\frac 1s}}
\end{equation}

for every (some) $r>0$.

Thanks to the parameter $\frac 14$ in the factor $e^{\frac 14|z-w|^2}$
rather than $\frac 12$, the estimates (<ref>) are much stronger
than the estimates (<ref>) with $\sigma = s$.
Corollary <ref> can thus be seen as a
consequence of Theorems <ref>
and <ref>, and  <cit.>.

The estimates for $a_0^{\aw}$ in Theorem
<ref> may seem
weak since the dominating factor $e^{\frac 14|z-w|^2}$
is present in (<ref>) and (<ref>)
but absent in the original estimates (<ref>)
and (<ref>) for $a_0$.

On the other hand, Wick symbols for operators with
continuity involving the spaces $\maclA _s(\cc d)$ and
$\maclA _s'(\cc d)$, as well as  $\maclA _{0,s}(\cc d)$ and
$\maclA _{0,s}'(\cc d)$, usually satisfies conditions of the

|a(z,w)| \lesssim e^{\frac 12|z-w|^2 \pm r_1|z+w|^{\frac 12}\pm |z-w|^{\frac 1s}}

in view of <cit.>,
and Theorem <ref>. Here
the dominating factor is $e^{\frac 12|z-w|^2}$, which is larger
than the factor $e^{\frac 14|z-w|^2}$ in Theorem

This factor has a large impact on functions on $\rr d$ that are transformed back
by the inverse of the Bargmann transform.
For instance, if $\ep >0$, then the Bargmann image of any
non-trivial Gelfand-Shilov space and its distribution space

\begin{equation*}
\sets {F\in A(\cc d)}{|F(z)|\lesssim e^{(\frac 12-\ep )|z|^2}}
\end{equation*}

and are contained in

\begin{equation*}
\sets {F\in A(\cc d)}{|F(z)|\lesssim e^{(\frac 12+\ep )|z|^2}}.
\end{equation*}

The same holds true for the Bargmann images of
$\mascS (\rr d)$ and $\mascS '(\rr d)$.

Theorem <ref> is
a straight-forward consequence of the following two
propositions, which give more details on the relationships between
$r$ and $r_0$ in
(<ref>), (<ref>),  (<ref>)
and (<ref>).

Let $s\ge \frac 12$ and let $r_0,r\in (0,\infty )$ be
such that

r_0   ∈(0,∞)
and       r   <r_0/4(1+r_0),
when       s    =1/2,

r_0   ∈(0,∞)
and       r    ≤2^-1/sr_0,
when       s   ∈(1/2,∞),

with strict inequality in (<ref>)
when $s<1$.
If $a_0\in L^\infty _{loc}(\cc {d})$ satisfies
(<ref>) and $a_0^{\aw} \in \wideparen A(\cc {2d})$
is given by (<ref>)$'$, then

(<ref>) holds.

Let $s\ge \frac 12$ and $r_0,r\in (0,\infty )$ be
such that

r_0   ∈(0,1)
and       r   >r_0/4(1-r_0),
when       s    =1/2,
r_0   ∈(0,∞)
and       r    ≥2^-1/sr_0,
when       s   ∈(1/2,∞),

with strict inequality in (<ref>)$'$
when $s<1$.
If $a_0\in L^\infty _{loc}(\cc {d})$ satisfies
(<ref>) and $a_0^{\aw} \in \wideparen A(\cc {2d})$
is given by (<ref>)$'$, then

(<ref>) holds.

For the proofs of Propositions <ref>
and <ref>
we use the inequalities

|z|^θ-|w|^θ   ≤|z+w|^θ≤|z|^θ+|w|^θ,
θ   ∈(0,1],     z,w    ∈d

|z+w|^θ   ≤(1+)|z|^θ+(1+^-1)|w|^θ,
θ   ∈[1,2],     z,w    ∈d,

|z+w|^θ   ≥(1-)|z|^θ+(1-^-1)|w|^θ,
θ   ∈[1,2],     z,w    ∈d,

for every $\ep >0$.

Suppose that $a_0$ satisfies
(<ref>) for some $r_0 >0$.
First we consider the case $s>\frac 12$.
If $s < 1$ let $\ep _1 >  0$ and $\ep _2 = \ep_1^{-1}$,
and if $s \ge 1$ let $\ep _1 = 0$ and $\ep _2 = 2$,
and let $c=2^{-\frac 1s}$.
Then (<ref>)$'$,
and (<ref>) give

\begin{multline}\label{Eq:WickAWickComp1}
\lesssim
\int _{\cc d} e^{-r_0 |w_1|^{\frac 1s}}
e^{-\operatorname{Re}(z-w_1,w-w_1)}\, d\lambda (w_1)
\\[1ex]
e^{\frac 14|z+w|^2-\operatorname{Re}(z,w)}
\int _{\cc d} e^{-r_0 |w_1|^{\frac 1s} -|w_1-(z+w)/2|^2}
\, d\lambda (w_1)
\\[1ex]
e^{\frac 14|z-w|^2}
\int _{\cc d} e^{-r_0 |w_1+(z+w)/2|^{\frac 1s} -|w_1|^2}
\, d\lambda (w_1)
\\[1ex]
\le
e^{\frac 14|z-w|^2}e^{-cr_0(1-\ep _1) |z+w|^{\frac 1s}}
\int _{\cc d} e^{-r_0(1-\ep _2) |w_1|^{\frac 1s} -|w_1|^2}
\, d\lambda (w_1)
\\[1ex]
\asymp
e^{\frac 14|z-w|^2}e^{-cr_0(1-\ep _1) |z+w|^{\frac 1s}}.
\end{multline}

If $s\ge 1$, then $\ep _1=0$ and $\ep _2=2$,
and the result follows from (<ref>).
If instead $s<1$, then
the result follows by choosing $\ep _1>0$ small enough,
and we have proved the result in the case $s>\frac 12$.

Next suppose that $s=\frac 12$.
For $\ep _1 >  0$ and $\ep _2 = \ep_1^{-1}$
(<ref>) gives

\begin{equation*}
\lesssim
e^{\frac 14|z-w|^2}
e^{-\frac 14r_0(1-\ep _1) |z+w|^2}
\int _{\cc d} e^{-(r_0(1-\ep _2)+1) |w_1|^2}
\, d\lambda (w_1).
\end{equation*}

For any $\ep _2<\frac {1+r_0}{r_0}$ it follows that the integral
converges, and

1-\ep _1=1-\ep _2^{-1}<(1+r_0)^{-1}.

By the assumptions there is $\delta > 0$ such that

\begin{equation*}
r = \frac{r_0(1-\delta)}{4 (1+r_0)}.
\end{equation*}

1-\ep _1 \nearrow (1+r_0)^{-1}
\quad \text{as}\quad
\ep _2 \nearrow \frac {1+r_0}{r_0}

we may pick $0 < \ep _2<\frac {1+r_0}{r_0}$ such that

\begin{equation*}
\frac{1-\delta}{1+r_0} \le 1-\ep _1
\end{equation*}

and the result follows in the case $s=\frac 12$.

First we consider the case when
$s>\frac 12$.
Suppose that $a_0$ satisfies (<ref>)
for some $r_0 >0$, let $\ep _1,\ep _2\ge 0$ be
such that $\ep _1=\ep _2=0$ when $s\ge 1$ and
$\ep_1\ep _2=1$ when $s<1$, and let $c=2^{-\frac 1s}$. Then
(<ref>)$'$, (<ref>)
and (<ref>) give

\begin{multline}\label{Eq:WickAWickComp3}
\lesssim
\int _{\cc d} e^{r_0 |w_1|^{\frac 1s}}
e^{-\operatorname{Re}(z-w_1,w-w_1)}\, d\lambda (w_1)
\\[1ex]
e^{\frac 14|z+w|^2-\operatorname{Re}(z,w)}
\int _{\cc d} e^{r_0 |w_1|^{\frac 1s} -|w_1-(z+w)/2|^2}
\, d\lambda (w_1)
\\[1ex]
e^{\frac 14|z-w|^2}
\int _{\cc d} e^{r_0 |w_1+(z+w)/2|^{\frac 1s} -|w_1|^2}
\, d\lambda (w_1)
\\[1ex]
\le
e^{\frac 14|z-w|^2}e^{cr_0(1+\ep _1) |z+w|^{\frac 1s}}
\int _{\cc d} e^{r_0(1+\ep _2) |w_1|^{\frac 1s} -|w_1|^2}
\, d\lambda (w_1)
\\[1ex]
\asymp
e^{\frac 14|z-w|^2}e^{cr_0(1+\ep _1) |z+w|^{\frac 1s}}.
\end{multline}

If $s\ge 1$, then $\ep _1=\ep _2=0$, and the result follows
from (<ref>). If instead $s<1$, then
the result follows by choosing $\ep _1>0$ small enough,
and the result follows in the case $s>\frac 12$.

Next suppose that $s=\frac 12$. Then
(<ref>) gives

\begin{equation*}
\lesssim
e^{\frac 14|z-w|^2}
e^{\frac 14r_0(1+\ep _1) |z+w|^2}
\int _{\cc d} e^{r_0(1+\ep _2) |w_1|^2 -|w_1|^2}
\, d\lambda (w_1).
\end{equation*}

For any $\ep _2<\frac {1-r_0}{r_0}$ the integral converges, and

1+\ep _1=1+\ep _2^{-1}>(1-r_0)^{-1}.

1+\ep _1 \searrow (1-r_0)^{-1}
\quad \text{as}\quad
\ep _2 \nearrow \frac {1-r_0}{r_0},

the result follows in the case $s=\frac 12$
by letting $r=\frac {r_0(1+\ep _1)}4$.

§ A LOWER BOUND FOR WICK OPERATORS

In this section we apply the asymptotic expansions in the previous section
for Shubin-Wick operators to deduce a sharp Gårding inequality.

First we have the following result. We put
$\wideparen \maclA _{\Sh ,\rho}(\cc {2d})
\wideparen \maclA _{\Sh ,\rho}^{(\omega )}(\cc {2d})$ when $\omega =1$.

Let $\omega \in \mascP (\cc d)$, $p\in [1,\infty]$,
$a\in \wideparen \maclA _{\Sh ,0}(\cc {2d})$
and $a_0\in L^\infty (\cc d)$. Then $\op _{\mathfrak V}(a)$ and
$\op _{\mathfrak V}^{\aw}(a_0)$ are both continuous on
$A^p_{(\omega )}(\cc d)$.

The claimed continuity of $\op _{\mathfrak V}(a)$

is a straight-forward consequence of
<cit.>, in combination with
Proposition <ref> and the relationship
$ K(z,w)=a(z,w) e^{(z,w)} $ between the kernel and symbol of a
Wick operator (cf. (<ref>)). In order to be
self-contained we include an alternative and shorter proof.

Let $F\in A^p_{(\omega )}(\cc d)$, $G(z)
e^{-\frac 12|z|^2}|F(z)\omega (\sqrt 2\overline z)|$,

\begin{align*}
H_1(z) & = e^{-\frac 12|z|^2}|\op _{\mathfrak V}(a)F(z)\omega (\sqrt 2\overline z)|
\quad \text{and}\quad \\
H_2(z) & = e^{-\frac 12|z|^2}|\op _{\mathfrak V}^{\aw}(a_0)F(z)\omega (\sqrt 2\overline z)|.
\end{align*}

We have

\omega (\sqrt 2\overline z) \lesssim \omega (\sqrt 2\overline w)\eabs {z-w}^{N_0}

for some $N_0\ge 0$. By Theorem <ref> and (<ref>) we get

\begin{multline*}
H_1(z) \lesssim e^{-\frac 12|z|^2}\int _{\cc d} e^{\frac 12|z-w|^2}
\eabs {z-w}^{-N}|F(w)\omega (\sqrt 2\overline z)|
e^{\repart (z,w)-|w|^2}\, d\lambda (w)
\\[1ex]
= (\eabs \cdo ^{N_0-N}*G)(z),
\end{multline*}

for every $N\ge 0$. By choosing $N>2d+N_0$ and using Young's inequality we get
$\nm {H_1}{L^p}\lesssim \nm {G}{L^p}$ which means
$\nm {\op _{\mathfrak V}(a)F}{A^p_{(\omega )}}\lesssim \nm F{A^p_{(\omega )}}$,
and the asserted continuity for $\op _{\mathfrak V}(a)$ follows.

In the same way we get

\begin{multline*}
H_2(z) \lesssim \nm {a_0} {L^\infty} e^{-\frac 12|z|^2}
\int _{\cc d} |F(w)\omega (\sqrt 2\overline w)|\eabs {z-w}^{N_0}
e^{\repart (z,w)-|w|^2}\, d\lambda (w)
\\[1ex]
\asymp ((\eabs \cdo ^{N_0}e^{-\frac 12|\cdo |^2})*G)(z),
\end{multline*}

and another application of Young's inequality shows that
$\nm {H_2}{L^p_{(\omega )}}\lesssim \nm {G}{L^p_{(\omega )}}$ that is
$\nm {\op _{\mathfrak V}^{\aw}(a_0)F}{A^p_{(\omega )}}\lesssim \nm F{A^p_{(\omega )}}$.

We have finally a version of the sharp Gårding inequality.

Let $\rho >0$, $\omega (z) = \eabs z ^{2\rho}$ and let
$a\in \wideparen \maclA _{\Sh ,\rho}^{(\omega )}(\cc {2d})$ be such that
$a(w,w)\ge -C_0$ for all $w \in \cc d$, for some constant $C_0\ge 0$. Then

( (_𝔙(a)F,F)_A^2 )
F    ∈_(d)

| Im ( (_𝔙(a)F,F)_A^2 ) |
F    ∈_(d)

for some constant $C\ge 0$.

Let $b_0(w)=a(w,w)$. Then
$\op _{\mathfrak V}(a) = \op _{\mathfrak V}^{\aw}(b_0)+\op _{\mathfrak V}(a_1)$
for some $a_1\in \wideparen \maclA _{\Sh ,\rho}(\cc {2d})
\subseteq \wideparen \maclA _{\Sh ,0}(\cc {2d})$, in view of Proposition
Since $\Pi_A F = F$ for $F \in A^2(\cc d)$ (cf. (<ref>)),
the assumption $b_0\ge - C_0$ implies
$(\op _{\mathfrak V}^{\aw}(b_0)F,F)_{A^2}\ge - C_0
\| F \|_{A^2}^2 $ for every $F\in \maclA _{\mascS}(\cc d)$.
The operator $\op _{\mathfrak V}(a_1)$ is continuous on $A^2(\cc d)$
in view of Proposition <ref>. A combination of these facts
gives the result.

§ ELLIPTICITY AND HYPOELLIPTICITY FOR SHUBIN AND
WICK OPERATORS

In this section we show that the Bargmann assignment
$\mathsf S_{\mathfrak V}$ maps the sets of hypoelliptic symbols and
weakly elliptic symbols
in the Shubin class $\Sh _\rho ^{(\omega )}(\rr {2d})$
bijectively into the sets of hypoelliptic symbols and
weakly elliptic Wick symbols
in $\wideparen \maclA _{\Sh ,\rho}^{(\omega )}(\cc {2d})$, respectively.
Then we explain some consequences for polynomial symbols.

§.§ Transition of weakly elliptic symbols

For symbols in
$\wideparen \maclA _{\Sh ,\rho} ^{(\omega )}(\cc {2d})$
we define ellipticity and weak ellipticity as follows.

Let $\rho >0$, $\omega \in \mascP _{\Sh ,\rho}(\cc d)$ and
$a\in \wideparen \maclA _{\Sh ,\rho} ^{(\omega)}(\cc {2d})$.
Then $a$ is called weakly elliptic

of order $\rho _0\ge 0$, or $\rho _0$-weakly elliptic, if for some $R > 0$

|a(z,z)|\gtrsim \eabs z^{-\rho _0}\omega (\sqrt 2\overline z), \quad |z| \ge R.

If $a$ is weakly elliptic of order $0$ then $a$ is called elliptic.

Let $\omega \in \mascP (\rr {2d})\simeq \mascP (\cc d)$,
$\rho >0$ and $\fka \in \Sh _\rho ^{(\omega )}(\rr {2d})$.
Then the following is true:

* if $z=x+i\xi$, $x,\xi \in \rr d$, then

\begin{equation}\label{Eq:RelAssignDiag}
| \mathsf S_{\mathfrak V} \fka (z ,z) -\fka (\sqrt 2\, x,-\sqrt 2\, \xi )|
\lesssim
\omega (\sqrt 2\, \overline z)
\eabs z^{-2\rho}\text ;
\end{equation}

* if $\rho _0\in [0,2\rho )$, then $\mathsf S_{\mathfrak V}$ is
bijective from the set of weakly elliptic symbols in
$\Sh _\rho ^{(\omega )}(\rr {2d})$ of order $\rho _0$ to the set of
weakly elliptic symbols in
$\wideparen A _{\Sh ,\rho}^{(\omega )}(\cc {2d})$ of order $\rho _0$.

As a consequence of (2) in the previous theorem we get the following.

Let $\fka$ be as in Theorem <ref>. Then
the following is true:

* if $\rho _0\in [0,2\rho )$, then $\fka \in \Sh _\rho ^{(\omega )}(\rr {2d})$ is weakly elliptic of order
$\rho _0$, if and only if
$\mathsf S_{\mathfrak V}\fka \in \wideparen A _{\Sh ,\rho}^{(\omega )}(\cc {2d})$ is weakly elliptic of order $\rho _0$;

* $\fka \in \Sh _\rho ^{(\omega )}(\rr {2d})$ is elliptic if and only if
$\mathsf S_{\mathfrak V}\fka \in \wideparen A _{\Sh ,\rho}^{(\omega )}(\cc {2d})$ is elliptic.

For the proof of Theorem <ref> we need the following proposition,
related to Propositions <ref> and

Let $N\ge 0$ be an integer, $\rho \ge 0$, $\omega \in \mascP _{\Sh ,\rho}(\rr {2d})\simeq
\mascP  _{\Sh ,\rho}(\cc d)$, $\omega _k(x,\xi )=\omega (x,\xi )\eabs {(x,\xi )}^{-2\rho k}$
and $\fka \in \Sh _\rho ^{(\omega )}(\rr {2d})$. Then for some
$\fkc _N\in \Sh _\rho ^{(\omega _{N+1})}(\rr {2d})$ and
constants $\{ c_{\alpha} \}_{|\alpha| \le 2N}$ with $c_{0}=1$, it holds

\begin{equation}\label{Eq:WickShubinTaylorExp}
\mathsf S_{\mathfrak V}\fka  (2^{-\frac 12}z ,2^{-\frac 12}z)
\sum _{k=0}^N\fka _k(x,-\xi ) + \fkc _N(x,-\xi ),
\qquad
\fka _k = \sum _{|\alpha |=2k} c_{\alpha} \partial ^\alpha \fka .
\end{equation}

Let $\psi$ be as in Proposition <ref>.
If we put $z=w$, then (<ref>) and
Taylor's formula give

\begin{multline}\label{Eq:TaylorNonWick}
(2\pi )^{d} \mathsf S_{\mathfrak V}\fka  (2^{-\frac 12}z ,2^{-\frac 12}z)
(2\pi )^{\frac {3d}2} \cT _\psi \fka  (x,-\xi ,0,0)
\\[1ex]
2^d\iint _{\rr {2d}} \fka (t+x,\tau -\xi )e^{-(|t|^2+|\tau |^2)}\, dtd\tau
\sum _{k=0}^{2N+1} \fkb _k(x,-\xi ) +\fkc (x,-\xi )
\end{multline}

\begin{align*}
\fkb _k(x,\xi ) &= \frac {2^d}{k!}\iint _{\rr {2d}}
\eabs {\fka ^{(k)}(x,\xi);(t,\tau ),\dots ,(t,\tau )}
e^{-(|t|^2+|\tau |^2)}\, dtd\tau
\intertext{and}
\fkc (x,\xi )&=
\frac 1{(2N+1)!}\int _0^1 (1-\theta )^{2N+1} \fkc _\theta (x,\xi )\, d\theta ,
\intertext{with}
\fkc _\theta (x,\xi ) &= 2^d\iint _{\rr {2d}}
\eabs {\fka ^{(2N+2)}(x+\theta t,\xi +\theta \tau  );(t,\tau ),\dots ,(t,\tau )}
e^{-(|t|^2+|\tau |^2)}\, dtd\tau .
\end{align*}

If $k$ is odd, then

(t,\tau )\mapsto \eabs {\fka ^{(k)}(x,\xi);(t,\tau ),\dots ,(t,\tau )}
e^{-(|t|^2+|\tau |^2)}

is odd which implies that the integral is zero. Hence $\fkb _k(x,\xi )=0$ when
$k$ is odd. For $k=0$ we observe that the integral for $\fkb _0$ becomes

2^d\iint _{\rr {2d}} e^{-(|t|^2+|\tau |^2)}\, dtd\tau
(2\pi )^d,

and it follows from these relations that

(2\pi )^{-d}\sum _{k=0}^{2N+1}\fkb _k = \sum _{k=0}^N\fka _k,

with $\fka _k$ as in (<ref>) and $c_{0}=1$. Hence
the result follows if we prove that the last term in (<ref>) satisfies $\fkc _N \in \Sh _\rho ^{(\omega _{N+1})}(\rr {2d})$.

For $\theta \in [0,1]$ and $\alpha \in \nn {2d}$ we have

\begin{multline*}
|\partial ^\alpha \fkc _\theta (x,\xi )|
\lesssim
\iint _{\rr {2d}} |\partial^\alpha \fka ^{(2N+2)}(x+\theta t,\xi +\theta \tau  )|
\eabs {(t,\tau )}^{2N+2} e^{-(|t|^2+|\tau |^2)}\, dtd\tau
\\[1ex]
\lesssim
\iint _{\rr {2d}} \omega (x+\theta t,\xi +\theta \tau  )
\eabs {(x+\theta t,\xi +\theta \tau  )}^{-(2N+2+|\alpha |)\rho}
\eabs {(t,\tau )}^{2N+2}
e^{-(|t|^2+|\tau |^2)}\, dtd\tau
\\[1ex]
\lesssim
\omega (x,\xi)
\eabs {(x,\xi )}^{-(2N+2+|\alpha |)\rho}
\iint _{\rr {2d}}
\eabs {(t,\tau )}^{N_0}
e^{-(|t|^2+|\tau |^2)}\, dtd\tau
\\[1ex]
\asymp
\omega (x,\xi)
\eabs {(x,\xi )}^{-(2N+2+|\alpha |)\rho}
\end{multline*}

for some $N_0>0$. In the last inequality we have used the fact that $\omega$
is polynomially moderate.

This implies

|\partial ^\alpha \fkc  (x,\xi )|\lesssim \int _0^1 |\partial ^\alpha \fkc _\theta (x,\xi )|
\, d\theta
\lesssim
\omega (x,\xi)
\eabs {(x,\xi )}^{-(2N+2+|\alpha |)\rho},

which shows that $\fkc, \fkc _N \in \Sh _\rho ^{(\omega _{N+1})}(\rr {2d})$.

Let $\psi$ be as in Proposition <ref> and $N=0$
in Proposition <ref>. Then

\begin{equation}\tag*{(\ref{Eq:RelAssignDiag})$'$}
| \mathsf S_{\mathfrak V}\fka (2^{-\frac 12}z ,2^{-\frac 12}z) -\fka (x,-\xi )|
\lesssim
\omega (x,-\xi)
\eabs {(x,-\xi )}^{-2\rho},
\end{equation}

and (1) follows.

Suppose $\rho _0\in [0,2\rho)$. Then it follows from the latter inequality that

| \mathsf S_{\mathfrak V}\fka  (z ,z)|
\gtrsim \eabs z^{-\rho _0}\omega (\sqrt 2\, \overline z),
\qquad |z|\ge R

for some $R>0$, if and only if

|\fka (x,\xi )|\gtrsim \eabs {(x,\xi )}^{-\rho _0}\omega (x,\xi ), \qquad |z|\ge R

for some $R>0$, and the asserted equivalence in (2) follows.

§.§ Shubin hypoellipticity in a Wick setting

Let $\rho >0$, $\rho _0\ge 0$,
$\omega \in \mascP _{\Sh ,\rho}(\cc d)$ and
$a\in \wideparen \maclA _{\Sh ,\rho}^{(\omega )}(\cc {2d})$.
Then $a$ is called hypoelliptic (in the Shubin-Wick sense
in $\wideparen \maclA _{\Sh ,\rho}^{(\omega )}(\cc {2d})$) of order $\rho _0$,
if there is an $R>0$
such that for every $\alpha ,\beta \in \nn d$, it holds

|z|    ≥R .

|a(z,z)|    ≳ω_0(√(2) z)
|z|    ≥R.

According to Definition <ref>,
if $\omega$, $\rho$ and $\rho _0$ are as in
the definition, then $\fka \in \Sh _\rho ^{(\omega )}(\rr {2d})$
is hypoelliptic

of order $\rho _0$
means that
there is an $R>0$ such that for every $\alpha  \in \nn {2d}$, it holds

|(x,ξ)|    ≥R .

|(x,ξ)|    ≳ω(x,ξ)
|(x,ξ)|    ≥R.

Similar to Theorem <ref> we have the following.

Let $\rho >0$, $\rho _0\ge 0$,
$\omega \in \mascP _{\Sh ,\rho}(\rr {2d})\simeq
\mascP _{\Sh ,\rho}(\cc d)$,
$\fka \in \Sh _\rho ^{(\omega )}(\rr {2d})$ and $a=\mathsf S_{\mathfrak V}\fka$.
Then $\fka$ is hypoelliptic of order $\rho _0$ in
$\Sh _\rho ^{(\omega )}(\rr {2d})$, if and only if
$a$ is hypoelliptic of order $\rho _0$
in $\wideparen \maclA _{\Sh ,\rho}^{(\omega )}(\cc {2d})$.

Suppose that $\fka \in \Sh _\rho ^{(\omega )}(\rr {2d})$ is hypoelliptic of order $\rho _0$, and choose $N\ge 0$ such that
$2N\rho >\rho _0$. Suppose that
$R>0$ is chosen such that (<ref>)
and (<ref>) are fulfilled.
Then Proposition <ref> gives for $z=x+i\xi$ with
$|z|\ge R$ where $R > 0$ is sufficiently large

\begin{multline*}
|a(2^{-\frac 12}z,2^{-\frac 12}z)|
\gtrsim
|\fka (x,-\xi )|-
\sum _{k=1}^N\sum _{|\alpha |=2k}
(|\partial ^\alpha \fka (x,-\xi )| +
|\fkc (x,-\xi )| )
\\[1ex]
\gtrsim
|\fka (x,-\xi )| -|\fka (x,-\xi )|\eabs {(x,-\xi )}^{-2\rho}-
\omega (x,-\xi )\eabs {(x,-\xi )}^{-\rho (2N+2)}
\\[1ex]
\gtrsim
|\fka (x,-\xi )| -|\fka (x,-\xi )|\eabs {(x,-\xi )}^{-2\rho}
\\[1ex]
\gtrsim
|\fka (x,-\xi )| \gtrsim \omega (x,-\xi )\eabs {(x,-\xi )}^{-\rho _0},
\end{multline*}

and (<ref>) follows. In particular
it follows from the previous estimates that

\begin{equation}\label{Eq:IneqHypoShubinHypoWick}
|a(2^{-\frac 12}z,2^{-\frac 12}z)|
\gtrsim
|\fka (x,-\xi )|,
\qquad |z|\ge R.
\end{equation}

For fixed $\alpha ,\beta \in \nn d$, let
$\Omega _k$ be the set of all $(\gamma ,\delta )\in \nn {2d}\times \nn {2d}$
such that $|\gamma |=2k$ and $|\delta |=|\alpha +\beta |$. By Proposition <ref> and
(<ref>) we have for
some $R$ large enough and $|z|\ge R$,

\begin{multline*}
|(\partial _z^\alpha \overline \partial _w^\beta a)(2^{-\frac 12}z,2^{-\frac 12}z)|
\lesssim
\sum _{k=0}^N\sum _{(\gamma ,\delta )\in \Omega _k}
(|\partial ^{\gamma +\delta}\fka (x,-\xi )| +
|\partial ^\delta \fkc (x,-\xi )| )
\\[1ex]
\lesssim
\sum _{k=0}^N\sum _{(\gamma ,\delta )\in \Omega _k}
%\sum _{|\gamma |=2k}\sum _{|\delta |=|\alpha +\beta |}
\big ( |\fka (x,-\xi )|\eabs {(x,-\xi )}^{-\rho (2k+|\alpha +\beta |)} +
\omega (x,-\xi )\eabs {(x,-\xi )}^{-\rho (2N+|\alpha +\beta |)} \big )
\\[1ex]
\asymp
|\fka (x,-\xi )|\eabs {(x,-\xi )}^{-\rho |\alpha +\beta |} +
\omega (x,-\xi )\eabs {(x,-\xi )}^{-\rho (2N+|\alpha +\beta |)}
\\[1ex]
\lesssim
|\fka (x,-\xi )|\eabs {(x,-\xi )}^{-\rho |\alpha +\beta |} +
|\fka (x,-\xi )|\eabs {(x,-\xi )}^{\rho _0-\rho (2N+|\alpha +\beta |)}
\\[1ex]
\asymp
|\fka (x,-\xi )|\eabs {(x,-\xi )}^{-\rho |\alpha +\beta |}