Pseudo-differential operators with isotropic symbols, Wick and anti-Wick operators, and hypoellipticity

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 anti-Wick operators which leads to a sharp G<span></span>rding 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 {#sec0}
============

In the paper we investigate conjugation with the Bargmann transformation of pseudo-differential and Toeplitz operators on ${\mathbf R^{d}}$ 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 ${\mathbf C^{d}}$ that are called Wick or Berezin operators because of the fundamental contributions by F. Berezin @Berezin71 [@Berezin72], which in turns goes back to some ideas in @Wi by G. C. Wick.

Let $a$ be a suitable locally bounded function on ${\mathbf C^{2d}}$ such that $z\mapsto a(z,w)$ is analytic, $z,w\in {\mathbf C^{d}}$. Then the Wick operator ${\operatorname{Op}}_{\mathfrak V}(a)$ with symbol $a$ is the operator which takes an appropriate entire function $F$ on ${\mathbf C^{d}}$ into the entire function $$\label{Eq:AnalPseudoIntro}
{\operatorname{Op}}_{\mathfrak V}(a)F (z) = \pi ^{-d}
\int _{{\mathbf C^{d}}} a(z,w)F(w)e^{(z-w,w)}\, d\lambda (w),$$ where $d\lambda$ is the Lebesgue measure and $({\, \cdot \, },{\, \cdot \, })$ is the scalar product on ${\mathbf C^{d}}$. (See @Ho1 and Section \[sec1\] 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 $$F\mapsto {\langle z,\nabla _z\rangle}F +cF,
\quad
F\mapsto z_jF
\quad \text{and}\quad
F\mapsto \partial _{z_j}F,$$ respectively, for some constant $c$, in the Wick formulation (see @B1).

An advantage of the Wick calculus compared to corresponding operators on functions and distributions defined on ${\mathbf R^{d}}$ 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 @Teofanov2, 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 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 ${\operatorname{Op}}_{\mathfrak V}(a)={\operatorname{Op}}_{\mathfrak V}(a_0)$. Consequently it is no restriction to assume that $a(z,w)$ in 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 Pilipovi<span>ć</span> space, to the corresponding distribution spaces, respectively, is in a unique way transformed into a Wick operator by the Bargmann transform (see @Teofanov2).

Several operators in quantum mechanics are so-called Shubin operators, i.e. pseudo-differential operators $${\operatorname{Op}}({\mathfrak a})f(x) = (2\pi )^{-\frac d2}\int _{{\mathbf R^{d}}}{\mathfrak a}(x,\xi )
\widehat f(\xi )e^{i{\langle x,\xi\rangle}}\, d\xi ,
\qquad f\in {\mathscr S}({\mathbf R^{d}}),$$ where the symbol ${\mathfrak a}$ belongs to the Shubin class ${\operatorname{Sh}}_\rho ^{(\omega )}({\mathbf R^{2d}})$, the set of all ${\mathfrak a}\in C^\infty ({\mathbf R^{2d}})$ such that $$|\partial _x^\alpha \partial _\xi ^\beta {\mathfrak a}(x,\xi )|
\lesssim
\omega (x,\xi )(1+|x|+|\xi |)^{-\rho |\alpha +\beta |},\quad \alpha ,\beta \in {\mathbf N^{d}}.$$ Here $\omega$ is a suitable weight function on ${\mathbf R^{2d}}$ 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 \[sec2\] we prove that the Bargmann image of Shubin operators with symbols in ${\operatorname{Sh}}_\rho ^{(\omega )}({\mathbf R^{2d}})$ is the set of all Wick operators in such that $a$ belongs to $\wideparen {\mathcal A}_{{\operatorname{Sh}},\rho}^{(\omega )}({\mathbf C^{2d}})$. This means that ${\mathbf C^{2d}} \ni (z,w)\mapsto a(z,\overline w)$ is an entire function that satisfies $$\label{Eq:AnalShubinIntro}
|\partial _z^\beta \overline \partial _w^\gamma a(z,w)|
\lesssim
e^{\frac 12|z-w|^2}\omega (\sqrt 2 \overline z)
{\langle z+w\rangle}^{-\rho |\beta +\gamma |} {\langle z-w\rangle}^{-N}$$ 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 ${\mathbf C^{d}}$, its anti-Wick operator is given by $$\tag*{(\ref{Eq:AnalPseudoIntro})$'$}
{\operatorname{Op}}_{\mathfrak V}^{{\operatorname{aw}}}(a_0)F (z) = \pi ^{-d}
\int _{{\mathbf C^{d}}} a_0(w)F(w)e^{(z-w,w)}\, d\lambda (w).$$ Again $F$ is a suitable entire function on ${\mathbf C^{d}}$. The anti-Wick operators can also be described as the Bargmann image of Toeplitz operators on ${\mathbf R^{d}}$. (See e.g. @LieSol [@Shubin1; @To11] 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. @LieSol [@Lieb1; @Lieb2].) An operator $T={\operatorname{Op}}_{\mathfrak V}(a)$ with $a$ satisfying for every $N\ge 0$, is called *positive* (*non-negative*), if there is a constant $C> 0$ ($C\ge 0$) such that $$(TF,F)_{A^2} \ge C{\Vert F\Vert _{A^2}}^2,$$ for every analytic polynomial $F$ on ${\mathbf C^{d}}$, where $({\, \cdot \, },{\, \cdot \, })_{A^2}$ is the scalar product induced by the Hilbert norm $${\Vert F\Vert _{A^2}} =\pi ^{-\frac d2}
\left ( \int _{{\mathbf C^{d}}} |F(z)|^2e^{-|z|^2}\, d\lambda (z) \right )^{\frac 12}.$$

The implication from non-negative symbols to non-negative operators is not relevant for Wick operators in 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. @Berezin71 [@Berezin72; @Fo]). By choosing $$d=1,\quad a(z,w) = 1-2z\overline w +2z^2\overline w^2
\quad \text{and}\quad
F(z)=z$$ we obtain $$a(w,w)=(1-|w|^2)^2+|w|^4> 0
\quad \text{but}\quad
({\operatorname{Op}}_{\mathfrak V}(a)F,F)_{A^2} =-1<0.$$ Consequently ${\operatorname{Op}}_{\mathfrak V}(a)$ 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 \[sec3\] a weaker positivity result for Wick operators, which is equivalent to the sharp G<span>Ã¥</span>rding inequality in isotropic pseudo-differential calculus on ${\mathbf R^{d}}$ (see Theorem 18.6.7 and the proof of Theorem 18.6.8 in @Ho1). That is for $a\in \wideparen {\mathcal A}_{{\operatorname{Sh}},\rho}^{(\omega )}({\mathbf C^{2d}})$ with $\omega (z)={\langle z\rangle}^{2\rho}$ and $\rho > 0$ we prove $$\begin{aligned}
{{\operatorname{Re}}}({\operatorname{Op}}_{\mathfrak V}(a)F,F)_{A^2}
&\ge -C{\Vert F\Vert _{A^2}}^2
\label{Eq:SharpGardingIntro}
\intertext{and}
|{{\operatorname{Im}}}({\operatorname{Op}}_{\mathfrak V}(a)F,F)_{A^2}|
&\le C{\Vert F\Vert _{A^2}}^2,
\quad \text{when}\quad
a(w,w) \ge 0
\label{Eq:SharpGardingIntroImPart}\end{aligned}$$ (cf. Theorem \[Thm:ShGarding\]). 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 with $a\in \wideparen {\mathcal A}_{{\operatorname{Sh}},\rho}^{(\omega )}({\mathbf C^{2d}})$ the remarkable identity $$\label{Eq:WickToAntiWickIntro}
{\operatorname{Op}}_{\mathfrak V}(a)
=
\sum _{|\alpha |<N} \frac {(-1)^{|\alpha |}}{\alpha !}
{\operatorname{Op}}_{\mathfrak V}^{{\operatorname{aw}}}(b_\alpha ) + {\operatorname{Op}}_{\mathfrak V}(c_N)
\quad \text{where} \quad
b_\alpha (w)
=
\partial _z^\alpha \overline \partial _{w}^\alpha a(w,w),$$ for some $c_N\in {\mathcal A}_{{\operatorname{Sh}},\rho}^{(\omega _N)}({\mathbf C^{2d}})$ with $\omega _N(z)= \omega (z){\langle z\rangle}^{-2N\rho}$. Here we again assume $\rho > 0$. The decay conditions on $b_\alpha$ and $c_N$ are, respectively, $$\begin{aligned}
|\partial _w^\beta \overline \partial _w^\gamma b_\alpha (w)|
& \lesssim
\omega (\sqrt 2 \overline w){\langle w\rangle}^{-\rho |2\alpha +\beta +\gamma |},
\qquad
\alpha, \beta, \gamma  \in {\mathbf N^{d}},
\label{Eq:WickAntiWickbalphaIntro}
\intertext{and}
|\partial _z^\beta \overline \partial _w^\gamma c_N(z,w)|
& \lesssim
e^{\frac 12|z-w|^2}\omega (\sqrt 2 \overline z){\langle z\rangle}^{-2N\rho}
{\langle z+w\rangle}^{-\rho |\beta +\gamma |} {\langle z-w\rangle}^{-N}.\end{aligned}$$ Consequently, many Wick operators can essentially be expressed as linear combinations of anti-Wick operators. The expansion is deduced in Section \[sec3\] using Taylor expansion and integration by parts, see Proposition \[Prop:WickToAntiWick\] and Remark \[Rem:WickToAntiWick2\].

The conditions on $b_\alpha$ are the same as the conditions on $a$ , 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 when $|z-w|\gtrsim |z|$ or $|z-w|\gtrsim |w|$. The right-hand side of becomes as large as possible when $\alpha =\beta =\gamma =0$, that is $b_0$ is the dominating term in the sum .

The conditions on $c_N$ are the same as the estimates again with improved decay due to the factor ${\langle z\rangle}^{-2N\rho}$.

For polynomial symbols, agree with the integral formula @Berezin71 [Theorem 3] due to Berezin which carry over Wick operators into anti-Wick operators. For the general case, is analogous to the approximation technique of pseudo-differential operators on ${\mathbf R^{d}}$ in terms of Toeplitz operators given in @Shubin1 [Theorem 24.1] and its proof, by Shubin.

The anti-Wick symbols in $b_\alpha (w)=\partial _z^\alpha \overline \partial _w^\alpha a (w,w)$ 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 on Wick symbols.

The sharp G<span>Ã¥</span>rding inequality is reached by using the fact that ${\operatorname{Op}}_{\mathfrak V}^{{\operatorname{aw}}}(b_0)$ is non-negative, and that if $T$ is either ${\operatorname{Op}}^{{\operatorname{aw}}}_{\mathfrak V}(b_\alpha )$ or ${\operatorname{Op}}_{\mathfrak V}(c_N)$ for $\alpha \neq 0$, then ${\Vert TF\Vert _{A^2}}\lesssim {\Vert F\Vert _{A^2}}$ when $F\in A({\mathbf C^{d}})$ is a polynomial.

In Section \[sec5\] 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 @Shubin1). More specifically, we say that the symbol ${\mathfrak a}\in {\operatorname{Sh}}_\rho ^{(\omega )}({\mathbf R^{2d}})$ is hypoelliptic of order $\rho _0\ge 0$, whenever there is an $R>0$ such that $$|{\mathfrak a}(x,\xi )|\gtrsim \omega (x,\xi ){\langle (x,\xi )\rangle}^{-\rho _0}
\quad \text{and}\quad |\partial ^\alpha {\mathfrak a}(x,\xi )|
\lesssim |{\mathfrak a}(x,\xi )|{\langle (x,\xi )\rangle}^{-\rho |\alpha |}$$ when $|(x,\xi )|\ge R$.

A linear operator $T$ from ${\mathscr S}'({\mathbf R^{d}})$ to ${\mathscr S}'({\mathbf R^{d}})$ is called globally hypoelliptic if $$Tf=g,\ f\in {\mathscr S}'({\mathbf R^{d}}),\ g\in {\mathscr S}({\mathbf R^{d}})
\quad \Rightarrow \quad
f\in {\mathscr S}({\mathbf R^{d}}).$$ (See e.g. @BogBuzRod.) It can be proved that a pseudo-differential operator with hypoelliptic symbol in Shubin's sense is globally hypoelliptic as operator (see e.g. @Shubin1 [Corollary 25.1]).

We show, similarly to our investigations of the sharp G<span>Ã¥</span>rding inequality and for expansion , that ellipticity, hypoellipticity and to some degree weak ellipticity for the Shubin symbol ${\mathfrak a}$ can be characterized by certain conditions for the corresponding Wick symbol $a(z,w)$ along the diagonal $z=w$. For example, let ${\mathfrak a}$ be a polynomial on ${\mathbf R^{d}}$ with principal symbol ${\mathfrak a}_p$, and let $a(z,w)$ be a polynomial in $z, \overline w \in {\mathbf C^{d}}$ with principal part $a_p$. Then ${\mathfrak a}$ is elliptic means that ${\mathfrak a}_p(x,\xi )\neq 0$ when $(x,\xi )\neq (0,0)$, and $a$ is elliptic means that $a_p(z,z)\neq 0$ when $z\neq 0$. For such ${\mathfrak a}$ we prove $${\mathfrak a}\ \text{is elliptic}
\quad \Leftrightarrow \quad
a\ \text{is elliptic},$$ when $a(z,w)$ is the Wick symbol corresponding to ${\mathfrak a}$ (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 ${\operatorname{Op}}_{\mathfrak V}(a)$ of operators ${\operatorname{Op}}({\mathfrak a})$ of infinite order in @AbCaTo, and characterize their images under the Bargmann transform (see Theorem \[Thm:GevreySymbolsBargmTransfer\]). Then we deduce in Subsections \[subsec3.2\] and \[subsec3.3\] continuity results for anti-Wick operators which holds for the symbols $b_\alpha$ in when ${\operatorname{Op}}_{\mathfrak V}(a)$ is the Bargmann image of an operator of infinite order.

In fact, in Subsection \[subsec3.2\] we show that ${\operatorname{Op}}_{\mathfrak V}^{{\operatorname{aw}}}(b_\alpha )$ possess several other continuity properties than what is valid for ${\operatorname{Op}}_{\mathfrak V}(a)$ in the expansion (see Propositions \[prop:contAW\] and \[prop:contAW2\]). In Subsection \[subsec3.3\] we deduce estimates of the Wick symbol $b_{\alpha}^{{\operatorname{aw}}}$ to the anti-Wick operator ${\operatorname{Op}}_{\mathfrak V}^{{\operatorname{aw}}}(b_\alpha )$, i.e. the unique element $b_{\alpha}^{{\operatorname{aw}}}\in \wideparen A({\mathbf C^{2d}})$ such that ${\operatorname{Op}}_{\mathfrak V}(b_{\alpha}^{{\operatorname{aw}}})
={\operatorname{Op}}_{\mathfrak V}^{{\operatorname{aw}}}(b_\alpha )$. We show that usually, $b_{\alpha}^{{\operatorname{aw}}}$ satisfies stronger conditions than $a$ when ${\operatorname{Op}}_{\mathfrak V}(a)$ is a Wick operator of infinite order (see Theorems \[Thm:WickSymbolAntiWickOpFolland\], \[Thm:WickSymbolAntiWickOpGS\] and \[Thm:WickSymbolAntiWickOpSOmega\]).

The paper is organized as follows. In Section \[sec1\] 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 \[sec2\] 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 \[Prop:SBaTaRel\]).

In Section \[sec2\] 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 \[sec3\] 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 \[sec4\] we discuss lower bounds for Wick operators and deduce the sharp G<span>å</span>rding's inequality. Section \[sec5\] concerns ellipticity, hypoellipticity and weak ellipticity.

Finally we observe in Section \[sec6\] 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 {#sec1}
=============

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 \[subsec1.1\] concerns weight functions and Subsection \[subsec1.2\] treats Gelfand-Shilov spaces. In Subsection \[subsec1.3\] we introduce the Bargmann transform and topological spaces of entire functions on ${\mathbf C^{d}}$, and in Subsection \[subsec1.4\] we recall the definitions and some facts on pseudo-differential operators on ${\mathbf R^{d}}$ as well as Wick and anti-Wick operators on ${\mathbf C^{d}}$. Subsection \[subsec1.5\] defines certain symbol classes for pseudo-differential operators on ${\mathbf R^{d}}$.

Weight functions {#subsec1.1}
----------------

A *weight* on ${\mathbf R^{d}}$ is a positive function $\omega \in  L^\infty _{loc}({\mathbf R^{d}})$ such that $1/\omega \in  L^\infty _{loc}({\mathbf R^{d}})$. The weight $\omega$ is called *moderate* if there is a positive locally bounded function $v$ such that $$\label{eq:2}
\omega(x+y)\le C\omega(x)v(y),\quad x,y\in{\mathbf R^{d}},$$ for some constant $C\ge 1$. If $\omega$ and $v$ are weights such that holds, then $\omega$ is also called *$v$-moderate*. The set of all moderate weights on ${\mathbf R^{d}}$ is denoted by ${\mathscr P}_E({\mathbf R^{d}})$. The set ${\mathscr P}({\mathbf R^{d}})$ consists of weights that are $v$-moderate for a polynomially bounded weight, that is a weight of the form $v(x) = {{\langle x\rangle}}^s$ where ${\langle x\rangle} = (1+|x|^2)^{\frac 12}$ and $s \ge 0$. The bracket notation is also used for complex arguments as ${\langle z\rangle}=(1+|z|^2)^{\frac 12}$ when $z\in {\mathbf C^{d}}$. If $s\in \mathbf R$ then $x\mapsto {\langle x\rangle}^s$ belongs to ${\mathscr P}({\mathbf R^{d}})$, due to Peetre's inequality $$\label{eq:Peetre}
{\langle x+y\rangle}^s {\leqslant}2^{|s|} {\langle x\rangle}^s{\langle y\rangle}^{|s|}\qquad x,y\in{\mathbf R^{d}},
\qquad s \in {\mathbf R}.$$

The weight $v$ is called *submultiplicative* if it is even and holds for $\omega =v$. If holds and $v$ is submultiplicative then $$\label{eq:2Next}
\begin{gathered}
\frac {\omega (x)}{v(y)} \lesssim \omega(x+y) \lesssim \omega(x)v(y),
\\[1ex]
\quad
v(x+y) \lesssim v(x)v(y)
\quad \text{and}\quad v(x)=v(-x),
\quad x,y\in{\mathbf R^{d}}.
\end{gathered}$$ 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 @To11 there is a submultiplicative weight $v$ such that and hold. If $v$ is submultiplicative then $$\label{Eq:CondSubWeights}
1\lesssim v(x) \lesssim e^{r|x|}$$ for some constant $r>0$ (cf. @Groch). In particular, if $\omega$ is moderate, then $$\label{Eq:ModWeightProp}
\omega (x+y)\lesssim \omega (x)e^{r|y|}
\quad \text{and}\quad
e^{-r|x|}\lesssim \omega (x)\lesssim e^{r|x|},\quad
x,y\in {\mathbf R^{d}}$$ for some $r>0$. If not otherwise specified the symbol $v$ always denote a submultiplicative weight.

Gelfand-Shilov spaces {#subsec1.2}
---------------------

Let $s ,\sigma > 0$. The Gelfand-Shilov space ${\mathcal S}_s^\sigma ({\mathbf R^{d}})$ ($\Sigma _s^\sigma ({\mathbf R^{d}})$) of Roumieu (Beurling) type consists of all $f\in C^\infty ({\mathbf R^{d}})$ such that $$\label{gfseminorm}
{\Vert f\Vert _{{\mathcal S}_{s,h}^\sigma}}\equiv \sup \frac {|x^\alpha \partial ^\beta
f(x)|}{h^{|\alpha  + \beta |}\alpha !^s \, \beta !^\sigma}$$ is finite for some (every) $h>0$. The supremum refers to all $\alpha ,\beta \in \mathbf N^d$ and $x\in {\mathbf R^{d}}$. The seminorms ${\Vert {\, \cdot \, }\Vert _{{\mathcal S}_{s,h}^\sigma}}$ induce an inductive limit topology for the space ${\mathcal S}_s^\sigma ({\mathbf R^{d}})$ and a projective limit topology for $\Sigma _s^\sigma ({\mathbf R^{d}})$. The latter space is a Fr<span>é</span>chet space under this topology. The space ${\mathcal S}_s^\sigma ({\mathbf R^{d}})\neq \{ 0\}$ ($\Sigma _s^\sigma ({\mathbf R^{d}})\neq \{0\}$), if and only if $s+\sigma \ge 1$ ($s+\sigma \ge 1$ and $(s,\sigma) \neq (\frac 12, \frac 12)$). We write ${\mathcal S}_s ({\mathbf R^{d}}) = {\mathcal S}_s^s ({\mathbf R^{d}})$ and $\Sigma _s ({\mathbf R^{d}}) = \Sigma _s^s ({\mathbf R^{d}})$.

The *Gelfand-Shilov distribution spaces* $({\mathcal S}_s^\sigma )'({\mathbf R^{d}})$ and $(\Sigma _s^\sigma )'({\mathbf R^{d}})$ are the dual spaces of ${\mathcal S}_s^\sigma ({\mathbf R^{d}})$ and $\Sigma _s^\sigma ({\mathbf R^{d}})$, respectively.

The embeddings $$\begin{gathered}
\label{GSembeddings}
{\mathcal S}_{s_1}^{\sigma _1} ({\mathbf R^{d}}) \hookrightarrow \Sigma _{s_2}^{\sigma _2}({\mathbf R^{d}})
\hookrightarrow
{\mathcal S}_{s_2}^{\sigma _2} ({\mathbf R^{d}})
\hookrightarrow
{\mathscr S}({\mathbf R^{d}})
\\[1ex]
\hookrightarrow {\mathscr S}'({\mathbf R^{d}}) \hookrightarrow  ({\mathcal S}_{s_2}^{\sigma _2})'({\mathbf R^{d}})
\hookrightarrow  (\Sigma _{s_2}^{\sigma _2})'({\mathbf R^{d}})
\hookrightarrow ({\mathcal S}_{s_1}^{\sigma _1}) '({\mathbf R^{d}}),
\\[1ex]
s_1+\sigma _1 \ge 1,\ s_1<s_2,\ \sigma _1<\sigma _2,\end{gathered}$$ are dense. For topological spaces $A$ and $B$, $A\hookrightarrow B$ means that the inclusion $A\subseteq B$ is continuous.

The spaces ${\mathcal S}_s$ and $\Sigma_s$, and their duals spaces, admit characterizations in terms of coefficients with respect to expansions with respect to the Hermite functions $$h_\alpha (x) = \pi ^{-\frac d4}(-1)^{|\alpha |}
(2^{|\alpha |}\alpha !)^{-\frac 12}e^{\frac {|x|^2}2}
(\partial ^\alpha e^{-|x|^2}),\quad \alpha \in {\mathbf N^{d}}.$$ The set of Hermite functions on ${\mathbf R^{d}}$ is an orthonormal basis for $L^2({\mathbf R^{d}})$. We use ${\mathcal H}_0({\mathbf R^{d}})$ to denote the space of finite linear combinations of Hermite functions. Then ${\mathcal H}_0({\mathbf R^{d}})$ is dense in the Schwartz space ${\mathscr S}({\mathbf R^{d}})$, as well as in ${\mathscr S}'({\mathbf R^{d}})$, with respect to its weak$^*$ topology. The same conclusion is true for $\Sigma _s({\mathbf R^{d}})$ when $s>\frac 12$, ${\mathcal S}_s({\mathbf R^{d}})$ when $s\ge \frac 12$ and their distribution dual spaces $\Sigma _s'({\mathbf R^{d}})$ and ${\mathcal S}_s'({\mathbf R^{d}})$. An $f$ in any of these spaces possess an expansion of the form $$\label{Eq:HermiteExpansions}
f=\sum _{\alpha \in {\mathbf N^{d}}}c(f,\alpha )h_\alpha ,
\quad
c (f,\alpha )=(f,h_\alpha ), \quad \alpha \in {\mathbf N^{d}}.$$ Here $({\, \cdot \, }, {\, \cdot \, })$ denotes the unique extensions of the $L^2$ form, which is linear in the first variable and conjugate linear in the second variable, from ${\mathcal H}_0({\mathbf R^{d}})\times {\mathcal H}_0({\mathbf R^{d}})$ to ${\mathcal S}_s'({\mathbf R^{d}})\times {\mathcal S}_s({\mathbf R^{d}})$ or $\Sigma _s'({\mathbf R^{d}})\times \Sigma _s({\mathbf R^{d}})$. We recall that (cf. @RS [Chapter V.3 ]) $$\begin{alignedat}{5}
f&\in {\mathscr S}({\mathbf R^{d}}) & \quad &\Leftrightarrow &\quad
|c(f,\alpha )| &\lesssim {\langle \alpha\rangle} ^{-N} &
\ &\text{for every}& \ N&\ge 0 ,
\\[1ex]
f&\in {\mathscr S}'({\mathbf R^{d}}) & \quad &\Leftrightarrow &\quad
|c(f,\alpha )| &\lesssim {\langle \alpha\rangle} ^{N} &
\ &\text{for some}& \ N&\ge 0.
\end{alignedat}$$ The topology on ${\mathscr S}({\mathbf R^{d}})$ is equivalent to the Fréchet space topology defined by the sequence space seminorms $${\mathscr S}({\mathbf R^{d}}) \ni f \mapsto \sum_{\alpha \in {\mathbf N^{d}}}
{\langle \alpha\rangle} ^{2N} |c(f,\alpha )|^2, \quad N \ge 0.$$ For $f \in {\mathscr S}'({\mathbf R^{d}})$ the sum in 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 $$\mathscr Ff (\xi )= \widehat f(\xi ) \equiv (2\pi )^{-\frac d2}\int _{{\mathbf R^{d}}} f(x)e^{-i{\langle x,\xi\rangle} }\, dx, \quad \xi \in {\mathbf R^{d}},$$ when $f\in L^1({\mathbf R^{d}})$. Here ${\langle {\, \cdot \, },{\, \cdot \, }\rangle}$ denotes the scalar product on ${\mathbf R^{d}}$. In fact $$H_dh_\alpha = (2|\alpha |+d)h_\alpha, \quad \alpha \in {\mathbf N^{d}}.$$

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

Let $\phi \in {\mathscr S}({\mathbf R^{d}}) \setminus 0$ be fixed. We use the transform $$\label{eq:cTdef}
\begin{aligned}
{\mathcal{T}}_\phi  f(x,\xi) &= (2\pi)^{-\frac d2}e^{i \langle x, \xi \rangle}
(f,e^{i{\langle {\, \cdot \, },\xi\rangle}}\phi ({\, \cdot \, }-x))
\\[1ex]
&=
e^{i \langle x, \xi \rangle}{\mathscr F}(f\cdot \overline {\phi ({\, \cdot \, }-x)})(\xi )
=
{\mathscr F}(f({\, \cdot \, }+x)\overline \phi )(\xi ), \quad x, \xi \in {\mathbf R^{d}},
\end{aligned}$$ where $f \in {\mathscr S}'({\mathbf R^{d}})$ and $\phi  \in {\mathscr S}({\mathbf R^{d}}) \setminus 0$ (cf. @Cappiello1). If $f ,\phi \in {\mathscr S}({\mathbf R^{d}})$ then $$\begin{aligned}
{\mathcal{T}}_\phi f(x,\xi )
&=
(2\pi )^{-\frac d2}e^{i \langle x, \xi \rangle}
\int _{{\mathbf R^{d}}} f(y)\overline {\phi
(y-x)}e^{-i{\langle y,\xi\rangle}}\, dy
\\[1ex]
&=
(2\pi )^{-\frac d2}
\int _{{\mathbf R^{d}}} f(y+x)\overline {\phi
(y)}e^{-i{\langle y,\xi\rangle}}\, dy, \quad x,\xi \in {\mathbf R^{d}}.\end{aligned}$$

We notice that the short-time Fourier transform $V_{\phi}f$ of $f$ is given by $$\label{Eq:LinkSTFTandTmap}
V_{\phi}f(x,\xi) = e^{-i \langle x, \xi \rangle} {\mathcal{T}}_\phi  f(x,\xi).$$ Thus by @To11 [Theorem 2.3] it follows that the definition of the map $(f,\phi)\mapsto {\mathcal{T}}_{\phi} f$ from ${\mathscr S}({\mathbf R^{d}}) \times {\mathscr S}({\mathbf R^{d}})$ to ${\mathscr S}({\mathbf R^{2d}})$ is uniquely extendable to a continuous map from ${\mathcal S}_s'({\mathbf R^{d}})\times {\mathcal S}_s'({\mathbf R^{d}})$ to ${\mathcal S}_s'({\mathbf R^{2d}})$, and restricts to a continuous map from ${\mathcal S}_s ({\mathbf R^{d}})\times {\mathcal S}_s ({\mathbf R^{d}})$ to ${\mathcal S}_s({\mathbf R^{2d}})$. The same conclusion holds with $\Sigma _s$ in place of ${\mathcal S}_s$, at each place.

The adjoint ${\mathcal{T}}_\phi ^*$ is given by $$({\mathcal{T}}_\phi^* F, g)_{L^2({\mathbf R^{d}})}
=
(F, {\mathcal{T}}_\phi g)_{L^2({\mathbf R^{2d}})}$$ for $F \in {\mathcal S}_s'({\mathbf R^{2d}})$ and $g \in {\mathcal S}_s({\mathbf R^{d}})$, and similarly with $\Sigma _s$ or with ${\mathscr S}$ in place of ${\mathcal S}_s$ at each occurrence. When $F$ is a polynomially bounded measurable function we write $$\label{Eq:Moyalsformula}
{\mathcal{T}}_\phi^* F(y) = (2\pi)^{-\frac d2} \iint _{{\mathbf R^{2d}}} 
F(x,\xi) \, e^{i{\langle y-x,\xi\rangle}}\phi (y-x)
\, d x d \xi ,$$ where the integral is defined weakly so that $({\mathcal{T}}_\phi^* F, g)_{L^2({\mathbf R^{d}})}
=
(F, {\mathcal{T}}_\phi g)_{L^2({\mathbf R^{2d}})}$ for $g \in {\mathscr{S}}({\mathbf R^{d}})$. The identity is called Moyal's formula.

We have $$\label{eq:reproducing}
({\mathcal{T}}_\psi ^*\circ {\mathcal{T}}_\phi )f
=(\psi ,\phi ) f, \qquad f \in {\mathcal S}_s'({\mathbf R^{d}}),\ \phi ,\psi \in {\mathcal S}_s({\mathbf R^{d}}),$$ and similarly with $\Sigma _s$ or with ${\mathscr S}$ in place of ${\mathcal S}_s$ at each occurrence.

Two important features of ${\mathcal{T}}_\phi$ which distinguish it from the short-time Fourier transform are the differential identities $$\begin{aligned}
\label{eq:diffident}
\partial_x^\alpha {\mathcal{T}}_\phi f (x,\xi) & = {\mathcal{T}}_\phi (\partial^\alpha f) (x,\xi),
\qquad \alpha \in {\mathbf N^{d}}
\intertext{and}
\label{eq:diffidentstar}
D_{\xi}^\beta {\mathcal{T}}_\phi f (x,\xi) & = {\mathcal{T}}_{g_\beta} f(x,\xi), \qquad \beta \in {\mathbf N^{d}}, \qquad
\phi _\beta (x) = (-x)^\beta \phi (x).\end{aligned}$$

By 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 ${\mathcal{T}}_\phi$ in place of $V_\phi$. For example we have the following (see e.g. @GZ [@Teof] for the proof of (1) and @Toft18 for the proof of (2)). See also @CPRT10 for related results.

\[stftGelfand2\] Let $s,\sigma >0$, $\phi \in {\mathcal S}_s^\sigma ({\mathbf R^{d}})\setminus 0$ ($\phi \in \Sigma _s^\sigma ({\mathbf R^{d}})\setminus 0$) and let $f\in ({\mathcal S}_s^\sigma )'({\mathbf R^{d}})$ ($f\in (\Sigma _s^\sigma )'({\mathbf R^{d}})$). Then the following is true:

1.  $f\in {\mathcal S}_s^\sigma ({\mathbf R^{d}})$ ($f\in \Sigma _s^\sigma ({\mathbf R^{d}})$) if and only if $$\label{stftexpest2}
    |{\mathcal{T}}_\phi f(x,\xi )| \lesssim  e^{-r (|x|^{\frac 1s}+|\xi |^{\frac 1\sigma})}, \quad x,\xi \in {\mathbf R^{d}},$$ for some (every) $r > 0$.

2.  $f\in ({\mathcal S}_s^\sigma )'({\mathbf R^{d}})$ ($f\in (\Sigma _s^\sigma )'({\mathbf R^{d}})$) if and only if $$\label{stftexpest2Dist}
    |{\mathcal{T}}_\phi f(x,\xi )| \lesssim  e^{r(|x|^{\frac 1s}+|\xi |^{\frac 1\sigma})}, \quad
    x,\xi \in {\mathbf R^{d}},$$ for every (some) $r > 0$.

The Bargmann transform and spaces of analytic functions {#subsec1.3}
-------------------------------------------------------

If $\Omega \subseteq {\mathbf C^{d}}$ is open then $A(\Omega)$ consists of all (complex-valued) analytic functions on $\Omega$. Complex derivatives are denoted, with $z = x+iy \in \Omega$, $$\partial_{z_j} = \frac{1}{2} \left( \partial_{x_j} - i \partial_{y_j} \right), \quad 
\overline{\partial}_{z_j} = \frac{1}{2} \left( \partial_{x_j} + i \partial_{y_j} \right)$$ 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 $$\mathfrak A_d(z,y)=\pi ^{-\frac d4} \exp \Big ( -\frac 12({\langle z,z\rangle}+|y|^2)+2^{1/2}{\langle z,y\rangle}\Big ), \quad z \in {\mathbf C^{d}}, \quad y \in {\mathbf R^{d}},$$ where $${\langle z,w\rangle} = \sum _{j=1}^dz_jw_j\quad \text{and} \quad
(z,w)= {\langle z,\overline w\rangle}$$ when $$z=(z_1,\dots ,z_d) \in {\mathbf C^{d}}\quad  \text{and} \quad w=(w_1,\dots ,w_d)\in {\mathbf C^{d}}.$$ Sometimes ${\langle {\, \cdot \, },{\, \cdot \, }\rangle}$ 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 $f\in {\mathcal S}_{1/2}'({\mathbf R^{d}})$ is the entire function $$\label{bargdistrform}
\mathfrak V_d f (z) ={\langle f,\mathfrak A_d(z,{\, \cdot \, })\rangle},
\quad z \in {\mathbf C^{d}}.$$ The right-hand side is a well defined element in $A({\mathbf C^{d}})$, since $y\mapsto \mathfrak A_d(z,y)$ belongs to ${\mathcal S}_{1/2} ({\mathbf R^{d}})$ for $z \in {\mathbf C^{d}}$ fixed, and $\mathfrak A_d({\, \cdot \, },y)$ is entire for all $y \in {\mathbf R^{d}}$. Let $p\in [1,\infty ]$ and $\omega \in {\mathscr P}_E({\mathbf R^{d}})$. Then $L^p_{(\omega )}({\mathbf R^{d}})$ consists of all $f\in L^1_{loc}({\mathbf R^{d}})$ such that ${\Vert f\Vert _{L^p_{(\omega )}}}\equiv {\Vert f\cdot \omega \Vert _{L^p}}$ is finite. If $f\in L^p_{(\omega )}({\mathbf R^{d}})$, then $$\begin{gathered}
\mathfrak V_d f (z) =\int_{{\mathbf R^{d}}} \mathfrak A_d(z,y)f(y)\, dy
\\[1ex]
=
\pi ^{-\frac d4}\int _{{\mathbf R^{d}}}\exp \Big ( -\frac 12({\langle z,z\rangle}+|y|^2)+2^{1/2}{\langle z,y\rangle} \Big )f(y)\, dy,\quad z \in {\mathbf C^{d}}.\end{gathered}$$ (Cf. @B1 [@B2; @To11; @Toft18].)

For $p\in (0,\infty]$, $\omega \in {\mathscr P}_E({\mathbf C^{d}})$ and $\omega _0(z)=\omega (\sqrt 2\overline z)$, let $A^p_{(\omega )}({\mathbf C^{d}})$ be the set of all $F\in A({\mathbf C^{d}})$ such that $${\Vert F\Vert _{A^p_{(\omega )}}}
\equiv \pi ^{-\frac dp}
{\Vert F\cdot e^{-\frac 12|{\, \cdot \, }|^2}\cdot \omega _0\Vert _{L^p}},$$ and set $A^p=A^p_{(\omega )}$ when $\omega =1$. It was proved by Bargmann @B1 that $$\label{Eq:L2A2Isometry}
\mathfrak V_d : L^2({\mathbf R^{d}}) \to A^2({\mathbf C^{d}})$$ is bijective and isometric. The space $A^2({\mathbf C^{d}})$ is the Hilbert space of entire functions with scalar product $$(F,G)_{A^2}\equiv  \int _{{\mathbf C^{d}}} F(z)\overline {G(z)}\, d\mu (z),\quad F,G\in A^2({\mathbf C^{d}}),$$ where $d\mu (z)=\pi ^{-d} e^{-|z|^2}\, d\lambda (z)$ and $d\lambda (z)$ is the Lebesgue measure on ${\mathbf C^{d}}$. The space $A^2({\mathbf C^{d}})$ is known as the Fock space in quantum mechanics (see@Fo).

In @B1 it was proved that the Bargmann transform maps the Hermite functions to monomials as $$\label{Eq:BargmannHermiteMap}
\mathfrak V_dh_\alpha = e_\alpha ,\qquad e_\alpha (z)= \frac {z^\alpha}{\alpha !^{\frac 12}},
\quad z\in {\mathbf C^{d}},\quad \alpha \in {\mathbf N^{d}}.$$ The orthonormal basis $\{ h_\alpha \}_{\alpha \in {\mathbf N^{d}}} \subseteq L^2({\mathbf R^{d}})$ is thus mapped to the orthonormal basis $\{ e_\alpha \} _{\alpha \in {\mathbf N^{d}}}\subseteq A^2({\mathbf C^{d}})$. Bargmann also proved that there is a reproducing formula for $A^2({\mathbf C^{d}})$. Let $\Pi _A$ be the operator from $L^2(d\mu )$ to $A({\mathbf C^{d}})$, given by $$\label{eq:projection}
\Pi _A F (z)= \int _{{\mathbf C^{d}}} F(w)e^{(z,w)}\, d\mu (w),
\quad z \in {\mathbf C^{d}}.$$ Then $\Pi _A$ is the orthogonal projection from $L^2(d\mu)$ to $A^2({\mathbf C^{d}})$ (cf. @B1).

When we discuss extensions and restrictions of the Bargmann transform to Gelfand-Shilov spaces and their distribution spaces, we use $$\label{Eq:GevreyQuasinormMIxed}
|z|_{s,\sigma} = |\operatorname{Re} z|^{\frac 1s}
+
|\operatorname{Im} z|^{\frac 1\sigma},
\qquad
z\in {\mathbf C^{d}},$$ and consider the seminorms $$\begin{aligned}
{2}
{\Vert F\Vert _{{\mathcal A}_{{\mathscr S};r}}}
&\equiv
{\Vert F\cdot e^{-\frac 12|{\, \cdot \, }|^2}{\langle {\, \cdot \, }\rangle} 
^{r}\Vert _{L^\infty}},&
\quad
{\Vert F\Vert _{{\mathcal A}_{{\mathscr S};r}'}}
&\equiv
{\Vert F\cdot e^{-\frac 12|{\, \cdot \, }|^2}{\langle {\, \cdot \, }\rangle} 
^{-r}\Vert _{L^\infty}}
\intertext{and}
{\Vert F\Vert _{{\mathcal A}_{{\mathcal S}_{s;r}^\sigma}}}
&\equiv
{\Vert F\cdot e^{-\frac 12|{\, \cdot \, }|^2+r|{\, \cdot \, }|_{s,\sigma}}\Vert _{L^\infty}}, &
\quad
{\Vert F\Vert _{{\mathcal A}_{{\mathcal S}_{s;r}^\sigma}'}}
&\equiv
{\Vert F\cdot e^{-\frac 12|{\, \cdot \, }|^2-r|{\, \cdot \, }|_{s,\sigma}}\Vert _{L^\infty}}\end{aligned}$$ when $F\in A({\mathbf C^{d}})$, $r>0$ and $s,\sigma \ge  \frac 12$. Then ${\mathcal A}_{0,s}^\sigma ({\mathbf C^{d}})$ for $s, \sigma > \frac 12$, ${\mathcal A}_{{\mathscr S}} ({\mathbf C^{d}})$ and $({\mathcal A}_s^\sigma )' ({\mathbf C^{d}})$ for $s,\sigma \ge \frac 12$ are the sets of all $F\in A({\mathbf C^{d}})$ such that $${\Vert F\Vert _{{\mathcal A}_{{\mathcal S}_{s;r}^\sigma}}}<\infty,
\quad
{\Vert F\Vert _{{\mathcal A}_{{\mathscr S};r}}}<\infty 
\quad \text{and}\quad
{\Vert F\Vert _{{\mathcal A}_{{\mathcal S}_{s;r}^\sigma}'}}<\infty,$$ 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 ${\mathcal A}_{s}^\sigma ({\mathbf C^{d}})$ for $s,\sigma \ge \frac 12$, ${\mathcal A}_{{\mathscr S}}' ({\mathbf C^{d}})$ and $({\mathcal A}_{0,s}^\sigma )' ({\mathbf C^{d}})$ for $s,\sigma >\frac 12$ be the sets of all $F\in A({\mathbf C^{d}})$ such that $${\Vert F\Vert _{{\mathcal A}_{{\mathcal S}_{s;r}^\sigma}}}<\infty, 
\quad
{\Vert F\Vert _{{\mathcal A}_{{\mathscr S};r}'}}<\infty
\quad \text{and}\quad
{\Vert F\Vert _{{\mathcal A}_{{\mathcal S}_{s;r}^\sigma}'}}<\infty ,$$ 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 $${\mathcal A}_{0,s} ={\mathcal A}_{0,s}^s
\quad \text{and}\quad
{\mathcal A}_{s} ={\mathcal A}_{s}^s.$$ Then $$\begin{aligned}
{5}
{\mathfrak V}_d \, &: &\,
{\mathscr S}({\mathbf R^{d}}) &\to {\mathcal A}_{{\mathscr S}}({\mathbf C^{d}}), &
\qquad
{\mathfrak V}_d \, &: &\,
{\mathscr S}'({\mathbf R^{d}}) &\to {\mathcal A}_{{\mathscr S}}'({\mathbf C^{d}}), & &
\\[1ex]
\mathfrak V_d \, &: &\,
{\mathcal S}_s^\sigma ({\mathbf R^{d}}) &\to {\mathcal A}_s^\sigma ({\mathbf C^{d}}), &
\qquad
\mathfrak V_d \, &: &\,
({\mathcal S}_s^\sigma )'({\mathbf R^{d}})
&\to ({\mathcal A}_s^\sigma )' ({\mathbf C^{d}}) &
\quad
s,\sigma &\ge \frac 12
\intertext{and}
\mathfrak V_d \, &: &\,
\Sigma _s^\sigma ({\mathbf R^{d}})
&\to
{\mathcal A}_{0,s}^\sigma ({\mathbf C^{d}}), &
\qquad
\mathfrak V_d \, &: &\,
(\Sigma _s^\sigma )'({\mathbf R^{d}})
&\to ({\mathcal A}_{0,s}^\sigma )' ({\mathbf C^{d}}), &
\quad
s,\sigma &> \frac 12\end{aligned}$$ are homeomorphisms @Toft18.

From these homeomorphisms, the fact that the map is a homeomorphism and duality properties for Gelfand-Shilov spaces, it follows that $({\, \cdot \, },{\, \cdot \, })_{A^2}$ on ${\mathcal A}_{1/2}({\mathbf C^{d}})\times
{\mathcal A}_{1/2}({\mathbf C^{d}})$ is uniquely extendable to a continuous sesqui-linear form on $({\mathcal A}_s^\sigma )'({\mathbf C^{d}})\times {\mathcal A}_s^\sigma ({\mathbf C^{d}})$. The dual of ${\mathcal A}_s^\sigma ({\mathbf C^{d}})$ can be identified with $({\mathcal A}_s^\sigma )'({\mathbf C^{d}})$ through this form. Similar facts hold for ${\mathcal A}_{0,s}^\sigma$ in place of ${\mathcal A}_s^\sigma$ at each occurrence. (Cf. e.g. @To11 [@Toft18].)

Finally let ${\mathcal A}_{\flat _1;r}({\mathbf C^{d}})$ and ${\mathcal A}_{\flat _\infty ;r}({\mathbf C^{d}})$ for $r > 0$ be the Banach spaces which consist of all $F\in A({\mathbf C^{d}})$ such that $${\Vert F\Vert _{{\mathcal A}_{\flat _1;r}}}\equiv {\Vert F\cdot e^{-r|{\, \cdot \, }|}\Vert _{L^\infty}}
\quad \text{respectively}\quad
{\Vert F\Vert _{{\mathcal A}_{\flat _\infty ;r}}}\equiv {\Vert F\cdot e^{-r|{\, \cdot \, }|^2}\Vert _{L^\infty}}$$ is finite, and let ${\mathcal A}_{\flat _1}({\mathbf C^{d}})$ be the inductive limit of ${\mathcal A}_{\flat _1;r}({\mathbf C^{d}})$ with respect to $r>0$. Also let ${\mathcal A}_{0,\flat _\infty}({\mathbf C^{d}})$ and ${\mathcal A}_{0,\flat _\infty}'({\mathbf C^{d}})$ be the projective respectively inductive limit topologies of ${\mathcal A}_{\flat _\infty ;r}({\mathbf C^{d}})$ with respect to $r>0$.

It is evident that ${\mathcal A}_{\flat _1}({\mathbf C^{d}})$ is densely embedded in ${\mathcal A}_s^\sigma ({\mathbf C^{d}})$ for every $s,\sigma \ge \frac 12$, as well as in ${\mathcal A}_{0,s}^\sigma ({\mathbf C^{d}})$ for every $s,\sigma > \frac 12$. The form $({\, \cdot \, },{\, \cdot \, })_{A^2}$ on ${\mathcal A}_{\flat _1}({\mathbf C^{d}})\times
{\mathcal A}_{\flat _1}({\mathbf C^{d}})$ is uniquely extendable to a continuous sesqui-linear form on $A({\mathbf C^{d}})\times {\mathcal A}_{\flat _1} ({\mathbf C^{d}})$ and the dual of ${\mathcal A}_{\flat _1} ({\mathbf C^{d}})$ can be identified with $A({\mathbf C^{d}})$. The Fr<span>é</span>chet space topology of $A({\mathbf C^{d}})$ can be defined by the seminorms $$F\mapsto  \sup _{|z|\le N}|F(z)|,\qquad N=1,2,\dots .$$ (Cf. @Toft18.)

\[Rem:BargmannPilipovicSpaces\] The spaces ${\mathcal A}_{\flat _1}({\mathbf C^{d}})$ and ${\mathcal A}_{0,\flat _\infty}({\mathbf C^{d}})$ are examples of Bargmann images of special Pilipovi<span>ć</span> 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 @Toft18. For any $\sigma >0$, the Bargmann image of the Pilipovi<span>ć</span> spaces ${\mathcal H}_{\flat _\sigma}({\mathbf R^{d}})$ and ${\mathcal H}_{0,\flat _\sigma}({\mathbf R^{d}})$ are given by $$\begin{aligned}
{\mathcal A}_{\flat _\sigma}({\mathbf C^{d}})
& \equiv
{\{ \, F\in A({\mathbf C^{d}})\, ;\, |F(z)|\lesssim e^{r|z|^{\frac {2\sigma}{\sigma +1}}}
\ \text {for some}\ r>0 \, \} }
\intertext{respectively}
{\mathcal A}_{0,\flat _\sigma}({\mathbf C^{d}})
& \equiv
{\{ \, F\in A({\mathbf C^{d}})\, ;\, |F(z)|\lesssim e^{r|z|^{\frac {2\sigma}{\sigma +1}}}
\ \text {for every}\ r>0 \, \} }.\end{aligned}$$

If $\sigma >1$, then the (strong) duals of ${\mathcal A}_{\flat _\sigma}({\mathbf C^{d}})$ and ${\mathcal A}_{0,\flat _\sigma}({\mathbf C^{d}})$ are given by $$\begin{aligned}
{\mathcal A}_{\flat _\sigma}'({\mathbf C^{d}})
& \equiv
{\{ \, F\in A({\mathbf C^{d}})\, ;\, |F(z)|\lesssim e^{r|z|^{\frac {2\sigma}{\sigma -1}}}
\ \text {for every}\ r>0 \, \} }
\intertext{respectively}
{\mathcal A}_{0,\flat _\sigma}'({\mathbf C^{d}})
& \equiv
{\{ \, F\in A({\mathbf C^{d}})\, ;\, |F(z)|\lesssim e^{r|z|^{\frac {2\sigma}{\sigma -1}}}
\ \text {for some}\ r>0 \, \} }\end{aligned}$$ through a unique extension of the $A^2$ scalar product on ${\mathcal A}_{\flat _1}({\mathbf C^{d}})\times {\mathcal A}_{\flat _1}({\mathbf C^{d}})$. In particular, if $\sigma$ tends to $\infty$, it follows that some of these conditions tend to $$\begin{aligned}
{\mathcal A}_{0,\flat _\infty}({\mathbf C^{d}})
& \equiv
{\{ \, F\in A({\mathbf C^{d}})\, ;\, |F(z)|\lesssim e^{r|z|^2}
\ \text {for every}\ r>0 \, \} }
\intertext{respectively}
{\mathcal A}_{0,\flat _\infty}'({\mathbf C^{d}})
& \equiv
{\{ \, F\in A({\mathbf C^{d}})\, ;\, |F(z)|\lesssim e^{r|z|^2}
\ \text {for some}\ r>0 \, \} }.\end{aligned}$$ Note that in @Toft18 [@Teofanov2], the set ${\mathcal A}_{0,\flat _\infty}({\mathbf C^{d}})$ is denoted by ${\mathcal A}_{0,\frac 12}({\mathbf C^{d}})$, and its dual ${\mathcal A}_{0,\flat _\infty}'({\mathbf C^{d}})$ is denoted by ${\mathcal A}_{0,\frac 12}'({\mathbf C^{d}})$.

At many places it will be crucial to use the Gaussian window $$\label{Eq:phidef}
\phi (x)=\pi ^{-\frac d4}e^{-\frac 12|x|^2}, \quad x\in {\mathbf R^{d}},$$ in the transform ${\mathcal{T}}_\phi$. For this $\phi$ the relationship between the Bargmann transform and ${\mathcal{T}}_\phi$ is $$\label{bargstft1}
\mathfrak V_d = U_{\mathfrak V}\circ {\mathcal{T}}_\phi ,\quad \text{and}\quad
U_{\mathfrak V}^{-1} \circ \mathfrak V_d =  {\mathcal{T}}_\phi ,$$ where $U_{\mathfrak V}$ is the linear, continuous and bijective operator on $\mathscr D'({\mathbf R^{2d}})\simeq \mathscr D'({\mathbf C^{d}})$, given by $$\label{UVdef}
U_{\mathfrak V} F (x+i\xi ) = (2\pi )^{\frac d2} e^{\frac 12(|x|^2+|\xi |^2)}e^{i{\langle x,\xi\rangle}}
F(\sqrt 2\, x,-\sqrt 2\, \xi ), \quad x,\xi \in {\mathbf R^{d}},$$ cf. @To11 in combination with .

In analytic operator theory we need subspaces of $$\wideparen A({\mathbf C^{2d}})
\equiv
{\left \{ \, \Theta K\, ;\, K\in A({\mathbf C^{2d}})\, \right \} },$$ where the semi-conjugation operator is $$\label{Eq:ThetaOpDef}
(\Theta K) (z,w) = K(z,\overline w),\qquad z,w\in {\mathbf C^{d}}.$$ If $T$ is a linear and continuous operator from ${\mathcal S}_{1/2}({\mathbf R^{d}})$ to ${\mathcal S}_{1/2}'({\mathbf R^{d}})$, then there is a unique $K\in \wideparen A({\mathbf C^{2d}})$ such that $\Theta K\in {\mathcal A}_{1/2}'({\mathbf C^{2d}})$ and $\mathfrak V_d \circ T \circ \mathfrak V_d^{-1}$ is given by $$\label{Eq:CompIntOp}
F(z)\mapsto \int _{{\mathbf C^{d}}}K(z,w)F(w)\, d\mu (w).$$ (See e.g. @Teofanov2.) For these reasons we let $$\wideparen {\mathcal A}_{0,s}({\mathbf C^{2d}}),
\quad
\wideparen {\mathcal A}_s({\mathbf C^{2d}}),
\quad
\wideparen {\mathcal A}_{{\mathscr S}}({\mathbf C^{2d}}),
\quad
\wideparen {\mathcal A}_{{\mathscr S}}'({\mathbf C^{2d}}),
\quad
\wideparen {\mathcal A}_s'({\mathbf C^{2d}})
\quad \text{and}\quad
\wideparen {\mathcal A}_{0,s}'({\mathbf C^{2d}})$$ be the images of $${\mathcal A}_{0,s}({\mathbf C^{2d}}),
\quad
{\mathcal A}_s({\mathbf C^{2d}}),
\quad
{\mathcal A}_{{\mathscr S}}({\mathbf C^{2d}}),
\quad
{\mathcal A}_{{\mathscr S}}'({\mathbf C^{2d}}),
\quad
{\mathcal A}_s'({\mathbf C^{2d}})
\quad \text{and}\quad
{\mathcal A}_{0,s}'({\mathbf C^{2d}})$$ respectively, under the map $\Theta$. We also let $\wideparen A^p({\mathbf C^{2d}})$ and $\wideparen {\mathcal A}_{\flat _1}({\mathbf C^{2d}})$ be the images of $A^p({\mathbf C^{2d}})$ and ${\mathcal A}_{\flat _1}({\mathbf C^{2d}})$, 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 {#subsec1.4}
-----------------------------

Let $A$ be a real $d\times d$ matrix. The *pseudo-differential operator* ${\operatorname{Op}}_A({\mathfrak a})$ with *symbol* ${\mathfrak a}\in {\mathcal S}_{1/2} ({\mathbf R^{2d}})$ is the linear and continuous operator on ${\mathcal S}_{1/2} ({\mathbf R^{d}})$ given by $$\label{e0.5}
{\operatorname{Op}}_A({\mathfrak a})f (x)
=
(2\pi  ) ^{-d}\iint {\mathfrak a}(x-A(x-y),\xi )
f(y)e^{i{\langle x-y,\xi\rangle} }\,
dyd\xi, \quad x\in {\mathbf R^{d}}.$$ For ${\mathfrak a}\in {\mathcal S}_{1/2}'({\mathbf R^{2d}})$ the pseudo-differential operator ${\operatorname{Op}}_A({\mathfrak a})$ is defined as the continuous operator from ${\mathcal S}_{1/2}({\mathbf R^{d}})$ to ${\mathcal S}_{1/2}'({\mathbf R^{d}})$ with distribution kernel $$\label{atkernel}
K_{{\mathfrak a},A}(x,y)
=
(2\pi )^{-\frac d2} {\mathscr F}_2^{-1}{\mathfrak a}(x-A(x-y),x-y), \quad 
x,y \in {\mathbf R^{d}},$$ where ${\mathscr F}_2F$ is the partial Fourier transform of $F(x,y)\in
{\mathcal S}_{1/2}'({\mathbf R^{2d}})$ with respect to the $y$ variable. This definition makes sense since the mappings $$\label{homeoF2tmap}
{\mathscr F}_2\quad \text{and}\quad F(x,y)\mapsto F(x,x-y)$$ are homeomorphisms on ${\mathcal S}_{1/2}'({\mathbf R^{2d}})$. The map $a\mapsto K_{{\mathfrak a},A}$ is hence a homeomorphism on ${\mathcal S}_{1/2}'({\mathbf R^{2d}})$.

If $A$ and $B$ are real $d\times d$ matrices and ${\mathfrak a}\in {\mathcal S}_{1/2}'({\mathbf R^{2d}})$, then there is a unique ${\mathfrak b}\in {\mathcal S}_{1/2}'({\mathbf R^{2d}})$ such that ${\operatorname{Op}}_A({\mathfrak a})= {\operatorname{Op}}_B({\mathfrak b})$, and that ${\mathfrak b}$ can be obtained by $$\label{Eq:RealPseudoCalculiTransfer}
{\operatorname{Op}}_A({\mathfrak a})= {\operatorname{Op}}_B({\mathfrak b})
\quad \Leftrightarrow \quad
e^{i{\langle AD_\xi ,D_x\rangle}}{\mathfrak a}(x,\xi )
= 
e^{i{\langle BD_\xi ,D_x\rangle}}{\mathfrak b}(x,\xi )$$ (see @Ho1 [@CaTo]).

\[BijKernelsOps\] By Fourier's inversion formula, and the kernel theorem @LozPer [Theorem 2.2], @Teof2 [Theorem 2.5] for operators from Gelfand-Shilov spaces to their duals, it follows that the map ${\mathfrak a}\mapsto {\operatorname{Op}}_A({\mathfrak a})$ is bijective from ${\mathcal S}_{1/2}'({\mathbf R^{2d}})$ to the set of all linear and continuous operators from ${\mathcal S}_{1/2}({\mathbf R^{d}})$ to ${\mathcal S}_{1/2}'({\mathbf R^{2d}})$.

If $A=0$ then ${\operatorname{Op}}_A({\mathfrak a}) = {\operatorname{Op}}_0({\mathfrak a}) = {\operatorname{Op}}({\mathfrak a}) = {\mathfrak a}(x,D)$ is the Kohn-Nirenberg or standard representation. If $A=\frac 12 I_d$ where $I_d$ is the $d\times d$ identity matrix then ${\operatorname{Op}}_A({\mathfrak a}) = {\operatorname{Op}}^w({\mathfrak a})$ is the Weyl quantization. In this paper we use mainly the Weyl quantization and we put $$K_{{\mathfrak a}}^w = K_{{\mathfrak a},I_d/2} \ .$$

The Weyl product ${\mathfrak a}{{\text{\footnotesize $\#$}}}{\mathfrak b}$ of two Weyl symbols ${\mathfrak a},{\mathfrak b}\in {\mathcal S}_{1/2}({\mathbf R^{2d}})$ is defined as the product of symbols corresponding to operator composition. Thus $${\operatorname{Op}}^w({\mathfrak a}{{\text{\footnotesize $\#$}}}{\mathfrak b}) = {\operatorname{Op}}^w({\mathfrak a}) \circ {\operatorname{Op}}^w({\mathfrak b})$$ 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 $a\in \wideparen A({\mathbf C^{2d}})$ satisfies $$\label{Eq:AntiWickL1Cond}
w\mapsto a(z,w)e^{r|w|-|w|^2} \in L^1({\mathbf C^{d}})$$ locally uniformly with respect to $z\in {\mathbf C^{d}}$ for every $r>0$. Then the *analytic pseudo-differential operator*, or *Wick operator* ${\operatorname{Op}}_{\mathfrak V}(a)$ with symbol $a$ and acting on $F\in {\mathcal A}_{\flat _1}({\mathbf C^{d}})$, is defined by $$\begin{aligned}
{\operatorname{Op}}_{\mathfrak V}(a)F(z) &= \int _{{\mathbf C^{d}}} a(z,w)F(w)e^{(z,w)}\, d\mu (w),
\quad z \in {\mathbf C^{d}}.
\label{Eq:AnalPseudo}\end{aligned}$$ (Cf. e.g. @Berezin71 [@Fo; @Teofanov2; @To11; @Toft18].) The condition and $F\in {\mathcal A}_{\flat _1}({\mathbf C^{d}})$ imply that the integrand on the right-hand side of is well defined. The locally uniform condition with respect to $z\in {\mathbf C^{d}}$ implies that ${\operatorname{Op}}_{\mathfrak V}(a)F \in A({\mathbf C^{d}})$.

In @Teofanov2 several extensions and restrictions of ${\operatorname{Op}}_{\mathfrak V}(a)$ are given. The following result follows from @Teofanov2 [Theorems 2.7 and 2.8]. Here $\mathcal L({\mathcal A}_{\flat _1}({\mathbf C^{d}}),A({\mathbf C^{d}}))$ is the space of all linear and continuous operators from ${\mathcal A}_{\flat _1}({\mathbf C^{d}})$ to $A({\mathbf C^{d}})$.

\[Prop:ContAnalPseudo\] The map $a\mapsto {\operatorname{Op}}_{\mathfrak V}(a)$ from $\wideparen {\mathcal A}_{\flat _1}({\mathbf C^{2d}})$ to $\mathcal L({\mathcal A}_{\flat _1}({\mathbf C^{d}}),A({\mathbf C^{d}}))$ is uniquely extendable to a bijective map from $\wideparen A({\mathbf C^{2d}})$ to $\mathcal L({\mathcal A}_{\flat _1}({\mathbf C^{d}}),A({\mathbf C^{d}}))$.

Let $L_A({\mathbf C^{2d}})$ be the set of all $a \in L^1_{\rm loc}({\mathbf C^{2d}})$ such that $z\mapsto a(z,w)$ is entire for almost every $w\in {\mathbf C^{d}}$ and $$\label{Eq:LACond}
w \mapsto
\sup _{\alpha \in {\mathbf N^{d}}}
\left |
\frac { \partial _z^\alpha a(z,w)\cdot e^{r|w|-|w|^2}}{h^{|\alpha |}\alpha !}
\right | 
\in L^1({\mathbf C^{d}})$$ for every $h,r>0$ and $z\in {\mathbf C^{d}}$. If $a\in \wideparen A({\mathbf C^{2d}})$ satisfies then $a\in L_A({\mathbf C^{2d}})$ as a consequence of Cauchy's integral formula. Thus $L_A({\mathbf C^{2d}})$ is a relaxation of the former condition.

If $a\in L_A({\mathbf C^{2d}})$ then ${\operatorname{Op}}_{\mathfrak V}(a):
{\mathcal A}_{\flat _1}({\mathbf C^{d}}) \to {\mathcal A}_{\flat _1}'({\mathbf C^{d}})=A({\mathbf C^{d}})$ is continuous. Hence the following result is a straight-forward consequence of Proposition \[Prop:ContAnalPseudo\] and the fact that $\wideparen {{\mathcal A}} _{\flat _1}'({\mathbf C^{2d}})=\wideparen A({\mathbf C^{2d}})$.

\[Prop:LAOpsIdent\] Let $a\in L_A({\mathbf C^{2d}})$. Then there is a unique $a_0\in \wideparen A({\mathbf C^{2d}})$ such that ${\operatorname{Op}}_{\mathfrak V}(a) = {\operatorname{Op}}_{\mathfrak V}(a_0)$ as mappings from ${\mathcal A}_{\flat _1}({\mathbf C^{d}})$ to ${\mathcal A}_{\flat _1}'({\mathbf C^{d}})$. It holds $$\begin{gathered}
\label{Eq:AntiWickAnalPseudoRel}
{\operatorname{Op}}_{\mathfrak V}(a) = {\operatorname{Op}}_{\mathfrak V}(a_0)
\\[1ex]
\text{where}\quad
a_0(z,w) = \pi ^{-d} \int _{{\mathbf C^{d}}}a(z,w_1)e^{-(z-w_1,w-w_1)}\, d\lambda (w_1).\end{gathered}$$

The operator $\Pi _A$ defined in is the orthogonal projection from $L^2(d\mu)$ to $A^2({\mathbf C^{d}})$ which is uniquely extendable to a continuous map from $$\label{Eq:AntiWickL1SymbClass}
L_{0,A}({\mathbf C^{d}}) \equiv
{\{ \, a_0 \in L^1_{\rm loc}({\mathbf C^{d}})\, ;\, w \mapsto a_0(w)e^{r|w|-|w|^2} \in L^1({\mathbf C^{d}})
\ \text{for every}\ r>0\, \} }$$ to $A({\mathbf C^{d}})$ (see e.g. @To11). Hence, if $F,G\in {\mathcal A}_{\flat _1}({\mathbf C^{d}})$ and $a_0$ is given by then $$\begin{gathered}
({\operatorname{Op}}_{\mathfrak V}(a)F,G)_{A^2}
=
(({\operatorname{Op}}_{\mathfrak V}(a)\circ \Pi _A)F,G)_{A^2}
\\[1ex]
= \left (
\int _{{\mathbf C^{d}}} \left ( \int _{{\mathbf C^{d}}}
a({\, \cdot \, },w_1)e^{({\, \cdot \, },w_1)}e^{(w_1,w)}\, d\mu (w_1)
\right )
F(w)\, d\mu (w),G \right )_{\!\! A^2}
\\[1ex]
=
\left (
\int _{{\mathbf C^{d}}} a_0({\, \cdot \, },w)e^{({\, \cdot \, },w)} F(w)\, d\mu (w),G
\right )_{\!\! A^2}
=
({\operatorname{Op}}_{\mathfrak V}(a_0)F,G)_{A^2},\end{gathered}$$ and thus ${\operatorname{Op}}_{\mathfrak V}(a) = {\operatorname{Op}}_{\mathfrak V}(a_0)$ follows. The assertion now follows from Proposition \[Prop:ContAnalPseudo\] and the fact that $a_0$ in the integral formula of defines an element in $\wideparen A({\mathbf C^{2d}})$.

We will also consider *anti-Wick operators* @Fo [@Berezin71; @Berezin72] defined by $$\label{Eq:AntiWick}
{\operatorname{Op}}_{\mathfrak V}^{{\operatorname{aw}}}(a_0)F(z) = \int _{{\mathbf C^{d}}} a_0(w)F(w)e^{(z,w)}
\, d\mu (w),\quad z \in {\mathbf C^{d}},$$ when $a_0\in L_{0,A}({\mathbf C^{d}})$ and $F$ belongs to ${\mathcal A}_0({\mathbf C^{d}})$, the space of analytic polynomials on ${\mathbf C^{d}}$. Then $a_0\in L_{0,A}({\mathbf C^{d}})$ if and only if $a(z,w)\equiv a_0(w)$ belongs to $L_A({\mathbf C^{2d}})$, and then ${\operatorname{Op}}_{\mathfrak V}^{{\operatorname{aw}}}(a_0) = {\operatorname{Op}}_{\mathfrak V}(a)$. Consequently, all results for Wick operators with symbols in $L_A({\mathbf C^{2d}})$ hold for anti-Wick operators. In particular, if $a_0\in L_{0,A}({\mathbf C^{d}})$, then ${\operatorname{Op}}_{\mathfrak V}^{{\operatorname{aw}}}(a_0):
{\mathcal A}_{\flat _1}({\mathbf C^{d}}) \to A({\mathbf C^{d}})$ is continuous. We denote the Wick symbol of the anti-Wick operator ${\operatorname{Op}}_{\mathfrak V}^{{\operatorname{aw}}}(a_0)$ by $a_0^{{\operatorname{aw}}}$. Then takes the form $$\begin{gathered}
\tag*{(\ref{Eq:AntiWickAnalPseudoRel})$'$}
{\operatorname{Op}}_{\mathfrak V}^{{\operatorname{aw}}}(a_0) = {\operatorname{Op}}_{\mathfrak V}(a_0^{{\operatorname{aw}}})
\\[1ex]
\text{where}\quad
a_0^{{\operatorname{aw}}} (z,w) = \pi ^{-d} \int _{{\mathbf C^{d}}}a_0(w_1)e^{-(z-w_1,w-w_1)}\, d\lambda (w_1).\end{gathered}$$

Pseudo-differential operators on ${\mathbf R^{d}}$ may be transferred to Wick operators on ${\mathbf C^{d}}$ by means of the Bargmann transform.

\[Def:AnalPseudo\] Let ${\mathfrak a}\in {\mathcal S}_{1/2}'({\mathbf R^{2d}})$.

1.  the *Bargmann assignment* $\mathsf S_{\mathfrak V}{\mathfrak a}$ of ${\mathfrak a}$ is the unique element $a\in \wideparen A ({\mathbf C^{2d}})$ which fulfills $$\label{Eq:BargmannAssignment}
    {\operatorname{Op}}_{\mathfrak V}(a) = \mathfrak V_d \circ {\operatorname{Op}}^w ({\mathfrak a})\circ \mathfrak V_d^*
    \quad \Leftrightarrow \quad a = \mathsf S_{\mathfrak V}{\mathfrak a}\text ;$$

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

By the definitions we have $$\label{SVandSVK}
K_{\mathfrak V,{\mathfrak a}}(z,w) = e^{(z,w)}\mathsf S_{\mathfrak V}{\mathfrak a}(z,w).$$

\[Ex:BargAssignment\] 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 $$\label{Eq:LinearWickOps}
F\mapsto z_jF
\quad \text{and}\quad
F\mapsto \partial _{z_j}F,$$ by the Bargmann transform (see @B1). The Wick symbols of the operators in are $z_j$ and $\overline w_j$, respectively @Berezin71 [@To11]. By combining these identities with the fact that the Weyl symbol of $i^{-1}\partial _{x_j}$ equals $\xi _j$ we get $$\label{Eq:BargmannAssignBasicMaps}
\begin{gathered}
\mathsf S_{\mathfrak V}(2^{-\frac 12}(x_j-i\xi _j)) = z_j, 
\qquad
\mathsf S_{\mathfrak V}(2^{-\frac 12}(x_j+i\xi _j)) = \overline w_j,
\\[1ex]
\mathsf S_{\mathfrak V}(x_j) = 2^{-\frac 12}(z_j+\overline w_j) 
\qquad \text{and}\qquad
\mathsf S_{\mathfrak V}(\xi _j) = 2^{-\frac 12}i(z_j-\overline w_j).
\end{gathered}
$$

We need to compare $K_{{\mathfrak a}}^w$ and $K_{\mathfrak V,{\mathfrak a}}$. On the one hand we have for $f,g \in {\mathscr S}({\mathbf R^{d}})$ $$({\operatorname{Op}}^w ({\mathfrak a}) f, g)_{L^2({\mathbf R^{d}})} 
= (K_{{\mathfrak a}}^w, g \otimes \overline f)_{L^2({\mathbf R^{2d}})}
= (\mathfrak V_{2d} K_{{\mathfrak a}}^w, \mathfrak V_{2d} (g \otimes \overline f)_{A^2({\mathbf C^{2d}})}$$ and on the other hand $$\begin{aligned}
({\operatorname{Op}}^w ({\mathfrak a}) f, g)_{L^2({\mathbf R^{d}})} 
& = ({\operatorname{Op}}_{\mathfrak V}(a) \mathfrak V_d f, \mathfrak V_d g)_{A^2({\mathbf C^{d}})}
\\[1ex]
& = ( K_{\mathfrak V,{\mathfrak a}}, \mathfrak V_d g \otimes \overline{\mathfrak V_d f})_{A^2 ({\mathbf C^{2d}})}
\\[1ex]
& = ( \Theta K_{\mathfrak V,{\mathfrak a}},
\Theta (\mathfrak V_d g \otimes \overline{\mathfrak V_d f} ))_{A^2 ({\mathbf C^{2d}})}. \end{aligned}$$ Since $$\begin{aligned}
\Theta (\mathfrak V_d g \otimes \overline{\mathfrak V_d f} )(z,w)
= \mathfrak V_d g(z) \overline{\mathfrak V_d f(\overline w)}
= \mathfrak V_{2d} (g \otimes \overline f) (z,w)\end{aligned}$$ we obtain $$\label{eq:kernelreal2complex}
K_{\mathfrak V,{\mathfrak a}} = \Theta \mathfrak V_{2d} K_{{\mathfrak a}}^w
= \mathfrak V_{\Theta,d} K_{{\mathfrak a}}^w.$$

Symbol classes for pseudo-differential operators on ${\mathbf R^{d}}$ {#subsec1.5}
---------------------------------------------------------------------

In order to define a generalized family of Shubin symbol classes @Shubin1, we need to add a restriction of the involved weights. Let $\rho \in [0,1]$, and let ${\mathscr P}_{{\operatorname{Sh}},\rho }({\mathbf R^{d}})$ be the set of all $\omega \in {\mathscr P}({\mathbf R^{d}})\cap C^\infty ({\mathbf R^{d}})$ such that for every multi-index $\alpha \in {\mathbf N^{d}}$, $$|\partial ^\alpha \omega (x)|\lesssim \omega (x){\langle x\rangle}^{-\rho |\alpha |},
\quad x\in {\mathbf R^{d}}.$$ For $\omega \in {\mathscr P}_{{\operatorname{Sh}},\rho }({\mathbf R^{d}})$ the Shubin symbol class ${\operatorname{Sh}}_\rho ^{(\omega )}({\mathbf R^{d}})$ is the set of all $f \in C^\infty ({\mathbf R^{d}})$ such that for every $\alpha \in {\mathbf N^{2d}}$, $$|\partial ^\alpha f(x)|\lesssim \omega (x) {\langle x\rangle}^{-\rho |\alpha |},
\qquad x\in {\mathbf R^{d}}.$$

Let $\rho \in [0,1]$, $\omega \in {\mathscr P}_{{\operatorname{Sh}},\rho }({\mathbf R^{2d}})$ and $A$ be a real $d\times d$ matrix. Then it follows from @Shubin1 or @Ho1 [Section 18.5] that $e^{i{\langle AD_\xi ,D_x\rangle}}$ is a homeomorphism on ${\operatorname{Sh}}_\rho ^{(\omega )}({\mathbf R^{2d}})$, which implies that the set $${\{ \, {\operatorname{Op}}_A({\mathfrak a})\, ;\, {\mathfrak a}\in {\operatorname{Sh}}_\rho ^{(\omega )}({\mathbf R^{2d}})\, \} }$$ is independent of the choice of $A$, in view of . If $B$ is another real $d\times d$ matrix and ${\mathfrak a},{\mathfrak b}\in {\operatorname{Sh}}_\rho ^{(\omega )}({\mathbf R^{2d}})$ satisfy , then it follows from @Ho1 [Section 18.5] that $$\begin{gathered}
{\mathfrak a}- {\mathfrak b}\in {\operatorname{Sh}}_\rho ^{(\omega _\rho)}({\mathbf R^{2d}}),
\quad \text{where}\quad
\omega _\rho (x,\xi )= \omega (x,\xi ){\langle (x,\xi )\rangle}^{-2\rho}.
\label{Eq:DifferencesShubinSymbols}
\intertext{In particular}
|{\mathfrak a}(x,\xi ) - {\mathfrak b}(x,\xi ) |
\lesssim \omega (x,\xi ){\langle (x,\xi )\rangle}^{-2\rho}.
\label{Eq:DifferencesShubinSymbols2}\end{gathered}$$

We also need the symbol classes defined in @AbCaTo [Definition 1.8] with symbols satisfying estimates of the form $$\label{eq:symbolgevrey}
|\partial _x^\alpha \partial _\xi^\beta  {\mathfrak a}(x,\xi )|
\lesssim h^{|\alpha +\beta |} \alpha !^\sigma \beta !^s
e^{r (|x|^{\frac1{s}}+|\xi |^{\frac1{\sigma}})}, \qquad x,\xi  \in {\mathbf R^{d}}.$$ (See also @CaTo for the restricted case when $s=\sigma$.)

\[Def:GevreyGSSymbols\] Let $s,\sigma >0$. Then

1.  $\Gamma ^{\sigma ,s;0}_{s,\sigma}({\mathbf R^{2d}})$ consists of all ${\mathfrak a}\in C^\infty ({\mathbf R^{d}})$ such that holds for every $h > 0$ and some $r > 0$;

2.  $\Gamma ^{\sigma ,s}_{s,\sigma ;0}({\mathbf R^{2d}})$ consists of all ${\mathfrak a}\in C^\infty ({\mathbf R^{d}})$ such that holds for some $h > 0$ and every $r > 0$;

3.  $\Gamma ^{\sigma ,s}_{s,\sigma}({\mathbf R^{2d}})$ consists of all ${\mathfrak a}\in C^\infty ({\mathbf R^{d}})$ such that holds for some $h > 0$ and some $r > 0$.

The symbol classes ${\operatorname{Sh}}_\rho ^{(\omega )}({\mathbf R^{2d}})$ have isotropic behaviour with respect to phase space $T^* {\mathbf R^{d}}\simeq {\mathbf R^{2d}}$, and the same holds for the symbol classes in Definition \[Def:GevreyGSSymbols\] when $\sigma = s$. See also @CaTo for the restricted case when $s=\sigma$, and @AbCoTe for a bilinear extension. Important classes similar to those given by Definition \[Def:GevreyGSSymbols\] are considered in @Prangoski.

Pseudo-differential operators with symbols in the classes in Definition \[Def:GevreyGSSymbols\] are examples of so called operators of infinite order. These operators are continuous on appropriate Gelfand-Shilov (distribution) spaces @AbCaTo [@CaTo]. The next result characterizes the symbol classes in Definition \[Def:GevreyGSSymbols\] by means of estimates of form $$\label{Eq:STFTgevrey}
|{\mathcal{T}}_\psi {\mathfrak a}(x,\xi , \eta ,y)|
\lesssim
e^{r_1(|x|^{\frac1{s}}+|\xi|^{\frac1{\sigma}}) - r_2(|\eta |^{\frac1{\sigma}} + |y|^{\frac1{s}}) },
\quad x,\xi ,y, \eta \in {\mathbf R^{d}}.$$ We omit the proof since the result is a special case of @AbCaTo [Proposition 2.1$'$]. We refer to @AbCaTo [Subsection 1.1] for the definition of the Gelfand-Shilov spaces ${\mathcal S}_{s,\sigma}^{\sigma ,s}({\mathbf R^{2d}})$, $\Sigma _{s,\sigma}^{\sigma ,s}({\mathbf R^{2d}})$ and their distribution spaces.

\[Prop:GSclassesChar\] Let $s,\sigma >0$ and let ${\mathfrak a}\in C^\infty ({\mathbf R^{2d}})$. Then the following is true:

1.  if $\psi \in {\mathcal S}_{s,\sigma}^{\sigma ,s} ({\mathbf R^{2d}})\setminus 0$, then ${\mathfrak a}\in \Gamma ^{\sigma ,s}_{s,\sigma ;0}({\mathbf R^{2d}})$ if and only if holds for every $r_1 > 0$ and some $r_2 > 0$;

2.  if $\psi \in \Sigma _{s,\sigma}^{\sigma ,s} ({\mathbf R^{2d}})\setminus 0$, then ${\mathfrak a}\in \Gamma ^{\sigma ,s;0}_{s,\sigma }({\mathbf R^{2d}})$ if and only if holds for some $r_1 > 0$ and all $r_2 > 0$;

3.  if $\psi \in \Sigma _{s,\sigma}^{\sigma ,s} ({\mathbf R^{2d}})\setminus 0$, then ${\mathfrak a}\in \Gamma ^{\sigma ,s}_{s,\sigma}({\mathbf R^{2d}})$ if and only if holds for some $r_1 > 0$ and some $r_2 > 0$.

Elliptic, weakly elliptic and hypoelliptic elements in ${\operatorname{Sh}}^{(\omega )}_\rho ({\mathbf R^{d}})$
---------------------------------------------------------------------------------------------------------------

Let $\rho \ge 0$ and $\omega \in {\mathscr P}_{{\operatorname{Sh}},\rho}({\mathbf R^{d}})$. Then $f\in {\operatorname{Sh}}^{(\omega )}_\rho ({\mathbf R^{d}})$ is called *weakly elliptic* of order $\rho _0\ge 0$, (in ${\operatorname{Sh}}^{(\omega )}_\rho ({\mathbf R^{d}})$), or *$\rho _0$-weakly elliptic*, if there is an $R>0$ such that $$|f(x)| \gtrsim {\langle x\rangle}^{-\rho _0}\omega (x),\qquad |x|\ge R.$$ A weakly elliptic function of order $0$ is called *elliptic*.

Let $A$ and $B$ be real $d\times d$ matrices, $\rho >0$, $\rho _0\in [0,2\rho )$, $\omega \in {\mathscr P}_{{\operatorname{Sh}},\rho}({\mathbf R^{2d}})$ and suppose that ${\mathfrak a},{\mathfrak b}\in
{\operatorname{Sh}}_\rho ^{(\omega )}({\mathbf R^{2d}})$ satisfy . It follows from that ${\mathfrak a}$ is weakly elliptic of order $\rho _0$, if and only if ${\mathfrak b}$ is weakly elliptic of order $\rho _0$. In particular, ${\mathfrak a}$ is elliptic, if and only if ${\mathfrak b}$ is elliptic.

Next we define Shubin hypoelliptic symbols (cf. Definitions 5.1 and 25.1 in @Shubin1).

\[Def:Hypoelliptic\] Let $\rho >0$, $\rho _0 \ge 0$, $\omega _0\in {\mathscr P}_{{\operatorname{Sh}},\rho}({\mathbf R^{d}})$ and $f\in {\operatorname{Sh}}_\rho ^{(\omega _0)}({\mathbf R^{d}})$. Then $f$ is called *hypoelliptic* (in *Shubin's sense* in ${\operatorname{Sh}}_\rho ^{(\omega _0)}({\mathbf R^{d}})$) of order $\rho _0$, if there is an $R>0$ such that for every $\alpha \in {\mathbf N^{d}}$, it holds $$\begin{aligned}
{2}
|\partial ^\alpha f(x)| &\lesssim |f(x)|{\langle x\rangle}^{-\rho |\alpha |}, &
\qquad |x|&\ge R,
\intertext{and}
|f(x)| &\gtrsim \omega _0(x){\langle x\rangle}^{-\rho _0}, &
\qquad |x| &\ge R .\end{aligned}$$

Elliptic and hypoelliptic symbols are important since they give rise to parametrices. For $\rho$, $\omega$ as above and ${\mathfrak a}\in {\operatorname{Sh}}^{(\omega )}_\rho ({\mathbf R^{2d}})$ elliptic, there is an elliptic symbol ${\mathfrak b}\in {\operatorname{Sh}}^{(1/\omega )}_\rho ({\mathbf R^{2d}})$ such that $${\operatorname{Op}}_A({\mathfrak a})\circ {\operatorname{Op}}_A({\mathfrak b}) =I+{\operatorname{Op}}_A({\mathfrak c}_1)
\quad \text{and}\quad
{\operatorname{Op}}_A({\mathfrak b})\circ {\operatorname{Op}}_A({\mathfrak a}) =I+{\operatorname{Op}}_A({\mathfrak c}_2)$$ for some ${\mathfrak c}_1,{\mathfrak c}_2\in {\mathscr S}({\mathbf R^{2d}})$. An operator ${\operatorname{Op}}({\mathfrak c})$ with $c\in {\mathscr S}({\mathbf R^{2d}})$ is regularizing in the sense that ${\operatorname{Op}}({\mathfrak c})$ is continuous from ${\mathscr S}'({\mathbf R^{d}})$ to ${\mathscr S}({\mathbf R^{d}})$. (Cf. e.g. @BonChe [@Shubin1].)

Reformulation of pseudo-differential calculus using the Bargmann transform {#sec2}
==========================================================================

In this section we characterize the Bargmann assignment of pseudo-differential operator symbols from Subsection \[subsec1.5\], using estimates of complex derivatives. In Subsection \[subsec2.1\] we show how pseudo-differential operators on ${\mathbf R^{d}}$ with Shubin symbols are transformed to Wick operators by the Bargmann transform. In Subsection \[subsec2.2\] we deduce similar links between pseudo-differential operators of infinite order, given in the second part of Subsection \[subsec1.5\], and suitable classes of Wick operators. Subsection \[subsec2.3\] treats composition formulae for symbols of Wick operators, which leads to algebraic properties for operators in Subsection \[subsec2.1\] and \[subsec2.2\]. As an application we obtain short proofs of composition results for pseudo-differential operators on ${\mathbf R^{d}}$ from Subsection \[subsec1.5\].

Wick symbols of Shubin pseudo-differential operators {#subsec2.1}
----------------------------------------------------

The following proposition is essential in the characterization of Shubin type pseudo-differential operators on ${\mathbf R^{d}}$ by means of the corresponding Wick symbols. The Shubin classes can be characterized using the transform ${\mathcal{T}}_\phi$ by means of estimates of the form $$\begin{aligned}
|\partial_x^\alpha \partial_\xi^\beta {\mathcal{T}}_\phi f(x,\xi)|
&\lesssim
\omega(x) {\langle x\rangle}^{- \rho |\alpha|} {\langle \xi\rangle}^{-N},
\label{eq:Gineq1}
\\[1ex]
|\partial_x^\alpha{\mathcal{T}}_\phi f(x,\xi)|
&\lesssim
\omega(x) {\langle x\rangle}^{- \rho |\alpha|} {\langle \xi\rangle}^{-N}
\label{eq:Gineq2}
\intertext{and}
|{\mathcal{T}}_\phi f(x,\xi)|
&\lesssim
\omega(x) {\langle \xi\rangle}^{-N}. 
\label{eq:Gineq3}\end{aligned}$$ The proof of the following result is similar to the proof of @Cappiello1 [Proposition 3.2].

\[prop:symbchar1\] Let $0 \le \rho {\leqslant}1$, let $\omega \in {\mathscr P}_{{\operatorname{Sh}},\rho }({\mathbf R^{d}})$, and suppose $f\in {\mathscr{S}}'({\mathbf R^{d}})$ and $\phi \in{\mathscr{S}}({\mathbf R^{d}}) \setminus 0$. The following conditions are equivalent:

1.  $f \in {\operatorname{Sh}}_\rho^{(\omega )}({\mathbf R^{d}})$,

2.  holds true for any $N {\geqslant}0$ and $\alpha, \beta \in {\mathbf N^{d}}$,

3.  holds true for any $N {\geqslant}0$ and $\alpha \in {\mathbf N^{d}}$,

and the following conditions are equivalent:

1.  $f \in {\operatorname{Sh}}_0^{(\omega )}({\mathbf R^{d}})$,

2.  holds true for any $N {\geqslant}0$.

First we prove that (1) implies (2). Suppose $f \in {\operatorname{Sh}}_\rho^{(\omega )}({\mathbf R^{d}})$ and let $\alpha,\beta, \gamma \in {\mathbf N^{d}}$ be arbitrary. We will show $$|\xi^\gamma \partial_x^\alpha \partial_\xi^\beta {\mathcal{T}}_\phi f(x,\xi)|
\lesssim
\omega(x) {\langle x\rangle}^{-\rho|\alpha|}.$$

By , and integration by parts we get $$\begin{gathered}
|\xi^\gamma \partial_x^\alpha \partial_\xi^\beta {\mathcal{T}}_\phi f(x,\xi)|
 =
\left
| \xi^\gamma {\mathcal{T}}_{\phi _\beta}(\partial^\alpha f)(x,\xi)
\right |
\\[1ex]
 =
(2 \pi)^{-\frac d2} \left| \int_{{\mathbf R^{d}}} \left((i\partial_{y})^\gamma e^{- i {\langle}\xi,y {\rangle}}\right)
\overline{\phi _\beta(y)} \, \partial^\alpha f(x+y)\, dy \right |
\\[1ex]
\lesssim
\int_{{\mathbf R^{d}}} \left| \partial_{y}^\gamma \left[\overline{\phi _\beta(y)} \,
\partial^\alpha f(x+y)\right] \right|\, dy
\\[1ex]
 =
\int_{{\mathbf R^{d}}} \left| \sum_{\kappa {\leqslant}\gamma} \binom{\gamma}{\kappa}
\partial^{\gamma-\kappa}
\overline{\phi _\beta(y)} \, \partial^{\alpha+\kappa} f(x+y)\right|\, dy 
\\[1ex]
 \lesssim
\sum_{\kappa {\leqslant}\gamma} \binom{\gamma}{\kappa}  \int_{{\mathbf R^{d}}} \left |
\partial^{\gamma-\kappa} 
\phi _\beta(y) \right| \, \omega(x+y) {\langle x+y\rangle}^{-\rho|\alpha+\kappa|}\, dy.\end{gathered}$$ Since $\omega$ is polynomially moderate, Peetre's inequality and the fact that $\phi \in {\mathscr{S}}$ give $$\begin{gathered}
|\xi^\gamma \partial_x^\alpha \partial_\xi^\beta {\mathcal{T}}_\phi f(x,\xi)|
\\[1ex]
\lesssim
\omega(x) {\langle x\rangle}^{-\rho |\alpha |} \sum_{\kappa {\leqslant}\gamma}
\binom{\gamma}{\kappa}
\int_{{\mathbf R^{d}}} \left| \partial^{\gamma-\kappa} \phi _\beta(y) \right| \, \omega (y)
\, {\langle y\rangle}^{|m|+\rho |\alpha+\kappa|}\, dy
\\[1ex]
\asymp \omega(x) {\langle x\rangle}^{- \rho |\alpha|}. \end{gathered}$$ Thus $f\in {\operatorname{Sh}}_\rho^{(\omega )}({\mathbf R^{d}})$ implies implies , and as a special case , and $f \in {\operatorname{Sh}}_0^{(\omega )}({\mathbf R^{d}})$ implies . We have proved that (1) implies (2) which in turn implies (3), and that (1)$'$ implies (2)$'$.

Conversely, suppose (3), that is $f \in {\mathscr{S}}'({\mathbf R^{d}})$ and holds for all $ N \ge 0$ and all $\alpha \in {\mathbf N^{d}}$, which is a weaker assumption than (2). We obtain from $$\begin{aligned}
f(x)
& = \|\phi \|_{L^2}^{-2} \, {\mathcal{T}}_\phi^* {\mathcal{T}}_\phi f(x)
\\[1ex]
& = \|\phi \|_{L^2}^{-2} \, (2 \pi)^{-\frac d2} \iint _{{\mathbf R^{2d}}}
{\mathcal{T}}_\phi f(y,\xi) \, e^{i {\langle}\xi,x-y {\rangle}} \, \phi (x-y) \, dyd\xi\end{aligned}$$ which is an absolutely convergent integral due to and the fact that $\phi \in{\mathscr{S}}({\mathbf R^{d}})$. We may differentiate under the integral, so integration by parts, and Peetre's inequality give for some $N_0 \ge 0$, any $\alpha \in {\mathbf N^{d}}$ and any $x \in {\mathbf R^{d}}$ $$\begin{gathered}
\left |
\partial^\alpha f(x)
\right |
 =
\|\phi \|_{L^2}^{-2} \, (2 \pi)^{-\frac d2}  \left| \iint _{{\mathbf R^{2d}}} {\mathcal{T}}_\phi f(y,\xi)
\, \partial_y^\alpha
\left ( e^{i {\langle}\xi,x-y {\rangle}} \, \phi (x-y) \right ) \, dyd\xi
\right |
\\[1ex]
 =
\|\phi \|_{L^2}^{-2} \, (2 \pi)^{-\frac d2}  \left| \iint _{{\mathbf R^{2d}}} \partial_y^\alpha
{\mathcal{T}}_\phi f(y,\xi) \, e^{i {\langle}\xi,x-y {\rangle}} \, \phi (x-y) \, dyd\xi \right |
\\[1ex]
 =
\|\phi \|_{L^2}^{-2} \, (2 \pi)^{-\frac d2}  \left| \iint _{{\mathbf R^{2d}}} \partial_y^\alpha
{\mathcal{T}}_\phi f(x-y,\xi) \, e^{i {\langle}\xi,y {\rangle}} \, \phi (y) \, dyd\xi \right |
\\[1ex]
 \lesssim
\iint _{{\mathbf R^{2d}}}  \omega(x-y) {\langle x-y\rangle}^{- \rho |\alpha|} \,
{\langle \xi\rangle}^{-d-1} \, |\phi (y)| \, dyd\xi
\\[1ex]
 \lesssim
\omega (x) {\langle x\rangle}^{- \rho |\alpha|} \iint _{{\mathbf R^{2d}}}  {\langle \xi\rangle}^{-d-1}
{\langle y\rangle}^{N_0+\rho |\alpha|} \, |\phi (y)| \, dyd\xi
\\[1ex]
\asymp
\omega(x) {\langle x\rangle}^{- \rho |\alpha|}.  \end{gathered}$$ Thus $f \in {\operatorname{Sh}}_\rho^{(\omega )}({\mathbf R^{d}})$ and we have proved the equivalence of (1), (2) and (3).

It remains to show that (2)$'$ implies (1)$'$, that is for all $N \ge 0$ implies $f \in {\operatorname{Sh}}_0^{(\omega )}({\mathbf R^{d}})$. We have for some $N_0 {\geqslant}0$, any $\alpha \in {\mathbf N^{d}}$, $x \in {\mathbf R^{d}}$ and $N {\geqslant}0$, $$\begin{aligned}
\left |
\partial^\alpha f(x)
\right |
& =
\|\phi \|_{L^2}^{-2} \, (2 \pi)^{-\frac d2}  \left| \iint _{{\mathbf R^{2d}}} {\mathcal{T}}_\phi f(y,\xi) \, \partial _x^\alpha
\left ( e^{i {\langle}\xi,x-y {\rangle}} \, \phi (x-y) \right ) \, dyd\xi
\right |
\\[1ex]
& \lesssim
\sum_{\beta {\leqslant}\alpha}
\binom{\alpha}{\beta}
\iint _{{\mathbf R^{2d}}} \left|  {\mathcal{T}}_\phi f(y,\xi) \right| {\langle \xi\rangle}^{|\beta|} 
\left| \partial^{\alpha-\beta} \phi (x-y) \right| \, dyd\xi
\\[1ex]
& \lesssim
\sum_{\beta {\leqslant}\alpha}
\binom{\alpha}{\beta}
\iint _{{\mathbf R^{2d}}} \omega(y) {\langle \xi\rangle}^{|\alpha|-N} 
\left| \partial^{\alpha-\beta} \phi (x-y) \right| \, dyd\xi
\\[1ex]
& \lesssim 
\omega (x) \sum_{\beta {\leqslant}\alpha}
\binom{\alpha}{\beta}
\iint _{{\mathbf R^{2d}}} {\langle \xi\rangle}^{|\alpha|-N} 
{\langle x-y\rangle}^{N_0} \left| \partial^{\alpha-\beta} \phi (x-y) \right| \, dyd\xi
\\[1ex]
& \lesssim
\omega(x)\end{aligned}$$ provided $N$ is sufficiently large, since $\phi \in {\mathscr{S}}$. This shows that $f \in {\operatorname{Sh}}_0^{(\omega )}({\mathbf R^{d}})$.

We may now characterize the Shubin classes ${\operatorname{Sh}}_\rho^{(\omega )}({\mathbf R^{2d}})$ by estimates on their Bargmann (kernel) assignments of the forms $$\begin{aligned}
\big | 
( \partial_z  + \overline \partial _w)^\alpha (\partial _z- \overline \partial _w)
^\beta & {\mathsf S}_{\mathfrak V} {\mathfrak a}(z,w) 
\big |
\notag
\\[1ex]
&\lesssim
e^{\frac1{2} |z-w|^2}\omega (\sqrt 2\, \overline z)
{\langle z+w\rangle}^{- \rho |\alpha + \beta|} {\langle z-w\rangle}^{-N} ,
\label{eq:characShubinBargmann}
\\[1ex]
\left| \partial_z^\alpha \overline \partial _w
^\beta \mathsf S_{\mathfrak V} {\mathfrak a}(z,w) \right |
&\lesssim
e^{ \frac1{2} |z-w|^2}\omega (\sqrt 2\, \overline z){\langle z+w\rangle}^{- \rho |\alpha + \beta|}
{\langle z-w\rangle}^{-N} ,
\label{eq:characShubinBargmann2}
\\[1ex]
\left | \mathsf S_{\mathfrak V} {\mathfrak a}(z,w) \right|
&\lesssim
e^{ \frac1{2} |z-w|^2} \omega (\sqrt 2\, \overline z)
{\langle z-w\rangle}^{-N}
\label{eq:characShubinBargmann0}
\intertext{and}
\left | K_{\mathfrak V,{\mathfrak a}} (z,w) \right|
&\lesssim
\omega (\sqrt 2\, \overline z)
{\langle z-w\rangle}^{-N} e^{ \frac1{2} \left( |z|^2 + |w|^2 \right)}. 
\label{eq:characShubinBargmannkernel0}\end{aligned}$$

\[Thm:ShubinAnalChar\] Let $0 \le \rho {\leqslant}1$, $\omega \in {\mathscr P}_{{\operatorname{Sh}},\rho}({\mathbf R^{2d}})$ and ${\mathfrak a}\in {\mathscr S}'({\mathbf R^{2d}})$. The following conditions are equivalent:

1.  ${\mathfrak a}\in {\operatorname{Sh}}^{(\omega )}_\rho ({\mathbf R^{2d}})$,

2.  holds true for every $N\ge 0$, $z,w\in {\mathbf C^{d}}$ and $\alpha,\beta \in {\mathbf N^{d}}$,

3.  holds true for every $N\ge 0$, $z,w\in {\mathbf C^{d}}$ and $\alpha,\beta \in {\mathbf N^{d}}$,

and the following conditions are equivalent:

1.  ${\mathfrak a}\in {\operatorname{Sh}}^{(\omega )}_0 ({\mathbf R^{2d}})$,

2.  holds true for any $N \in {\mathbf N}$ and $z,w\in {\mathbf C^{d}}$,

3.  holds true for any $N \in {\mathbf N}$ and $z,w\in {\mathbf C^{d}}$.

For the proof we need the following proposition of independent interest. Here we recall that $\mathsf S_{\mathfrak V}$ is bijective from ${\mathcal S}_{1/2}'({\mathbf R^{2d}})$ to the set $$\label{Eq:SBGelfandShilovLargeDist}
{\{ \, a\in \wideparen A({\mathbf C^{2d}})\, ;\, |a(z,w)|\lesssim e^{(\frac 12+r)|z-w|^2}\ \text{for every}\ r>0 \, \} }.$$

\[Prop:SBaTaRel\] Let $\psi (x,\xi ) = (\frac 2\pi )^{\frac d2} e^{-(|x|^2+|\xi |^2)}$, $x,\xi \in {\mathbf R^{d}}$, ${\mathfrak a}\in {\mathcal S}_{1/2}'({\mathbf R^{2d}})$ and $a$ belongs to the set in . Then $$\begin{aligned}
\mathsf S_{\mathfrak V} {\mathfrak a}(z,w)
&=
(2 \pi)^{\frac d2} e^{\frac1{2} |z-w|^2} 
{\mathcal{T}}_\psi {\mathfrak a}\left( \frac{x+y}{\sqrt{2}}, - \frac{\xi+\eta}{\sqrt{2}},
\sqrt{2}(\eta-\xi),  \sqrt{2} (y-x) \right),
\label{eq:BargmannsymbolSTFT}
\intertext{and}
(\mathsf S_{\mathfrak V}^{-1}a)(x,-\xi )
&=
\left( \frac{2}{\pi} \right)^d \int _{{\mathbf C^{d}}}
a \left( \frac z{\sqrt 2} - w, \frac z{\sqrt 2} + w \right)
e^{- 2 |w|^2}\, d\lambda (w),
\label{eq:InvBargmannsymbolSTFT}\end{aligned}$$ with $z=x+i\xi $, $w=y+i\eta$ and $x,y,\xi ,\eta \in {\mathbf R^{d}}$.

Let $\phi (x,y) = \pi^{-\frac d2} e^{-\frac1{2}(|x|^2+|y|^2)}$ for $x,y \in {\mathbf R^{d}}$, and let $K_{{\mathfrak a}}^w$ be the kernel of ${\operatorname{Op}}^w ({\mathfrak a})$. Then $\psi ={\mathscr F}_2(\phi \circ \kappa)$, where $\kappa (x,y)=(x+y/2,x-y/2)$. By (or @Teofanov2 [Eq. (1.35)]) and @Cappiello1 [Lemma 4.1] we have $$\begin{aligned}
& \mathfrak V _{\Theta ,d} K_{{\mathfrak a}}^w (z,w) 
= \mathfrak V _{2d} K_{{\mathfrak a}}^w (z,\overline w) 
= \mathfrak V _{2d} K_{{\mathfrak a}}^w ( (x,y) + i (\xi,-\eta))
\\[1ex]
& = (2 \pi)^d e^{\frac1{2} \left( |z|^2 + |w|^2 \right) + i \left( \langle x, \xi \rangle
- \langle y, \eta \rangle \right)} 
{\mathcal{T}}_\phi K_{{\mathfrak a}}^w \left( \sqrt{2} (x,y), - \sqrt{2} (\xi,-\eta) \right)
\\[1ex]
& = (2 \pi)^{\frac d2} e^{\frac1{2} \left( |z|^2 + |w|^2 \right) + i \left( \langle y, \xi \rangle -
\langle x, \eta \rangle \right)} 
{\mathcal{T}}_\psi {\mathfrak a}\left( \frac{x+y}{\sqrt{2}}, - \frac{\xi+\eta}{\sqrt{2}}, \sqrt{2}(\eta-\xi),
\sqrt{2} (y-x) \right),
\\[1ex]
& = (2 \pi)^{\frac d2} e^{\frac1{2} \left( |z|^2 + |w|^2 \right) + i \, {{\operatorname{Im}}}(z,w)} 
{\mathcal{T}}_\psi {\mathfrak a}\left( \frac{x+y}{\sqrt{2}}, - \frac{\xi+\eta}{\sqrt{2}}, \sqrt{2}(\eta-\xi),
\sqrt{2} (y-x) \right).\end{aligned}$$ Together with the identity $$|z|^2 + |w|^2 + 2i \, {{\operatorname{Im}}}(z,w) = |z-w|^2 +2(z,w)$$ this gives $$\begin{gathered}
\mathfrak V _{\Theta ,d} K_{{\mathfrak a}}^w (z,w)
\\[1ex]
= (2 \pi)^{\frac d2} e^{\frac1{2}|z-w|^2 + (z,w)} 
{\mathcal{T}}_\psi {\mathfrak a}\left( \frac{x+y}{\sqrt{2}}, - \frac{\xi+\eta}{\sqrt{2}}, \sqrt{2}(\eta-\xi),
\sqrt{2} (y-x) \right ).\end{gathered}$$ A combination of this identity with and gives .

In order to prove , we use Moyal's formula , and the fact that ${\Vert \psi\Vert _{L^2}}=1$. This implies that the inverse of ${\mathcal{T}}_\psi$ is given by $$\begin{gathered}
({\mathcal{T}}_\psi ^{-1}F) (x,\xi ) = ({\mathcal{T}}_\psi ^*F) (x,\xi )
\\[1ex]
=
(2\pi )^{-d}\iiiint _{{\mathbf R^{4d}}}
F(x_1,\xi _1,\eta _1,y_1)\psi (x-x_1,\xi -\xi _1)
e^{i({\langle x-x_1,\eta _1\rangle}+{\langle y_1,\xi -\xi _1\rangle}}\, dx_1d\xi _1d\eta _1dy_1.\end{gathered}$$ Writing $$G(z,w) = F(x,\xi ,\eta ,y),\qquad z=x+i\xi ,\ w=y+i\eta,$$ we obtain $$\label{Eq:TInvOp1}
{\mathcal{T}}_\psi ^*F(x,\xi )
 =
2^d(2\pi )^{-\frac {3d}2}\iint _{{\mathbf C^{2d}}} G(w_1,w_2)e^{-|z-w_1|^2}
e^{i{{\operatorname{Im}}}{\langle z-w_1,w_2\rangle}}\, d\lambda (w_1)d\lambda (w_2).$$

If ${\mathfrak a}= {\mathcal{T}}_\psi ^*F$ and $a=\mathsf S_{\mathfrak V} {\mathfrak a}$, then shows that $$a(z,w)
=
(2\pi )^{\frac d2}e^{\frac 12|z-w|^2}G \left( \frac {\overline z+\overline w}{\sqrt 2},\sqrt 2(w-z) \right)$$ which gives $$G(z,w)
=
(2\pi )^{-\frac d2}e^{-\frac 14|w|^2}
a \left( \frac {2\overline z -w}{2\sqrt 2},\frac {2\overline z +w}{2\sqrt 2} \right).$$

Inserting this into we get $$\begin{gathered}
{\mathcal{T}}_\psi ^*F (x,-\xi )
\\[1ex]
=
\frac 1{2^d\pi ^{2d}}\iint _{{\mathbf C^{2d}}}
a \left( \frac {2\overline w_1 -w_2}{2\sqrt 2},\frac {2\overline w_1 +w_2}{2\sqrt 2} \right)
e^{-|\overline z-w_1|^2}e^{-\frac 14|w_2|^2}
e^{i{{\operatorname{Im}}}{\langle \overline z-w_1,w_2\rangle}}\, d\lambda (w_1)d\lambda (w_2),\end{gathered}$$ 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 $$\begin{aligned}
& {\mathcal{T}}_\psi ^*F (x,-\xi )
\\[1ex]
& =
\frac {2^d}{\pi ^{2d}}\iint _{{\mathbf C^{2d}}}
a \left( w_1+\frac z{\sqrt 2},w_2+\frac z{\sqrt 2} \right)
e^{-(|w_1|^2+|w_2|^2)}
e^{2i{{\operatorname{Im}}}({w_1},{w_2})}\, d\lambda (w_1)d\lambda (w_2)
\\[1ex]
& =
2^d \iint _{{\mathbf C^{2d}}}
a \left( w_1+\frac z{\sqrt 2},w_2+\frac z{\sqrt 2} \right)
e^{2i{{\operatorname{Im}}}({w_1},{w_2})}\, d\mu (w_1)d\mu (w_2)
\\[1ex]
& =
2^d \int _{{\mathbf C^{d}}} \left( \int _{{\mathbf C^{d}}}
a \left( w_1+\frac z{\sqrt 2},w_2+\frac z{\sqrt 2} \right)
e^{ (w_1,w_2)} \, e^{ (-w_2,w_1)} \, d\mu (w_1) \right) d\mu (w_2)
\\[1ex]
& =
2^d \int _{{\mathbf C^{d}}} 
a \left( -w_2+\frac z{\sqrt 2},w_2+\frac z{\sqrt 2} \right)
e^{ -|w_2|^2}  \, d\mu (w_2)
\\[1ex]
& =
\left( \frac{2}{\pi} \right)^d \int _{{\mathbf C^{d}}} 
a \left(\frac z{\sqrt 2} - w,\frac z{\sqrt 2} + w \right)
e^{ -2 |w|^2}  \, d\lambda (w). \end{aligned}$$

\[Proof of Theorem \[Thm:ShubinAnalChar\]\] Combining Propositions \[prop:symbchar1\] and \[Prop:SBaTaRel\], writing $z + w = 2 z + w - z$, we obtain that ${\mathfrak a}\in {\operatorname{Sh}}_\rho^{(\omega )}({\mathbf R^{2d}})$ if and only if for all $\alpha,\beta \in {\mathbf N^{d}}$ and $N \in {\mathbf N}$ we have $$\begin{gathered}
\left | (\partial_x + \partial_y)^\alpha (\partial_\xi + \partial_\eta)^\beta
\left( e^{ - \frac1{2} |z-w|^2} \mathsf S_{\mathfrak V} {\mathfrak a}(z,w) \right)\right |
\\[1ex]
\lesssim
\omega \left( \frac{\overline{z+w}}{\sqrt{2}} \right)
{\langle z+w\rangle}^{- \rho |\alpha + \beta|} {\langle z-w\rangle}^{-N}
\\[1ex]
\lesssim
\omega (\sqrt 2\, \overline z){\langle z+w\rangle}^{- \rho |\alpha + \beta|} {\langle z-w\rangle}^{-N+k}\end{gathered}$$ for some $k \in {\mathbf N}$ 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 $$\begin{gathered}
\left| (\partial_x + \partial_y)^\alpha (\partial_\xi + \partial_\eta)^\beta
\mathsf S_{\mathfrak V} {\mathfrak a}(z,w) \right |
\\[1ex]
\lesssim
\omega (\sqrt 2\, \overline z){\langle z+w\rangle}^{- \rho |\alpha + \beta|}
{\langle z-w\rangle}^{-N} e^{ \frac1{2} |z-w|^2} .\end{gathered}$$

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

The equivalence between (2) and (3) follows from the binomial formulae $$\begin{aligned}
(\partial _z+ t\overline \partial _w)^\alpha
&=
\sum _{\gamma \le \alpha}
{\alpha \choose \gamma}t^{|\gamma|}\partial _z^{\alpha -\gamma}
\overline \partial _w^{\gamma},\qquad t\in \{ -1,1\} ,
\\[1ex]
\partial _z^\alpha
&=
2^{-|\alpha |}
\sum _{\gamma \le \alpha}
{\alpha \choose \gamma}
(\partial _z+  \overline \partial _w)^{\alpha -\gamma}
(\partial _z-  \overline \partial _w)^{\gamma}
\intertext{and}
\overline \partial _w^\beta
&=
2^{-|\beta |}
\sum _{\gamma \le \beta}
{\beta \choose \gamma} (-1)^{|\gamma |}
(\partial _z+  \overline \partial _w)^{\beta -\gamma}
(\partial _z-  \overline \partial _w)^{\gamma}. \end{aligned}$$

It remains to consider the case $\rho = 0$. We obtain from Propositions \[prop:symbchar1\] and \[Prop:SBaTaRel\] that ${\mathfrak a}\in {\operatorname{Sh}}_0^{(\omega )}({\mathbf R^{2d}})$ if and only if for all $N \in {\mathbf N}$ we have $$\left | \mathsf S_{\mathfrak V} {\mathfrak a}(z,w) \right|
\lesssim
\omega (\sqrt 2\, \overline z)
{\langle z-w\rangle}^{-N} e^{ \frac1{2} |z-w|^2}, \quad z, \zeta \in {\mathbf C^{d}}.$$ This shows the equivalence between (1)$'$ and (2)$'$.

Finally the equivalence of (2)$'$ and (3)$'$ is an immediate consequence of and $$|e^{(|z|^2+|w|^2)/2} e^{-(z,w)}| = e^{(|z|^2-2 \, {{\operatorname{Re}}}(z,w)+|w|^2)/2}= e^{|z-w|^2/2}.
\qquad \qedhere$$

Let $\wideparen {\mathcal A}_{{\operatorname{Sh}},\rho}^{(\omega )}({\mathbf C^{2d}})$, be the set of all $a\in \wideparen A({\mathbf C^{2d}})$ such that $$\label{Eq:ShubinWickEstimates}
\left| \partial_z^\alpha \overline \partial _w
^\beta a(z,w) \right |
\le C
e^{ \frac1{2} |z-w|^2}\omega (\sqrt 2\, \overline z){\langle z+w\rangle}^{- \rho |\alpha + \beta|}
{\langle z-w\rangle}^{-N} ,\quad N\ge 0.$$ The smallest constant $C\ge 0$ defines a semi-norm parameterized by $\alpha$, $\beta$ and $N$, and we equip $\wideparen {\mathcal A}_{{\operatorname{Sh}},\rho}^{(\omega )}({\mathbf C^{2d}})$ with the Fr<span>é</span>chet space topology defined by these semi-norms. The following result is an immediate consequence of Theorem \[Thm:ShubinAnalChar\] and its proof.

Let $0 \le \rho {\leqslant}1$ and $\omega \in {\mathscr P}_{{\operatorname{Sh}},\rho}({\mathbf R^{2d}})$. Then ${\mathsf S}_{\mathfrak V}$ is a homeomorphism from ${\operatorname{Sh}}^{(\omega )}_\rho ({\mathbf R^{2d}})$ to $\wideparen
{\mathcal A}_{{\operatorname{Sh}},\rho}^{(\omega )}({\mathbf C^{2d}})$.

Wick operators corresponding to Gevrey type pseudo-differential operators {#subsec2.2}
-------------------------------------------------------------------------

Using and we obtain the following theorem expressed with estimates of the form $$\label{Eq:BargmannSTFT}
|a(z,w)|
\lesssim \exp \left( \frac1{2} |z-w|^2  + r_1 |z+w|_{s,\sigma} - r_2 |z-w|_{s,\sigma}  \right)$$ (cf. Definition \[Def:GevreyGSSymbols\]). The verification is left for the reader.

\[Thm:GevreySymbolsBargmTransfer\] The following is true:

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

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

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

The restrictions on $s$ and $\sigma$ in Theorem \[Thm:GevreySymbolsBargmTransfer\] are needed since we must choose $\psi$ in as the Gauss function in Proposition \[Prop:SBaTaRel\]. According to the proof of Theorem \[Thm:ShubinAnalChar\] this is necessary for the use of the formula that relates ${\mathcal{T}}_\phi K_{{\mathfrak a}}^w$ and the Bargmann transform $\mathfrak V _{2d} K_{{\mathfrak a}}^w$. For this $\psi$ we have $\psi \in {\mathcal S}_s^\sigma ({\mathbf R^{d}})$ ($\psi \in \Sigma _s^\sigma ({\mathbf R^{d}})$), if and only if $s,\sigma \ge \frac 12$ ($s,\sigma > \frac 12$).

Theorem \[Thm:GevreySymbolsBargmTransfer\] can be combined with continuity results in @AbCaTo to deduce continuity of Wick operators acting on the Bargmann images of $\Sigma _s^\sigma ({\mathbf R^{d}})$, ${\mathcal S}_s^\sigma ({\mathbf R^{d}})$, $({\mathcal S}_s^\sigma )'({\mathbf R^{d}})$ and $(\Sigma _s^\sigma )'({\mathbf R^{d}})$, respectively. The following result follows by a straight-forward combination of Theorems 3.8, 3.15 and 3.16 in @AbCaTo, and Theorem \[Thm:GevreySymbolsBargmTransfer\].

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

1.  if $s,\sigma \ge \frac 12$ and holds for all $r_1 > 0$ and some $r_2 > 0$, then ${\operatorname{Op}}_{\mathfrak V}(a)$ is continuous on ${\mathcal A}_s^\sigma ({\mathbf C^{d}})$ and on $({\mathcal A}_s^\sigma )'({\mathbf C^{d}})$;

2.  if $s,\sigma > \frac 12$ and holds for some $r_1 > 0$ and all $r_2 > 0$, then ${\operatorname{Op}}_{\mathfrak V}(a)$ is continuous on ${\mathcal A}_{0,s}^\sigma ({\mathbf C^{d}})$ and on $({\mathcal A}_{0,s}^\sigma )'({\mathbf C^{d}})$;

3.  if $s,\sigma >\frac 12$ and holds for some $r_1 > 0$ and some $r_2 > 0$, then ${\operatorname{Op}}_{\mathfrak V}(a)$ is continuous from ${\mathcal A}_{0,s}^\sigma ({\mathbf C^{d}})$ to ${\mathcal A}_{s}^\sigma ({\mathbf C^{d}})$, and from $({\mathcal A}_{s}^\sigma )'({\mathbf C^{d}})$ to $({\mathcal A}_{0,s}^\sigma )'({\mathbf C^{d}})$.

Composition of Wick operators {#subsec2.3}
-----------------------------

Let $a_1,a_2 \in \wideparen A({\mathbf C^{2d}})$. If composition is well defined then the complex twisted product $a_1 {{\text{\footnotesize $\#$}}}_{\mathfrak V}a_2$ is defined by $${\operatorname{Op}}_{\mathfrak V}(a_1)\circ {\operatorname{Op}}_{\mathfrak V}(a_2) =
{\operatorname{Op}}_{\mathfrak V}(a_1 {{\text{\footnotesize $\#$}}}_{\mathfrak V}a_2).$$ By straight-forward computations it follows that the product ${{\text{\footnotesize $\#$}}}_{\mathfrak V}$ is given by $$\label{Eq:DefCompTwistProd}
a_1 {{\text{\footnotesize $\#$}}}_{\mathfrak V} a_2 (z,w)
= \pi ^{-d}\int_{{\mathbf C^{d}}} a_1(z,u) a_2 (u,w) e^{-(z-u,w-u)}\, d \lambda (u),
\quad z,w \in {\mathbf C^{d}},$$ provided the integral is well defined. Inserting derivatives, takes the form $$\begin{gathered}
\tag*{(\ref{Eq:DefCompTwistProd})$'$}
(\partial _z^{\alpha _1} \overline \partial _w^{\beta _1} a_1) {{\text{\footnotesize $\#$}}}_{\mathfrak V}
(\partial _z^{\alpha _2} \overline \partial _w^{\beta _2}a_2) (z,w)
\\[1ex]
= \pi ^{-d}\int_{{\mathbf C^{d}}} (\partial _z^{\alpha _1} \overline \partial _u^{\beta _1} a_1)(z,u)
(\partial _u^{\alpha _2} \overline \partial _w^{\beta _2} a_2)(u,w) e^{-(z-u,w-u)}
\, d \lambda (u), \quad z,w \in {\mathbf C^{d}}.\end{gathered}$$

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

\[Lemma:DiffCompTwistProd\] Let $a_1,a_2\in \wideparen A({\mathbf C^{2d}})$ and suppose the integral in $'$ is well defined for all $z,w\in {\mathbf C^{d}}$ and all $\alpha _1,\alpha _2,\beta _1,\beta _2\in {\mathbf N^{d}}$ such that $$|\alpha _1+\alpha _2+\beta _1+\beta _2|\le 1.$$ Suppose also that the integrand in is harperblake@example.net. Then $$\begin{aligned}
{2}
\partial _{z_j}(a_1 {{\text{\footnotesize $\#$}}}_{\mathfrak V}a_2)
&=
(\partial _{z_j}a_1) {{\text{\footnotesize $\#$}}}_{\mathfrak V}a_2
+
a_1 {{\text{\footnotesize $\#$}}}_{\mathfrak V}(\partial _{z_j}a_2), &
\quad
j &= 1,\dots ,d
\label{Eq:DiffCompTwistProd1}
\intertext{and}
\overline \partial _{w_j}(a_1 {{\text{\footnotesize $\#$}}}_{\mathfrak V}a_2)
&=
(\overline \partial _{w_j}a_1) {{\text{\footnotesize $\#$}}}_{\mathfrak V}a_2
+
a_1 {{\text{\footnotesize $\#$}}}_{\mathfrak V}(\overline \partial _{w_j}a_2), &
\quad
j &= 1,\dots ,d.
\label{Eq:DiffCompTwistProd2}\end{aligned}$$

If $$F_{a_1,a_2}(z,w,u)
=
a_1(z,u) a_2 (u,w) e^{(z,u-w) + (u,w)}$$ then $$\pi ^d(a_1 {{\text{\footnotesize $\#$}}}_{\mathfrak V}a_2)(z,w)
= \int_{{\mathbf C^{d}}}F_{a_1,a_2}(z,w,u)e^{-|u|^2}\, d\lambda (u).$$ This gives $$\pi ^d \partial _{z_j}(a_1 {{\text{\footnotesize $\#$}}}_{\mathfrak V}a_2) (z,w) = b_1(z,w)
+b_2(z,w)-b_3(z,w),$$ where $$\begin{aligned}
b_1(z,w)
&=
\int_{{\mathbf C^{d}}} F_{\partial _{z_j}a_1,a_2}(z,w,u)e^{-|u|^2}\, d\lambda (u),
\\[1ex]
b_2(z,w)
&=
\int_{{\mathbf C^{d}}} F_{a_1,a_2}(z,w,u)\overline u_je^{-|u|^2}\, d\lambda (u)
\intertext{and}
b_3(z,w)
&=
\overline w_j\int_{{\mathbf C^{d}}} F_{a_1,a_2}(z,w,u)e^{-|u|^2}\, d\lambda (u)
\\[1ex]
&=
\overline w_j \pi^d (a_1 {{\text{\footnotesize $\#$}}}_{\mathfrak V}a_2)(z,w).\end{aligned}$$

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 $$\begin{gathered}
\partial _{u_j}F_{a_1,a_2}(z,w,u)
\\[1ex]
=
\left( a_1(z,u)  \partial _{u_j}a_2 (u,w) 
+
\overline w_ja_1(z,u) a_2 (u,w) \right) e^{(z,u-w) + (u,w)}
\\[1ex]
=
F_{a_1,\partial _{z_j}a_2}(z,w,u) + \overline w_jF_{a_1,a_2}(z,w,u).\end{gathered}$$ Consider $b_2(z,w)$. Integration by parts gives $$\begin{gathered}
b_2(z,w)
=
\int_{{\mathbf C^{d}}} F_{a_1,a_2}(z,w,u)\overline u_je^{-|u|^2}\, d\lambda (u)
\\[1ex]
= - \int_{{\mathbf C^{d}}} F_{a_1,a_2}(z,w,u) \, \partial _{u_j}e^{-|u|^2} \, d\lambda (u)
\\[1ex]
=
\int_{{\mathbf C^{d}}} \partial _{u_j}F_{a_1,a_2}(z,w,u) e^{-|u|^2}\, d\lambda (u)
\\[1ex]
=
\int_{{\mathbf C^{d}}} F_{a_1,\partial _{z_j}a_2}(z,w,u)
e^{-|u|^2}\, d\lambda (u)
+
\overline w_j \int_{{\mathbf C^{d}}} F_{a_1,a_2}(z,w,u)
e^{-|u|^2}\, d\lambda (u)
\\[1ex]
=
\int_{{\mathbf C^{d}}} F_{a_1,\partial _{z_j}a_2}(z,w,u)
e^{-|u|^2}\, d\lambda (u)
+
b_3(z,w).\end{gathered}$$

A combination of these identities now gives $$\begin{gathered}
\pi ^d \partial _{z_j} (a_1 {{\text{\footnotesize $\#$}}}_{\mathfrak V}a_2) (z,w)
\\[1ex]
=
\int_{{\mathbf C^{d}}} (
F_{\partial _{z_j}a_1,a_2}(z,w,u)
+
F_{a_1,\partial _{z_j}a_2}(z,w,u)
)e^{-|u|^2}\, d\lambda (u)
\\[1ex]
=\pi ^d(\partial _{z_j}a_1) {{\text{\footnotesize $\#$}}}_{\mathfrak V}a_2(z,w)
+
\pi ^d a_1 {{\text{\footnotesize $\#$}}}_{\mathfrak V}(\partial _{z_j}a_2) (z,w),\end{gathered}$$ and follows.

The assertion is proved by similar arguments.

The characterization in Theorem \[Thm:ShubinAnalChar\] (3) can be applied to prove the following composition result, which is a generalization of @Shubin1 [Theorem 23.6] to include the case when $\rho = 0$.

\[prop:composition\] Let $0 \le \rho {\leqslant}1$ and $\omega_j \in {\mathscr P}_{{\operatorname{Sh}},\rho}({\mathbf R^{2d}})$ for $j = 1,2$. If ${\mathfrak a}_j \in {\operatorname{Sh}}_\rho^{(\omega _j)}({\mathbf R^{2d}})$ for $j = 1,2$, then ${\mathfrak a}_1 {{\text{\footnotesize $\#$}}}{\mathfrak a}_2
\in {\operatorname{Sh}}_\rho^{(\omega _1\omega _2)}({\mathbf R^{2d}})$.

If ${\mathfrak a}_0 = {\mathfrak a}_1 {{\text{\footnotesize $\#$}}}{\mathfrak a}_2$ and $a_j = \mathsf S_{\mathfrak V} {\mathfrak a}_j$, $j=0,1,2$, then $a_0 = a_1 {{\text{\footnotesize $\#$}}}_{\mathfrak V} a_2$. From Lemma \[Lemma:DiffCompTwistProd\] and we obtain for $\alpha, \beta \in {\mathbf N^{d}}$, $$\begin{aligned}
& \partial_z  ^\alpha \overline \partial _w
^\beta a_0 (z,w)
\\[1ex]
& = \sum_{\gamma {\leqslant}\alpha} \sum_{\kappa {\leqslant}\beta}
\binom{\alpha}{\gamma} \binom{\beta}{\kappa}
\left( (\partial _z^{\alpha - \gamma} \overline \partial _w^{\beta-\kappa}
a_1) {{\text{\footnotesize $\#$}}}_{\mathfrak V} 
(\partial _z^{\gamma} \overline \partial _w^{\kappa}a_2) \right)(z,w)
\\[1ex]
& = \pi^{-d} \sum_{\gamma {\leqslant}\alpha} \sum_{\kappa {\leqslant}\beta}
\binom{\alpha}{\gamma} \binom{\beta}{\kappa}
\int_{{\mathbf C^{d}}} \partial _z^{\alpha-\gamma} {\overline \partial} _u^{\beta-\kappa} a_1(z,u)
\partial _u^{\gamma} \overline \partial _w^{\kappa} a_2 (u,w) e^{(z,u-w) + (u,w)} d \mu (u). \end{aligned}$$

Since $\omega _2 \in {\mathscr P}({\mathbf R^{2d}})\simeq {\mathscr P}({\mathbf C^{d}})$ is moderate, Theorem \[Thm:ShubinAnalChar\] gives for some $N_0 \ge 0$ and any $N_1, N_2 {\geqslant}0$ $$\begin{aligned}
|\partial _u^{\alpha-\gamma} \overline \partial _w^{\beta-\kappa} a_1(z,u)|
& \lesssim
\omega _1(\sqrt 2\, \overline z){\langle z+u\rangle}^{-\rho |\alpha+\beta-\gamma -\kappa |}
{\langle z-u\rangle}^{-N_1} e^{\frac1{2} | z - u |^2}
\intertext{and}
|\partial _u^\gamma \overline \partial _w^\kappa a_2(u,w)|
& \lesssim
\omega _2(\sqrt 2\, \overline z){\langle z-u\rangle}^{N_0}{\langle u+w\rangle}^{-\rho |\gamma +\kappa |}
{\langle u-w\rangle}^{-N_2} e^{\frac1{2} | u - w |^2}. \end{aligned}$$

This gives $$\begin{gathered}
\label{Eq:Derb0Est}
\left | \partial _z^\alpha \overline \partial _w
^\beta a_0 (z,w) \right|
\\[1ex]
\lesssim
\omega _1(\sqrt 2\, \overline z)\omega _2(\sqrt 2\, \overline z) e^{\frac 12|z-w|^2}
\int_{{\mathbf C^{d}}} 
F(z,w,u)e^{\Phi (z,w,u)}\, d \lambda (u)\end{gathered}$$ where for any $N_1 {\geqslant}0$ $$\begin{aligned}
F(z,w,u)
&=
{\langle z+u\rangle}^{-\rho|\alpha+\beta-\gamma-\kappa|} {\langle z-u\rangle}^{N_0-N_1}
{\langle u+w\rangle}^{-\rho|\gamma+\kappa|} {\langle u-w\rangle}^{-N_2}
\intertext{and}
\Phi (z,w,u)
&=
- \frac1{2} |z-w|^2 + \frac1{2} |z-u|^2 + \frac1{2} |u-w|^2 - |u|^2
\\[1ex]
& \qquad \qquad \qquad \qquad \qquad \qquad \qquad
+ {{\operatorname{Re}}}(z,u-w) + {{\operatorname{Re}}}(u,w) = 0.\end{aligned}$$ By Peetre's inequality and the facts that $\gamma \le \alpha$ and $\kappa \le \beta$ we get $$\begin{gathered}
{\langle z+u\rangle}^{\rho |\gamma +\kappa|}{\langle u+w\rangle}^{-\rho|\gamma+\kappa|}
\lesssim
{\langle z-w\rangle}^{\rho |\gamma +\kappa|}
\\[1ex]
\lesssim
{\langle z-u\rangle}^{\rho |\gamma +\kappa|}{\langle u-w\rangle}^{\rho |\gamma +\kappa|}
\\[1ex]
\le 
{\langle z-u\rangle}^{\rho |\alpha +\beta|}{\langle u-w\rangle}^{\rho |\alpha +\beta|}\end{gathered}$$ and $${\langle z+u\rangle}^{-\rho |\alpha +\beta |}
\lesssim
{\langle z+w\rangle}^{-\rho |\alpha +\beta |}{\langle u-w\rangle}^{\rho |\alpha +\beta |}$$ wherefrom $$\label{Eq:FuncFEst}
F(z,w,u)
\le 
{\langle z+w\rangle}^{-\rho |\alpha +\beta|}
{\langle z-u\rangle}^{\rho |\alpha +\beta| +N_0-N_1}
{\langle u-w\rangle}^{2\rho |\alpha +\beta |-N_2}.$$ Hence a combination of and gives for any $N {\geqslant}0$ $$\begin{aligned}
& (\omega _1(\sqrt 2\, \overline z)\omega _2(\sqrt 2\, \overline z))^{-1}
{\langle z+w\rangle}^{\rho|\alpha+\beta|}
\left| \partial _z^\alpha \overline \partial _w
^\beta a_0 (z,w) \right|
\\[1ex]
& \lesssim
e^{\frac1{2} |z-w|^2}
\int_{{\mathbf C^{d}}} {\langle z-u\rangle}^{\rho|\alpha+\beta| + N_0 - N_1} {\langle u-w\rangle}
^{2\rho|\alpha+\beta| - N_2}\, d \lambda (u)
\\[1ex]
& \lesssim
 {\langle z-w\rangle}^{-N} e^{\frac1{2} |z-w|^2}
\int_{{\mathbf C^{d}}} {\langle z-u\rangle}^{\rho|\alpha+\beta| + N_0+N - N_1} {\langle u-w\rangle}
^{2\rho|\alpha+\beta| + N - N_2}\, d \lambda (u). \end{aligned}$$ By letting $$N_1 {\geqslant}\rho|\alpha+\beta| + N_0 + N
\quad \text{and}\quad 
N_2 > 2\rho|\alpha+\beta| + N  + 2 d$$ we obtain $$\left| \partial _z^\alpha \overline \partial _w^\beta
a_0 (z,w) \right| 
\lesssim
\omega _1(\sqrt 2\, \overline z)\omega _2(\sqrt 2\, \overline z)
{\langle z+w\rangle}^{-\rho|\alpha+\beta|}
{\langle z-w\rangle}^{-N} e^{\frac1{2} |z-w|^2}.$$ According to Theorem \[Thm:ShubinAnalChar\] (3) this estimate implies that ${\mathfrak a}_0 \in {\operatorname{Sh}}_\rho^{(\omega _1\omega _2)}({\mathbf R^{2d}})$.

Eq. combined with Theorem \[Thm:GevreySymbolsBargmTransfer\] can be used to show composition results for pseudo-differential operators with symbols in $\Gamma ^{\sigma ,s}_{s,\sigma ;0}({\mathbf R^{2d}})$. In fact we may use an argument similar to the proof of Proposition \[prop:composition\], but simpler since derivatives can be avoided. We obtain $$\begin{aligned}
{3}
{\mathfrak a}_1 {{\text{\footnotesize $\#$}}}{\mathfrak a}_2 \in \Gamma ^{\sigma ,s}_{s,\sigma ;0}({\mathbf R^{2d}})
\quad \text{when}\quad
{\mathfrak a}_1,{\mathfrak a}_2 \in \Gamma ^{\sigma ,s}_{s,\sigma ;0}({\mathbf R^{2d}}),
\quad s,\sigma \ge \frac 12,\end{aligned}$$ 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 @AbCaTo [Theorem 3.18] for certain restrictions on $s$ and $\sigma$.

Relations and estimates for Wick and anti-Wick operators {#sec3}
========================================================

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 {#subsec3.1}
---------------------------------------------------------------------------

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

\[Prop:WickToAntiWick\] Suppose $s\ge \frac 12$, $a \in \wideparen {\mathcal A}_s'({\mathbf C^{2d}})$, let $N\ge 0$ be an integer, and let $$\begin{aligned}
a_\alpha (w) &= \partial _z^\alpha \overline \partial _w^\alpha a (w,w), \quad \alpha \in {\mathbf N^{d}}, 
\intertext{and}
b_\alpha (z,w) &= |\alpha | 
\int _0^1 (1-t)^{|\alpha |-1} \partial _z^\alpha \overline \partial _w^\alpha a
(w+t(z-w),w)\, dt, \quad \alpha \in {\mathbf N^{d}}\setminus 0.\end{aligned}$$ Then $$\label{Eq:WickToAntiWick}
{\operatorname{Op}}_{\mathfrak V}(a)
=
\sum _{|\alpha |\le N}
\frac {(-1)^{|\alpha |}{\operatorname{Op}}_{\mathfrak V}^{{\operatorname{aw}}}(a_\alpha )}{\alpha !}
+
\sum _{|\alpha | = N+1}\frac {(-1)^{|\alpha |}{\operatorname{Op}}_{\mathfrak V}(b_\alpha )}{\alpha !}.$$

Taylor expansion gives $$\begin{aligned}
a(z,w) & = \sum _{|\alpha |\le N} \frac {(-1)^{|\alpha |} c_{\alpha}(z,w)} {\alpha !}
+
\sum _{|\alpha | = N+1}\frac {(-1)^{|\alpha |}c_{0,\alpha}(z,w)} {\alpha !},
\intertext{where}
c_{\alpha} (z,w) &= (-1)^{|\alpha |} (z-w)^\alpha \partial _z^\alpha a (w,w)
\intertext{and}
c_{0,\alpha} (z,w) &= (-1)^{|\alpha |} |\alpha |(z-w)^\alpha 
\int _0^1 (1-t)^{|\alpha |-1} \partial _z^\alpha a (w+t(z-w),w)\, dt. \end{aligned}$$ Hence $${\operatorname{Op}}_{\mathfrak V}(a)
=
\sum _{|\alpha |\le N}\frac {(-1)^{|\alpha |}{\operatorname{Op}}_{\mathfrak V}(c_{\alpha} )}{\alpha !}
+
\sum _{|\alpha | = N+1}\frac {(-1)^{|\alpha |}{\operatorname{Op}}_{\mathfrak V}(c_{0,\alpha} )}{\alpha !},$$ and the result follows if we prove $$\label{Eq:WickAntiWickExpIdent}
{\operatorname{Op}}_{\mathfrak V}(c_{\alpha} )
=
{\operatorname{Op}}_{\mathfrak V}^{{\operatorname{aw}}}(a_\alpha ) 
\quad \text{and}\quad
{\operatorname{Op}}_{\mathfrak V}(c_{0,\alpha} )
=
{\operatorname{Op}}_{\mathfrak V}(b_\alpha ) .$$

It follows from that $${\operatorname{Op}}_{\mathfrak V}(b_\alpha ) = {\operatorname{Op}}_{\mathfrak V}(c_{1,\alpha} )
\quad \text{and}\quad
{\operatorname{Op}}_{\mathfrak V}(c_{0,\alpha} ) = {\operatorname{Op}}_{\mathfrak V}(c_{2,\alpha} )$$ where $$\begin{aligned}
c_{j,\alpha}(z,w)
&=
(-1)^{|\alpha |} \pi ^{-d} |\alpha |
\int _0^1 (1-t)^{|\alpha |-1} h_{j,\alpha}(a;t,z,w)\, dt, \label{eq:cjdef}
\intertext{$j=1,2$, with}
h_{1,\alpha}(a;t,z,w)
&=
(-1)^{|\alpha |} \int _{{\mathbf C^{d}}}
\partial _z^\alpha \overline \partial _w^\alpha a
(w_1+t(z-w_1),w_1)
e^{-(z-w_1,w-w_1)}\, d\lambda (w_1) \label{eq:h1def}
\intertext{and}
h_{2,\alpha}(a;t,z,w)
&=
\int _{{\mathbf C^{d}}}(z-w_1)^\alpha
\partial _z^\alpha a
(w_1+t(z-w_1),w_1)
e^{-(z-w_1,w-w_1)}\, d\lambda (w_1). \nonumber\end{aligned}$$ Since $$(z-w_1)^\alpha e^{-(z-w_1,w-w_1)}
=
\overline \partial _{w_1}^\alpha  e^{-(z-w_1,w-w_1)}$$ integration by parts yields $$\begin{gathered}
h_{2,\alpha}(a;t,z,w)
=
\int _{{\mathbf C^{d}}}
\partial _z^\alpha a
(w_1+t(z-w_1),w_1)
\overline \partial _{w_1}^\alpha e^{-(z-w_1,w-w_1)} \, d\lambda (w_1)
\\[1ex]
=
(-1)^{|\alpha |}
\int _{{\mathbf C^{d}}}
\partial _z^\alpha \overline \partial _{w}^\alpha a
(w_1+t(z-w_1),w_1)
e^{-(z-w_1,w-w_1)}\, d\lambda (w_1)
=
h_{1,\alpha}(a;t,z,w),\end{gathered}$$ and the second equality in follows. The first equality in follows by similar arguments. The details are left for the reader.

\[Rem:WickToAntiWick2\] Proposition \[Prop:WickToAntiWick\] and its proof show that $$\tag*{(\ref{Eq:WickToAntiWick})$'$}
{\operatorname{Op}}_{\mathfrak V}(a)
=
\sum _{|\alpha |\le N}
\frac {(-1)^{|\alpha |}{\operatorname{Op}}_{\mathfrak V}^{{\operatorname{aw}}}(a_{\alpha})}{\alpha !}
+
\sum _{|\alpha | = N+1}\frac {(-1)^{|\alpha |}{\operatorname{Op}}_{\mathfrak V}(c_{1,\alpha})}{\alpha !}$$ where $c_{1,\alpha}$ is defined by and .

In the following result we estimate $a_\alpha$ in Proposition \[Prop:WickToAntiWick\] and $c_{1,\alpha}$ in when $a =\mathsf S_{\mathfrak V} {\mathfrak a}$ satisfies for every $N\ge 0$ and $\alpha,\beta \in {\mathbf N^{d}}$. By Theorem \[Thm:ShubinAnalChar\] this means that ${\operatorname{Op}}_{\mathfrak V}(a)$ is the Bargmann transform of a Shubin type operator.

\[Prop:ShubinWickExpEst\] Let $0 \le \rho \le 1$, $\omega \in {\mathscr P}_{{\operatorname{Sh}},\rho}({\mathbf R^{2d}})$, $a \in \wideparen {\mathcal A}_{{\operatorname{Sh}},\rho}^{(\omega )}({\mathbf C^{2d}})$, and let $a_\alpha$ and $b_\alpha$ be as in Proposition \[Prop:WickToAntiWick\] for $\alpha \in {\mathbf N^{d}}$. Then ${\operatorname{Op}}_{\mathfrak V}(b_\alpha)
= {\operatorname{Op}}_{\mathfrak V}(c_{1,\alpha})$ for a unique $c_{1,\alpha}\in \wideparen A({\mathbf C^{2d}})$, $$\begin{aligned}
|\partial _w^\beta \overline \partial _w^\gamma a_\alpha (w)|
&\lesssim 
\omega (\sqrt 2\overline {w}) {\langle w\rangle}^{-\rho (2 |\alpha| +|\beta +\gamma |)},
\quad
\alpha ,\beta ,\gamma \in {\mathbf N^{d}},
\label{Eq:ShubinWickExpEst1}
\intertext{and}
|\partial _z^\beta \overline \partial _w^\gamma  c_{1,\alpha} (z,w)|
&\lesssim
e^{\frac{1}{2} |z-w|^2}
\omega (\sqrt 2\overline {z}) {\langle z+w\rangle}^{-\rho (2 |\alpha| + |\beta +\gamma|)}
{\langle z-w\rangle}^{-N},
\quad
\alpha ,\beta ,\gamma \in {\mathbf N^{d}}.
\label{Eq:ShubinWickExpEst2}\end{aligned}$$

\[Rem:UniquenessIdentRemainderTerm\] The Wick symbol $c_{1,\alpha}$ in Proposition \[Prop:ShubinWickExpEst\] is uniquely defined and given by in view of Proposition \[Prop:LAOpsIdent\], when $h_{1,\alpha}$ is defined by . The conditions in Proposition \[Prop:ShubinWickExpEst\] imply that $c_{1,\alpha} \in \wideparen {\mathcal A}_{{\operatorname{Sh}},\rho}^{(\omega _\alpha )} ({\mathbf C^{2d}})$ where $\omega _\alpha ={\langle {\, \cdot \, }\rangle} ^{-2\rho |\alpha |}\cdot \omega$.

\[Proof of Proposition \[Prop:ShubinWickExpEst\]\] The estimate is an immediate consequence of $$\partial _w^\beta \overline \partial _w^\gamma a_\alpha (w) 
= \partial _w^{\alpha+\beta} \overline \partial _w^{\alpha +\gamma} a (w,w)$$ and .

In order to prove we first note that the uniqueness assertion for $c_{1,\alpha}$ is a consequence of Remark \[Rem:UniquenessIdentRemainderTerm\]. Let $h_{1,\alpha}(a;z,w)$ be the same as in the proof of Proposition \[Prop:WickToAntiWick\]. 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 holds for $\beta =\gamma =0$.

The assumption $a \in \wideparen {\mathcal A}_{{\operatorname{Sh}},\rho}^{(\omega )}({\mathbf C^{2d}})$ combined with $\omega$ and ${\langle {\, \cdot \, }\rangle} ^{-|\alpha |}$ being moderate imply $$| \partial_z^\alpha \overline \partial_w^\beta a(z,w) |
\lesssim
e^{\frac1{2} |z-w|^2}
\omega (\sqrt 2\, \overline w){\langle w\rangle}^{- \rho |\alpha +\beta |}
{\langle z-w\rangle}^{-N}$$ for every $N\ge 0$. This gives $$\begin{gathered}
e^{{{\operatorname{Re}}}(z,w)} |h_{1,\alpha} (a;t,z,w)|
\\[1ex]
\lesssim
\int _{{\mathbf C^{d}}}
\omega (\sqrt 2\overline {w_1})
e^{\frac{t^2}{2} |z-w_1|^2}{\langle w_1\rangle}^{-2\rho |\alpha |}{\langle t(z-w_1)\rangle}^{-N}
e^{{{\operatorname{Re}}}(z+w-w_1,w_1)}\, d\lambda (w_1),\end{gathered}$$ that is $$\begin{gathered}
\label{Eq:LongEsthalpha}
e^{-\frac 14|z-w|^2} |h_{1,\alpha} (a;t,z,w)|
\\[1ex]
\lesssim
\int _{{\mathbf C^{d}}}
\omega (\sqrt 2\overline {w_1})
e^{\frac{t^2}{2} |z-w_1|^2}{\langle w_1\rangle}^{-2\rho |\alpha |}{\langle t(z-w_1)\rangle}^{-N}
e^{-|w_1-z_2|^2}\, d\lambda (w_1)
\\[1ex]
=
\int _{{\mathbf C^{d}}}
\omega (\sqrt 2(\overline {z_2+w_1}))
e^{\frac{t^2}{2} |z_1-w_1|^2}{\langle z_2+w_1\rangle}^{-2\rho |\alpha |}{\langle t(z_1-w_1)\rangle}^{-N}
e^{-|w_1|^2}\, d\lambda (w_1)\end{gathered}$$ 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 $$\begin{gathered}
e^{-|z_1|^2} |h_{1,\alpha} (a;t,z,w)|
\lesssim
\omega (\sqrt 2\overline {z_2}){\langle z_2\rangle}^{-2\rho |\alpha |}
\int _{{\mathbf C^{d}}}
e^{\frac 18|w_1|^2}
e^{\frac{1}{8} |z_1-w_1|^2}
e^{-|w_1|^2}\, d\lambda (w_1)
\\[1ex]
\lesssim
\omega (\sqrt 2\overline {z_2}){\langle z_2\rangle}^{-2\rho |\alpha |} e^{\frac{1}{4} |z_1|^2}\int _{{\mathbf C^{d}}}
e^{\frac{1}{4} |w_1|^2}
e^{-\frac 78|w_1|^2}\, d\lambda (w_1)
\\[1ex]
\lesssim
\omega (\sqrt 2\overline {z_2}){\langle z_2\rangle}^{-2\rho |\alpha |} e^{\frac{1}{2} |z_1|^2}
{\langle z_1\rangle}^{-N},\end{gathered}$$ for every $N\ge 0$. The moderateness of $\omega$ again gives $$\label{Eq:halphaEst}
|h_{1,\alpha} (a;t,z,w)|\lesssim e^{\frac{1}{2} |z-w|^2}
\omega (\sqrt 2\overline {z}) {\langle z+w\rangle}^{-2\rho |\alpha |}{\langle z-w\rangle}^{-N}$$ or every $N\ge 0$, when $t\in [0,\frac 12]$.

Suppose instead $t\in [\frac 12,1]$. Then ${\langle t(z_1-w_1)\rangle}^{-N}\asymp {\langle z_1-w_1\rangle}^{-N}$. Moderateness again gives $$\omega (\sqrt 2(\overline {z_2+w_1}))
{\langle z_2+w_1\rangle}^{-2\rho |\alpha |}{\langle z_1-w_1\rangle}^{-N_0}
\lesssim
\omega (\sqrt 2\overline z)
{\langle z\rangle}^{-2\rho |\alpha |}$$ for some $N_0$. Hence gives $$\begin{gathered}
e^{-|z_1|^2} \omega (\sqrt 2\overline z)^{-1}{\langle z\rangle}^{2\rho |\alpha |}
| h_{1,\alpha} (a;t,z,w)|
\\[1ex]
\lesssim
\int _{{\mathbf C^{d}}}
e^{\frac{1}{2} |z_1-w_1|^2}{\langle z_1-w_1\rangle}^{-N}
e^{-|w_1|^2}\, d\lambda (w_1)
\\[1ex]
=
e^{|z_1|^2}\int _{{\mathbf C^{d}}}
{\langle z_1-w_1\rangle}^{-N}
e^{- \frac{1}{2} |w_1+z_1|^2}\, d\lambda (w_1)
\asymp
e^{|z_1|^2}{\langle z_1\rangle}^{-N}\end{gathered}$$ for every $N\ge 0$. This gives also for $t\in [\frac 12,1]$.

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

The next result, analogous to Proposition \[Prop:ShubinWickExpEst\], will be useful in Section \[sec5\] when we discuss hypoellipticity for Shubin operators in the Wick setting.

\[Prop:AsymptoticExpansion1\] Let $\rho \ge 0$, $\omega \in {\mathscr P}_{{\operatorname{Sh}},\rho}({\mathbf C^{d}})$, $\omega _t=\omega \cdot {\langle {\, \cdot \, }\rangle} ^{-2\rho t}$ when $t\ge 0$, $a\in \wideparen {\mathcal A}_{{\operatorname{Sh}},\rho}^{(\omega )}({\mathbf C^{2d}})$, ${\mathfrak a}= \mathsf S_{\mathfrak V}^{-1}a$ and $N\ge 0$ be an integer. Then $$\label{Eq:ExpansionWickTaylor}
{\mathfrak a}(x,-\xi ) = \sum _{|\alpha |\le N}
\frac {(-1)^{|\alpha |}
(\partial _z^\alpha \overline \partial _w^\alpha a)(2^{-\frac 12}z,2^{-\frac 12}z)}
{2^{|\alpha |}\alpha !} +c_N(z), \quad z = x + i \xi,$$ where $$\label{Eq:EstExpansionWickTaylor}
\begin{alignedat}{2}
\partial _z^\alpha \overline \partial _w^\alpha a
\in
\wideparen {\mathcal A}_{{\operatorname{Sh}},\rho}^{(\omega _{|\alpha |})}({\mathbf C^{2d}})
\quad &\text{and} &\quad
(x,\xi )\mapsto c_N(x-i\xi ) \in {\operatorname{Sh}}_{\rho}^{(\omega _{N+1})}({\mathbf R^{2d}}). 
\end{alignedat}$$

The first claim in $\partial _z^\alpha \overline \partial _w^\alpha a \in
\wideparen {\mathcal A}_{{\operatorname{Sh}},\rho}^{(\omega _{|\alpha |})}({\mathbf C^{2d}})$ is an immediate consequence of the definition and Peetre's inequality.

By Taylor expanding the right-hand side of we obtain $$\label{Eq:ExpansionWickTaylor1}
{\mathfrak a}(x,-\xi ) = \sum _{|\alpha +\beta |\le 2N+1}
\frac {(2/\pi)^d I_{\alpha ,\beta}\cdot 
(\partial _z^\alpha \overline \partial _w^\beta a)(2^{-\frac 12}z,2^{-\frac 12}z)}
{\alpha !\beta !} +c_N(z),$$ where $$I_{\alpha ,\beta}
=
\int _{{\mathbf C^{d}}}(-w)^\alpha \overline w^\beta e^{- 2 |w|^2}
\, d\lambda (w),$$ and $$\label{Eq:cNdef}
c_N(z)
=
2(N+1)
\sum _{|\alpha +\beta |= 2N+2}
\frac {(-1)^{|\beta |}}{\alpha !\beta !}
\int _0^1(1-\theta )^{2N+1}
H_{\alpha ,\beta } (z ,\theta )\, d\theta$$ with $$\label{Eq:Hdef}
H_{\alpha ,\beta } (z ,\theta ) =
\left ( \frac 2\pi \right )^d \int _{{\mathbf C^{d}}}
(\partial _z^\alpha \overline \partial _w^\beta a)
\left ( \frac z{\sqrt 2}- \theta w,\frac z{\sqrt 2}+\theta w \right )
w^\alpha \overline w^\beta 
e^{- 2 |w|^2 }\, d\lambda (w).$$

The orthonormality of $\{ e_\alpha \} _{\alpha \in {\mathbf N^{d}}}\subseteq A^2({\mathbf C^{d}})$ (cf. ) yields $I_{\alpha,\beta} = 0$ if $\alpha \neq \beta$ and $$\begin{aligned}
I_{\alpha,\alpha} 
& = \int _{{\mathbf C^{d}}}(-w)^\alpha \overline w^\alpha e^{- 2 |w|^2} \, d\lambda (w)
\\[1ex]
& = (-1)^{|\alpha|} 2^{-d -|\alpha|} \alpha! \pi^d \int _{{\mathbf C^{d}}} |e_\alpha(w) |^2
\, d\mu (w)
\\[1ex]
& = (-1)^{|\alpha|} 2^{-d -|\alpha|} \alpha! \pi^d. \end{aligned}$$ Comparing with 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 ${\mathfrak c}(x,\xi )=c_N(x-i\xi )$ belongs to ${\operatorname{Sh}}_{\rho}^{(\omega _{N+1})}({\mathbf R^{2d}})$. If $$h_{\alpha ,\beta } (z ,w,\theta ) =
(\partial _z^\alpha \overline \partial _w^\beta a)
\left ( \frac z{\sqrt 2}- \theta w,\frac z{\sqrt 2}+\theta w \right )
w^\alpha \overline w^\beta e^{- 2 |w|^2 }$$ then $$H_{\alpha ,\beta } (z ,\theta )
=
\left ( \frac 2\pi \right )^d \int _{{\mathbf C^{d}}}
h_{\alpha ,\beta } (z ,w,\theta )
\, d\lambda (w).$$

First we notice that $$\begin{aligned}
\partial _z^\alpha \overline \partial _z^\beta c_N(z)
&=
2(N+1)
\sum _{|\gamma +\delta |= 2N+2}
\frac {(-1)^{|\delta |}}{\gamma !\delta !}
\int _0^1(1-\theta )^{2N+1}
\partial _z^{\alpha}\overline \partial _z^{\beta}
H_{\gamma ,\delta } (z ,\theta )\, d\theta, 
\\[1ex]
\partial _z^\alpha \overline \partial _z^\beta H_{\gamma ,\delta} (z,\theta )
&=
\left ( \frac 2\pi \right )^d
\int _{{\mathbf C^{d}}} \partial _z^\alpha \overline \partial _z^\beta h_{\gamma ,\delta}
(z,w,\theta ) \, d\lambda (w)
\intertext{and}
\partial _z^\alpha \overline \partial _z^\beta h_{\gamma ,\delta}
(z,w,\theta )
&=
2^{-\frac {|\alpha +\beta |}2}(\partial _z^{\alpha +\gamma}
\overline \partial _z^{\beta +\delta}
a)\left( \frac z{\sqrt 2} - \theta w,\frac z{\sqrt 2} + \theta w \right )w^\gamma
\overline w^\delta e^{- 2 |w|^2 }.\end{aligned}$$ From the definition this implies that for every $M\ge 0$ and some $M_0\ge 0$ we have $$\begin{gathered}
|\partial _z^\alpha \overline \partial _z^\beta h_{\gamma ,\delta}
(z,w,\theta )|
\\[1ex]
\lesssim
e^{-2(1-\theta ^2)|w|^2}\omega (\overline z-\sqrt 2\theta \overline w)
{\langle z\rangle}^{-\rho (|\alpha +\beta |+2N+2)}{\langle \theta w\rangle}^{-M-M_0}|w|^{2N+2}
\\[1ex]
\lesssim
e^{-2(1-\theta )|w|^2}\omega (\overline z)
{\langle z\rangle}^{-\rho (|\alpha +\beta |+2N+2)}{\langle \theta w\rangle}^{-M}|w|^{2N+2}.\end{gathered}$$ This gives $$|\partial _z^\alpha \overline \partial _z^\beta H_{\gamma ,\delta} (z,\theta )|
\lesssim
\omega (\overline z)
{\langle z\rangle}^{-\rho (|\alpha +\beta |+2N+2)}\cdot J(\theta ),$$ where $$J(\theta ) = \int _{{\mathbf C^{d}}} e^{-2(1-\theta )|w|^2}{\langle \theta w\rangle}^{-M}|w|^{2N+2}
\, d\lambda (w).$$

For $\theta \in [0,\frac 12]$ we get $$J(\theta ) = \int _{{\mathbf C^{d}}} e^{-|w|^2}|w|^{2N+2}\, d\lambda (w),$$ which is finite and independent of $\theta$. If instead $\theta \in [\frac 12,1]$, and choosing $M>2d+2N+2$, then $$J(\theta ) \le
\int _{{\mathbf C^{d}}} {\langle \theta w\rangle}^{-M}|w|^{2N+2}\, d\lambda (w),$$ which is again finite and independent of $\theta$.

A combination of these estimates give $$|\partial _z^\alpha \overline \partial _z^\beta H_{\gamma ,\delta} (z,\theta )|
\lesssim
\omega (\overline z)
{\langle z\rangle}^{-\rho (|\alpha +\beta |+2N+2)},$$ which in turn implies $$|\partial _z^\alpha \overline \partial _z^\beta c_N(z)|
\lesssim
\omega (\overline z)
{\langle z\rangle}^{-\rho (|\alpha +\beta |+2N+2)}.$$ This means that ${\mathfrak c}\in {\operatorname{Sh}}_{\rho}^{(\omega _{N+1})}({\mathbf R^{2d}})$.

Continuity of anti-Wick operators with exponentially bounded symbols {#subsec3.2}
--------------------------------------------------------------------

Next we consider anti-Wick symbols that satisfy exponential bounds of the form $$\begin{aligned}
|a_0 (w)|
&\lesssim
e^{-r_0 |w|^{\frac 1s}},
\label{Eq:NonAWGSEst1}
\intertext{or}
|a_0 (w)|
&\lesssim
e^{r_0 |w|^{\frac 1s}}.
\label{Eq:NonAWGSEst2}\end{aligned}$$

In order to formulate our results we introduce new spaces of entire functions. Let $s > \frac{1}{2}$, $t_0, r > 0$, and let ${\mathcal A}_{s, t_0,r}({\mathbf C^{d}})$ be the Banach space of all $F \in A({\mathbf C^{d}})$ such that $${\Vert F\Vert _{{\mathcal A}_{s, t_0,r}}} \equiv
{\Vert F\cdot e^{- t_0 |{\, \cdot \, }|^2 + r |{\, \cdot \, }|^{\frac{1}{s}}}\Vert _{L^\infty}} 
< \infty.$$ Set $${\mathcal A}_{0,(s, t_0)}({\mathbf C^{d}}) = \bigcap_{r > 0} {\mathcal A}_{s, t_0,r} ({\mathbf C^{d}})
\quad \text{and}\quad
{\mathcal A}_{(s, t_0)}'({\mathbf C^{d}}) = \bigcap_{r > 0} {\mathcal A}_{s, t_0,-r} ({\mathbf C^{d}})$$ equipped with the projective limit topology. Likewise we set $${\mathcal A}_{(s, t_0)}({\mathbf C^{d}}) = \bigcup_{r > 0} {\mathcal A}_{s, t_0,r} ({\mathbf C^{d}})
\quad \text{and}\quad
{\mathcal A}_{0,(s, t_0)}'({\mathbf C^{d}}) = \bigcup_{r > 0} {\mathcal A}_{s, t_0,-r} ({\mathbf C^{d}})$$ equipped with the inductive limit topology.

Referring to Section \[subsec1.3\] it is clear that the spaces ${\mathcal A}_{0,(s, t_0)}({\mathbf C^{d}})$, ${\mathcal A}_{(s, t_0)}({\mathbf C^{d}})$, ${\mathcal A}_{(s, t_0)}'({\mathbf C^{d}})$ and ${\mathcal A}_{0,(s, t_0)}'({\mathbf C^{d}})$ are generalizations of $$\begin{aligned}
{\mathcal A}_{0,(s, \frac{1}{2})}({\mathbf C^{d}}) &= \mathfrak V_d ( \Sigma_s ({\mathbf R^{d}}) ) = {\mathcal A}_{0,s}({\mathbf C^{d}})
\\[1ex]
{\mathcal A}_{(s, \frac{1}{2})}({\mathbf C^{d}}) &= \mathfrak V_d ( {\mathcal S}_s ({\mathbf R^{d}}) ) = {\mathcal A}_{s}({\mathbf C^{d}})
\\[1ex]
{\mathcal A}_{(s, \frac{1}{2})}'({\mathbf C^{d}}) &= \mathfrak V_d ( {\mathcal S}_s '({\mathbf R^{d}}) ) = {\mathcal A}_{s}'({\mathbf C^{d}})
\intertext{and}
{\mathcal A}_{0,(s, \frac{1}{2})}'({\mathbf C^{d}}) &= \mathfrak V_d ( \Sigma_s' ({\mathbf R^{d}}) ) = {\mathcal A}_{0,s}'({\mathbf C^{d}}),\end{aligned}$$ respectively.

\[prop:contAW\] Let $a_0\in L^\infty _{loc}({\mathbf C^{d}})$, $0 < t_0 < 1$ and $$\label{eq:t0t1}
t_1 = \frac{1}{4(1-t_0)}.$$ Then the following is true:

1.  if holds for some $r_0 > 0$ then $$\label{Eq:ContAW1}
    \begin{alignedat}{2}
    {\operatorname{Op}}_{\mathfrak V}^{{\operatorname{aw}}}(a_0) &: & {\mathcal A}_{0,(s, t_0)}({\mathbf C^{d}}) &\to {\mathcal A}_{0,(s, t_1)}({\mathbf C^{d}}),
    \\[1ex]
    {\operatorname{Op}}_{\mathfrak V}^{{\operatorname{aw}}}(a_0) &: & {\mathcal A}_{0,(s, t_0)}'({\mathbf C^{d}}) &\to {\mathcal A}_{0,(s, t_1)}'({\mathbf C^{d}})
    \end{alignedat}$$ are continuous;

2.  if holds for every $r_0 > 0$ then $$\label{Eq:ContAW2}
    \begin{alignedat}{2}
    {\operatorname{Op}}_{\mathfrak V}^{{\operatorname{aw}}}(a_0) &: &{\mathcal A}_{(s, t_0)}({\mathbf C^{d}}) &\to {\mathcal A}_{(s, t_1)}({\mathbf C^{d}}),
    \\[1ex]
    {\operatorname{Op}}_{\mathfrak V}^{{\operatorname{aw}}}(a_0) &: &{\mathcal A}_{(s, t_0)}'({\mathbf C^{d}}) &\to {\mathcal A}_{(s, t_1)}'({\mathbf C^{d}})
    \end{alignedat}$$ are continuous.

We only prove that the first map in 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 $F\in {\mathcal A}_{0,(s, t_0)}({\mathbf C^{d}})$. We have for $z \in {\mathbf C^{d}}$ $$\begin{aligned}
& |{\operatorname{Op}}_{\mathfrak V}^{{\operatorname{aw}}}(a_0)F(z)| e^{- t_1 |z |^2 + r_2 |z|^{\frac{1}{s}}}
\\[1ex]
& \lesssim e^{- t_1 |z |^2 + r_2 |z|^{\frac{1}{s}}} 
\int_{{\mathbf C^{d}}} |a_0 (w)| \, |F(w)| \, e^{{{\operatorname{Re}}}(z,w) - |w|^2}\,  d\lambda (w)
\\[1ex]
& \lesssim e^{- t_1 |z |^2 + r_2 |z|^{\frac{1}{s}}}  {\Vert F\Vert _{{\mathcal A}_{s, t_0,r_1}}}
\int_{{\mathbf C^{d}}} e^{r_0 |w|^{\frac{1}{s}}
+ t_0 |w|^2 - r_1 |w|^{\frac{1}{s}} + {{\operatorname{Re}}}(z,w) - |w|^2}\,  d\lambda (w)
\\[1ex]
& = e^{r_2 |z|^{\frac{1}{s}}}  {\Vert F\Vert _{{\mathcal A}_{s, t_0,r_1}}}
\int_{{\mathbf C^{d}}} e^{-(r_1-r_0) |w|^{\frac{1}{s}}  -(1- t_0) |w|^2 +
{{\operatorname{Re}}}(z,w) - t_1 |z |^2 }\,  d\lambda (w)
\\[1ex]
& = e^{r_2 |z|^{\frac{1}{s}}}  {\Vert F\Vert _{{\mathcal A}_{s, t_0,r_1}}}  
\int_{{\mathbf C^{d}}} e^{-(r_1-r_0)
|w|^{\frac{1}{s}}  - \left |  \sqrt{1- t_0} w - \frac{1}{2 \sqrt{1- t_0} } z \right|^2 }
\,  d\lambda (w)
\\[1ex]
& = e^{r_2 |z|^{\frac{1}{s}}}  {\Vert F\Vert _{{\mathcal A}_{s, t_0,r_1}}}  
\int_{{\mathbf C^{d}}} e^{-(r_1-r_0) \left| w + \frac{1}{2 (1- t_0) } z \right |
^{\frac{1}{s}}  - (1- t_0)\left|  w  \right|^2 }\,  d\lambda (w)
\\[1ex]
& \le
e^{(r_2 - c_1(r_1 - r_0)) |z|^{\frac{1}{s}}}  
{\Vert F\Vert _{{\mathcal A}_{s, t_0,r_1}}}  
\int_{{\mathbf C^{d}}} e^{ c_2 (r_1-r_0) | w |^{\frac{1}{s}} - (1- t_0)\left|  w  \right|^2 }\, d\lambda (w)
\\[1ex]
&\asymp 
{\Vert F\Vert _{{\mathcal A}_{s, t_0,r_1}}}  
e^{(r_2 - c_1(r_1 - r_0)) |z|^{\frac{1}{s}}}  \end{aligned}$$ for some constants $c_1, c_2>0$. By choosing $r_1$ sufficiently large we get $${\Vert {\operatorname{Op}}_{\mathfrak V}^{{\operatorname{aw}}}(a_0) F\Vert _{{\mathcal A}_{s, t_1,r_2}}}  
\lesssim
{\Vert F\Vert _{{\mathcal A}_{s, t_0,r_1}}}.$$ The estimates and imply ${\operatorname{Op}}_{\mathfrak V}^{{\operatorname{aw}}}(a_0)F \in A({\mathbf C^{d}})$.

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

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

\[cor:contBargmannGS\] Let $a_0\in L^\infty _{loc}({\mathbf C^{d}})$. If holds for some (every) $r_0 > 0$ then ${\operatorname{Op}}_{\mathfrak V}^{{\operatorname{aw}}}(a_0)$ is continuous on ${\mathcal A}_{0,s}({\mathbf C^{d}})$ (on ${\mathcal A}_{s}({\mathbf C^{d}})$).

With a technique similar to the proof of Proposition \[prop:contAW\] one shows the following result.

\[prop:contAW2\] Let $a_0\in L^\infty _{loc}({\mathbf C^{d}})$, $0 < t_0 < 1$ and suppose holds. Then the following is true:

1.  if holds for all $r_0 > 0$ then $$\label{Eq:RegAW1}
    {\operatorname{Op}}_{\mathfrak V}^{{\operatorname{aw}}}(a_0): {\mathcal A}_{0,(s, t_0)}'({\mathbf C^{d}}) \to {\mathcal A}_{0,(s, t_1)}({\mathbf C^{d}})$$ is continuous;

2.  if holds for some $r_0 > 0$ then $$\label{Eq:RegAW2}
    {\operatorname{Op}}_{\mathfrak V}^{{\operatorname{aw}}}(a_0): {\mathcal A}_{(s, t_0)}'({\mathbf C^{d}}) \to {\mathcal A}_{(s, t_1)}({\mathbf C^{d}})$$ is continuous.

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

\[cor:regBargmannGS\] Let $a_0\in L^\infty _{loc}({\mathbf C^{d}})$. Then the following is true:

1.  if holds for all $r_0 > 0$ then $${\operatorname{Op}}_{\mathfrak V}^{{\operatorname{aw}}}(a_0): {\mathcal A}_{0,s}'({\mathbf C^{d}}) \to {\mathcal A}_{0,s}({\mathbf C^{d}})$$ is continuous;

2.  if holds for some $r_0 > 0$ then $${\operatorname{Op}}_{\mathfrak V}^{{\operatorname{aw}}}(a_0): {\mathcal A}_{s}'({\mathbf C^{d}}) \to {\mathcal A}_{s}({\mathbf C^{d}})$$ is continuous.

Estimates of Wick symbols of anti-Wick operators with exponentially bounded symbols {#subsec3.3}
-----------------------------------------------------------------------------------

For anti-Wick operators in @Fo [Eq. (2.94)] we have the following result.

\[Thm:WickSymbolAntiWickOpFolland\] If $a_0\in L^\infty _{loc}({\mathbf C^{d}})$ satisfies $$\label{Eq:ExpQuadratic}
|a_0(w)| \lesssim e^{r|w|^2}, \quad w \in {\mathbf C^{d}}, \quad \text{for some} \quad r < 1,$$ then $a_0\in L_{0,A}({\mathbf C^{d}})$ and $'$ holds for some $a_0^{{\operatorname{aw}}}\in \wideparen A({\mathbf C^{2d}})$ with $$|a_0^{{\operatorname{aw}}}(z,w)|\lesssim e^{r_0 |z+w|^2-\operatorname{Re}(z,w)},
\qquad
r_0=4^{-1}(1-r)^{-1}.$$

The claim $a_0\in L_{0,A}({\mathbf C^{d}})$ is an immediate consequence of the assumption and the definition . The integral in $'$ can be estimated as $$\begin{gathered}
\left |
\int _{{\mathbf C^{d}}}a_0(w_1)e^{-(z-w_1,w-w_1)}\, d\lambda (w_1)
\right |
\\[1ex]
\lesssim
\int _{{\mathbf C^{d}}} e^{r|w_1|^2} \left | e^{-(z-w_1,w-w_1)} \right |
\, d\lambda (w_1)
\\[1ex]
=
e^{-\operatorname{Re}(z,w)}
\int _{{\mathbf C^{d}}} e^{-(1-r)|w_1|^2} e^{\operatorname{Re}(z+w,w_1)}
\, d\lambda (w_1)
\\[1ex]
=
e^{\frac 1{4(1-r)}|z+w|^2-\operatorname{Re}(z,w)}
\int _{{\mathbf C^{d}}} e^{-(1-r)|w_1-(z+w)/(2(1-r))|^2}
\, d\lambda (w_1)
\\[1ex]
\asymp e^{r_0|z+w|^2-\operatorname{Re}(z,w)} .\qedhere\end{gathered}$$

The condition on $a_0^{{\operatorname{aw}}}$ in Theorem \[Thm:WickSymbolAntiWickOpFolland\] implies that $a_0^{{\operatorname{aw}}}$ belongs to $\wideparen {{\mathcal A}} _{0,\frac 12}^\prime ({\mathbf C^{2d}})$ (see @Teofanov2). In particular it follows that ${\operatorname{Op}}_{\mathfrak V}^{{\operatorname{aw}}}(a_0)={\operatorname{Op}}_{\mathfrak V}(a_0^{{\operatorname{aw}}})$ is continuous from ${\mathcal A}_{0,\frac 12}({\mathbf C^{d}})$ to ${\mathcal A}_{0,\frac 12}'({\mathbf C^{d}})$ (cf. @Teofanov2 [Theorem 2.10] and Remark \[Rem:BargmannPilipovicSpaces\]).

The following result concerns exponentially moderate weight functions.

\[Thm:WickSymbolAntiWickOpSOmega\] Let $a_0\in L_{0,A}({\mathbf C^{d}})$, $a_0^{{\operatorname{aw}}} \in \wideparen A({\mathbf C^{2d}})$ is given by $'$ and $\omega \in {\mathscr P}_E({\mathbf C^{d}})$. If $$|a_0(w)|\lesssim  \omega (2w), \quad w \in {\mathbf C^{d}},$$ then $$|a_0^{{\operatorname{aw}}}(z,w)|\lesssim e^{\frac 14|z-w|^2} \omega (z+w), \quad z,w \in {\mathbf C^{d}}.$$

Let $r\ge 0$ be chosen such that $\omega (z+w)\lesssim \omega (z)e^{r|w|}$, $z,w\in {\mathbf C^{d}}$. From $'$ we get $$\begin{gathered}
|a_0^{{\operatorname{aw}}}(z,w)|
\lesssim
\int _{{\mathbf C^{d}}} \omega (2w_1)
e^{-\operatorname{Re}(z-w_1,w-w_1)}\, d\lambda (w_1)
\\[1ex]
=
e^{-\operatorname{Re}(z,w)}
\int _{{\mathbf C^{d}}} 
\omega (2w_1)
e^{\operatorname{Re}(z+w,w_1)-|w_1|^2}\, d\lambda (w_1)
\\[1ex]
=
e^{-\operatorname{Re}(z,w)+\frac 14|z+w|^2}
\int _{{\mathbf C^{d}}} 
\omega (2w_1)
e^{-|w_1-(z+w)/2|^2}\, d\lambda (w_1)
\\[1ex]
=
e^{\frac 14|z-w|^2}
\int _{{\mathbf C^{d}}} 
\omega (2w_1+z+w)
e^{-|w_1|^2}\, d\lambda (w_1)
\\[1ex]
\lesssim
e^{\frac 14|z-w|^2}
\omega (z+w)\int _{{\mathbf C^{d}}}  e^{2r|w_1|-|w_1|^2}\, d\lambda (w_1)
\asymp
e^{\frac 14|z-w|^2}
\omega (z+w). \qedhere\end{gathered}$$

The anti-Wick operators in Propositions \[prop:contAW\] and \[prop:contAW2\] can also be described as Wick operators with symbols that have smaller growth bounds than $\wideparen {\mathcal A}_s({\mathbf C^{2d}})$ and its dual. The following result extends Theorem \[Thm:WickSymbolAntiWickOpSOmega\] for weights of the form $e^{c |z|^{\frac1s}}$ with $c \in {\mathbf R}$ from $s \ge 1$ to $s  \ge \frac 12$.

\[Thm:WickSymbolAntiWickOpGS\] Let $s\ge \frac 12$ ($s> \frac 12$), $a_0\in L_{0,A}({\mathbf C^{d}})$ and let $a_0^{{\operatorname{aw}}}$ be given by $'$. Then the following is true:

1.  if holds for some (every) $r_0 > 0$ then $$\label{Eq:AWGSEst1}
    |a_0^{{\operatorname{aw}}} (z,w)|
    \lesssim
    e^{\frac 14|z-w|^2-r |z+w|^{\frac 1s}}$$ for some (every) $r>0$;

2.  if holds for every (some) $r_0>0$ then $$\label{Eq:AWGSEst2}
    |a_0^{{\operatorname{aw}}} (z,w)|
    \lesssim
    e^{\frac 14|z-w|^2+r |z+w|^{\frac 1s}}$$ 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 are much stronger than the estimates with $\sigma = s$. Corollary \[cor:contBargmannGS\] can thus be seen as a consequence of Theorems \[Thm:GevreySymbolsBargmTransfer\] and \[Thm:WickSymbolAntiWickOpGS\], and @CaTo [Definition 2.4, and Theorems 4.10 and 4.11].

\[Rem:WickSymbolAntiWickOpGS\] The estimates for $a_0^{{\operatorname{aw}}}$ in Theorem \[Thm:WickSymbolAntiWickOpGS\] may seem weak since the dominating factor $e^{\frac 14|z-w|^2}$ is present in and but absent in the original estimates and for $a_0$.

On the other hand, Wick symbols for operators with continuity involving the spaces ${\mathcal A}_s({\mathbf C^{d}})$ and ${\mathcal A}_s'({\mathbf C^{d}})$, as well as ${\mathcal A}_{0,s}({\mathbf C^{d}})$ and ${\mathcal A}_{0,s}'({\mathbf C^{d}})$, usually satisfies conditions of the form $$|a(z,w)| \lesssim e^{\frac 12|z-w|^2 \pm r_1|z+w|^{\frac 12}\pm |z-w|^{\frac 1s}}$$ in view of @Teofanov2 [Theorems 2.9 and 2.10], and Theorem \[Thm:GevreySymbolsBargmTransfer\]. 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 \[Thm:WickSymbolAntiWickOpGS\].

This factor has a large impact on functions on ${\mathbf R^{d}}$ that are transformed back by the inverse of the Bargmann transform. For instance, if ${\varepsilon}>0$, then the Bargmann image of any non-trivial Gelfand-Shilov space and its distribution space contain $${\{ \, F\in A({\mathbf C^{d}})\, ;\, |F(z)|\lesssim e^{(\frac 12-{\varepsilon})|z|^2}\, \} }$$ and are contained in $${\{ \, F\in A({\mathbf C^{d}})\, ;\, |F(z)|\lesssim e^{(\frac 12+{\varepsilon})|z|^2}\, \} }.$$ The same holds true for the Bargmann images of ${\mathscr S}({\mathbf R^{d}})$ and ${\mathscr S}'({\mathbf R^{d}})$.

Theorem \[Thm:WickSymbolAntiWickOpGS\] is a straight-forward consequence of the following two propositions, which give more details on the relationships between $r$ and $r_0$ in , , and .

\[Prop:WickSymbolAntiWickOpGS1\] Let $s\ge \frac 12$ and let $r_0,r\in (0,\infty )$ be such that $$\begin{aligned}
{5}
r_0&\in (0,\infty ) &
\quad &\text{and} & \quad
r&<\frac {r_0}{4(1+r_0)}, &
\quad
\quad &\text{when} & \quad
s &={\textstyle{\frac 12}},
\label{Eq:CondAntiWickOpGS1}
\intertext{and}
r_0&\in (0,\infty ) &
\quad &\text{and} & \quad
r &\le 2^{-\frac 1s}r_0, &
\quad
\quad &\text{when} & \quad
s&\in (\textstyle{\frac 12},\infty ),
\label{Eq:CondAntiWickOpGS2}\end{aligned}$$ with strict inequality in when $s<1$. If $a_0\in L^\infty _{loc}({\mathbf C^{d}})$ satisfies and $a_0^{{\operatorname{aw}}} \in \wideparen A({\mathbf C^{2d}})$ is given by $'$, then holds.

\[Prop:WickSymbolAntiWickOpGS2\] Let $s\ge \frac 12$ and $r_0,r\in (0,\infty )$ be such that $$\begin{aligned}
{5}
r_0&\in (0,1) &
\quad &\text{and} & \quad
r&>\frac {r_0}{4(1-r_0)}, &
\quad
\quad &\text{when} & \quad
s &={\textstyle{\frac 12}},
\tag*{(\ref{Eq:CondAntiWickOpGS1})$'$}
\intertext{and}
r_0&\in (0,\infty ) &
\quad &\text{and} & \quad
r &\ge 2^{-\frac 1s}r_0, &
\quad
\quad &\text{when} & \quad
s&\in (\textstyle{\frac 12},\infty ),
\tag*{(\ref{Eq:CondAntiWickOpGS2})$'$}\end{aligned}$$ with strict inequality in $'$ when $s<1$. If $a_0\in L^\infty _{loc}({\mathbf C^{d}})$ satisfies and $a_0^{{\operatorname{aw}}} \in \wideparen A({\mathbf C^{2d}})$ is given by $'$, then holds.

For the proofs of Propositions \[Prop:WickSymbolAntiWickOpGS1\] and \[Prop:WickSymbolAntiWickOpGS2\] we use the inequalities $$\begin{aligned}
{3}
|z|^\theta -|w|^\theta
&\le 
|z+w|^\theta
\le
|z|^\theta +|w|^\theta ,&
\qquad
\theta &\in (0,1], &\ z,w &\in {\mathbf C^{d}}
\label{Eq:TriangPowIneq1}
\\[1ex]
|z+w|^\theta
&\le
(1+{\varepsilon})|z|^\theta +(1+{\varepsilon}^{-1})|w|^\theta ,&
\qquad
\theta &\in [1,2], &\ z,w &\in {\mathbf C^{d}},
\label{Eq:TriangPowIneq2}
\intertext{and}
|z+w|^\theta
&\ge
(1-{\varepsilon})|z|^\theta +(1-{\varepsilon}^{-1})|w|^\theta ,&
\qquad
\theta &\in [1,2], &\ z,w &\in {\mathbf C^{d}},
\label{Eq:TriangPowIneq3}\end{aligned}$$ for every ${\varepsilon}>0$.

\[Proof of Proposition \[Prop:WickSymbolAntiWickOpGS1\]\] Suppose that $a_0$ satisfies for some $r_0 >0$. First we consider the case $s>\frac 12$. If $s < 1$ let ${\varepsilon}_1 >  0$ and ${\varepsilon}_2 = {\varepsilon}_1^{-1}$, and if $s \ge 1$ let ${\varepsilon}_1 = 0$ and ${\varepsilon}_2 = 2$, and let $c=2^{-\frac 1s}$. Then $'$, and give $$\begin{gathered}
\label{Eq:WickAWickComp1}
|a_0^{{\operatorname{aw}}}(z,w)|
\lesssim
\int _{{\mathbf C^{d}}} e^{-r_0 |w_1|^{\frac 1s}} 
e^{-\operatorname{Re}(z-w_1,w-w_1)}\, d\lambda (w_1)
\\[1ex]
=
e^{\frac 14|z+w|^2-\operatorname{Re}(z,w)}
\int _{{\mathbf C^{d}}} e^{-r_0 |w_1|^{\frac 1s} -|w_1-(z+w)/2|^2}
\, d\lambda (w_1)
\\[1ex]
=
e^{\frac 14|z-w|^2}
\int _{{\mathbf C^{d}}} e^{-r_0 |w_1+(z+w)/2|^{\frac 1s} -|w_1|^2}
\, d\lambda (w_1)
\\[1ex]
\le
e^{\frac 14|z-w|^2}e^{-cr_0(1-{\varepsilon}_1) |z+w|^{\frac 1s}}
\int _{{\mathbf C^{d}}} e^{-r_0(1-{\varepsilon}_2) |w_1|^{\frac 1s} -|w_1|^2}
\, d\lambda (w_1)
\\[1ex]
\asymp
e^{\frac 14|z-w|^2}e^{-cr_0(1-{\varepsilon}_1) |z+w|^{\frac 1s}}.\end{gathered}$$ If $s\ge 1$, then ${\varepsilon}_1=0$ and ${\varepsilon}_2=2$, and the result follows from . If instead $s<1$, then the result follows by choosing ${\varepsilon}_1>0$ small enough, and we have proved the result in the case $s>\frac 12$.

Next suppose that $s=\frac 12$. For ${\varepsilon}_1 >  0$ and ${\varepsilon}_2 = {\varepsilon}_1^{-1}$ gives $$|a_0^{{\operatorname{aw}}}(z,w)|
\lesssim
e^{\frac 14|z-w|^2}
e^{-\frac 14r_0(1-{\varepsilon}_1) |z+w|^2}
\int _{{\mathbf C^{d}}} e^{-(r_0(1-{\varepsilon}_2)+1) |w_1|^2}
\, d\lambda (w_1).$$ For any ${\varepsilon}_2<\frac {1+r_0}{r_0}$ it follows that the integral converges, and $$1-{\varepsilon}_1=1-{\varepsilon}_2^{-1}<(1+r_0)^{-1}.$$ By the assumptions there is $\delta > 0$ such that $$r = \frac{r_0(1-\delta)}{4 (1+r_0)}.$$ Since $$1-{\varepsilon}_1 \nearrow (1+r_0)^{-1}
\quad \text{as}\quad
{\varepsilon}_2 \nearrow \frac {1+r_0}{r_0}$$ we may pick $0 < {\varepsilon}_2<\frac {1+r_0}{r_0}$ such that $$\frac{1-\delta}{1+r_0} \le 1-{\varepsilon}_1$$ and the result follows in the case $s=\frac 12$.

\[Proof of Proposition \[Prop:WickSymbolAntiWickOpGS2\]\] First we consider the case when $s>\frac 12$. Suppose that $a_0$ satisfies for some $r_0 >0$, let ${\varepsilon}_1,{\varepsilon}_2\ge 0$ be such that ${\varepsilon}_1={\varepsilon}_2=0$ when $s\ge 1$ and ${\varepsilon}_1{\varepsilon}_2=1$ when $s<1$, and let $c=2^{-\frac 1s}$. Then $'$, and give $$\begin{gathered}
\label{Eq:WickAWickComp3}
|a_0^{{\operatorname{aw}}}(z,w)|
\lesssim
\int _{{\mathbf C^{d}}} e^{r_0 |w_1|^{\frac 1s}} 
e^{-\operatorname{Re}(z-w_1,w-w_1)}\, d\lambda (w_1)
\\[1ex]
=
e^{\frac 14|z+w|^2-\operatorname{Re}(z,w)}
\int _{{\mathbf C^{d}}} e^{r_0 |w_1|^{\frac 1s} -|w_1-(z+w)/2|^2}
\, d\lambda (w_1)
\\[1ex]
=
e^{\frac 14|z-w|^2}
\int _{{\mathbf C^{d}}} e^{r_0 |w_1+(z+w)/2|^{\frac 1s} -|w_1|^2}
\, d\lambda (w_1)
\\[1ex]
\le
e^{\frac 14|z-w|^2}e^{cr_0(1+{\varepsilon}_1) |z+w|^{\frac 1s}}
\int _{{\mathbf C^{d}}} e^{r_0(1+{\varepsilon}_2) |w_1|^{\frac 1s} -|w_1|^2}
\, d\lambda (w_1)
\\[1ex]
\asymp
e^{\frac 14|z-w|^2}e^{cr_0(1+{\varepsilon}_1) |z+w|^{\frac 1s}}.\end{gathered}$$ If $s\ge 1$, then ${\varepsilon}_1={\varepsilon}_2=0$, and the result follows from . If instead $s<1$, then the result follows by choosing ${\varepsilon}_1>0$ small enough, and the result follows in the case $s>\frac 12$.

Next suppose that $s=\frac 12$. Then gives $$|a_0^{{\operatorname{aw}}}(z,w)|
\lesssim
e^{\frac 14|z-w|^2}
e^{\frac 14r_0(1+{\varepsilon}_1) |z+w|^2}
\int _{{\mathbf C^{d}}} e^{r_0(1+{\varepsilon}_2) |w_1|^2 -|w_1|^2}
\, d\lambda (w_1).$$ For any ${\varepsilon}_2<\frac {1-r_0}{r_0}$ the integral converges, and $$1+{\varepsilon}_1=1+{\varepsilon}_2^{-1}>(1-r_0)^{-1}.$$ Since $$1+{\varepsilon}_1 \searrow (1-r_0)^{-1}
\quad \text{as}\quad
{\varepsilon}_2 \nearrow \frac {1-r_0}{r_0},$$ the result follows in the case $s=\frac 12$ by letting $r=\frac {r_0(1+{\varepsilon}_1)}4$.

A lower bound for Wick operators {#sec4}
================================

In this section we apply the asymptotic expansions in the previous section for Shubin-Wick operators to deduce a sharp G<span>Ã¥</span>rding inequality.

First we have the following result. We put $\wideparen {\mathcal A}_{{\operatorname{Sh}},\rho}({\mathbf C^{2d}})
=
\wideparen {\mathcal A}_{{\operatorname{Sh}},\rho}^{(\omega )}({\mathbf C^{2d}})$ when $\omega =1$.

\[Prop:ContApSpaces\] Let $\omega \in {\mathscr P}({\mathbf C^{d}})$, $p\in [1,\infty]$, $a\in \wideparen {\mathcal A}_{{\operatorname{Sh}},0}({\mathbf C^{2d}})$ and $a_0\in L^\infty ({\mathbf C^{d}})$. Then ${\operatorname{Op}}_{\mathfrak V}(a)$ and ${\operatorname{Op}}_{\mathfrak V}^{{\operatorname{aw}}}(a_0)$ are both continuous on $A^p_{(\omega )}({\mathbf C^{d}})$.

The claimed continuity of ${\operatorname{Op}}_{\mathfrak V}(a)$ is a straight-forward consequence of @Teofanov2 [Theorem 3.3], in combination with Proposition \[prop:symbchar1\] and the relationship $ K(z,w)=a(z,w) e^{(z,w)} $ between the kernel and symbol of a Wick operator (cf. ). In order to be self-contained we include an alternative and shorter proof.

Let $F\in A^p_{(\omega )}({\mathbf C^{d}})$, $G(z)
=
e^{-\frac 12|z|^2}|F(z)\omega (\sqrt 2\overline z)|$, $$\begin{aligned}
H_1(z) & = e^{-\frac 12|z|^2}|{\operatorname{Op}}_{\mathfrak V}(a)F(z)\omega (\sqrt 2\overline z)|
\quad \text{and}\quad \\
H_2(z) & = e^{-\frac 12|z|^2}|{\operatorname{Op}}_{\mathfrak V}^{{\operatorname{aw}}}(a_0)F(z)\omega (\sqrt 2\overline z)|.\end{aligned}$$ We have $$\omega (\sqrt 2\overline z) \lesssim \omega (\sqrt 2\overline w){\langle z-w\rangle}^{N_0}$$ for some $N_0\ge 0$. By Theorem \[Thm:ShubinAnalChar\] and we get $$\begin{gathered}
H_1(z) \lesssim e^{-\frac 12|z|^2}\int _{{\mathbf C^{d}}} e^{\frac 12|z-w|^2}
{\langle z-w\rangle}^{-N}|F(w)\omega (\sqrt 2\overline z)|
e^{{{\operatorname{Re}}}(z,w)-|w|^2}\, d\lambda (w)
\\[1ex]
= ({\langle {\, \cdot \, }\rangle} ^{N_0-N}*G)(z),\end{gathered}$$ for every $N\ge 0$. By choosing $N>2d+N_0$ and using Young's inequality we get ${\Vert H_1\Vert _{L^p}}\lesssim {\Vert G\Vert _{L^p}}$ which means ${\Vert {\operatorname{Op}}_{\mathfrak V}(a)F\Vert _{A^p_{(\omega )}}}\lesssim {\Vert F\Vert _{A^p_{(\omega )}}}$, and the asserted continuity for ${\operatorname{Op}}_{\mathfrak V}(a)$ follows.

In the same way we get $$\begin{gathered}
H_2(z) \lesssim {\Vert a_0\Vert _{L^\infty}} e^{-\frac 12|z|^2}
\int _{{\mathbf C^{d}}} |F(w)\omega (\sqrt 2\overline w)|{\langle z-w\rangle}^{N_0}
e^{{{\operatorname{Re}}}(z,w)-|w|^2}\, d\lambda (w)
\\[1ex]
\asymp (({\langle {\, \cdot \, }\rangle} ^{N_0}e^{-\frac 12|{\, \cdot \, }|^2})*G)(z),\end{gathered}$$ and another application of Young's inequality shows that ${\Vert H_2\Vert _{L^p_{(\omega )}}}\lesssim {\Vert G\Vert _{L^p_{(\omega )}}}$ that is ${\Vert {\operatorname{Op}}_{\mathfrak V}^{{\operatorname{aw}}}(a_0)F\Vert _{A^p_{(\omega )}}}\lesssim {\Vert F\Vert _{A^p_{(\omega )}}}$.

We have finally a version of the sharp G<span>Ã¥</span>rding inequality.

\[Thm:ShGarding\] Let $\rho >0$, $\omega (z) = {\langle z\rangle} ^{2\rho}$ and let $a\in \wideparen {\mathcal A}_{{\operatorname{Sh}},\rho}^{(\omega )}({\mathbf C^{2d}})$ be such that $a(w,w)\ge -C_0$ for all $w \in {\mathbf C^{d}}$, for some constant $C_0\ge 0$. Then $$\begin{aligned}
{2}
{{\operatorname{Re}}}\big ( ({\operatorname{Op}}_{\mathfrak V}(a)F,F)_{A^2} \big )
&\ge
-C{\Vert F\Vert _{A^2}}^2, &
\qquad
F &\in {\mathcal A}_{{\mathscr S}}({\mathbf C^{d}})
\label{Eq:ShGarding1}
\intertext{and}
\big | \operatorname{Im} \big ( ({\operatorname{Op}}_{\mathfrak V}(a)F,F)_{A^2} \big ) \big |
&\le
C{\Vert F\Vert _{A^2}}^2, &
\qquad
F &\in {\mathcal A}_{{\mathscr S}}({\mathbf C^{d}})
\label{Eq:ShGarding2}\end{aligned}$$ for some constant $C\ge 0$.

Let $b_0(w)=a(w,w)$. Then ${\operatorname{Op}}_{\mathfrak V}(a) = {\operatorname{Op}}_{\mathfrak V}^{{\operatorname{aw}}}(b_0)+{\operatorname{Op}}_{\mathfrak V}(a_1)$ for some $a_1\in \wideparen {\mathcal A}_{{\operatorname{Sh}},\rho}({\mathbf C^{2d}})
\subseteq \wideparen {\mathcal A}_{{\operatorname{Sh}},0}({\mathbf C^{2d}})$, in view of Proposition \[Prop:ShubinWickExpEst\]. Since $\Pi_A F = F$ for $F \in A^2({\mathbf C^{d}})$ (cf. ), the assumption $b_0\ge - C_0$ implies $({\operatorname{Op}}_{\mathfrak V}^{{\operatorname{aw}}}(b_0)F,F)_{A^2}\ge - C_0
\| F \|_{A^2}^2 $ for every $F\in {\mathcal A}_{{\mathscr S}}({\mathbf C^{d}})$. The operator ${\operatorname{Op}}_{\mathfrak V}(a_1)$ is continuous on $A^2({\mathbf C^{d}})$ in view of Proposition \[Prop:ContApSpaces\]. A combination of these facts gives the result.

Ellipticity and hypoellipticity for Shubin and Wick operators {#sec5}
=============================================================

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 ${\operatorname{Sh}}_\rho ^{(\omega )}({\mathbf R^{2d}})$ bijectively into the sets of hypoelliptic symbols and weakly elliptic Wick symbols in $\wideparen {\mathcal A}_{{\operatorname{Sh}},\rho}^{(\omega )}({\mathbf C^{2d}})$, respectively. Then we explain some consequences for polynomial symbols.

Transition of weakly elliptic symbols {#subsec4.1}
-------------------------------------

For symbols in $\wideparen {\mathcal A}_{{\operatorname{Sh}},\rho} ^{(\omega )}({\mathbf C^{2d}})$ we define ellipticity and weak ellipticity as follows.

\[Def:WickShubinEllipticity\] Let $\rho >0$, $\omega \in {\mathscr P}_{{\operatorname{Sh}},\rho}({\mathbf C^{d}})$ and $a\in \wideparen {\mathcal A}_{{\operatorname{Sh}},\rho} ^{(\omega)}({\mathbf C^{2d}})$. Then $a$ is called *weakly elliptic* of order $\rho _0\ge 0$, or *$\rho _0$-weakly elliptic*, if for some $R > 0$ $$|a(z,z)|\gtrsim {\langle z\rangle}^{-\rho _0}\omega (\sqrt 2\overline z), \quad |z| \ge R.$$ If $a$ is weakly elliptic of order $0$ then $a$ is called *elliptic*.

\[Thm:ElliptEquiv\] Let $\omega \in {\mathscr P}({\mathbf R^{2d}})\simeq {\mathscr P}({\mathbf C^{d}})$, $\rho >0$ and ${\mathfrak a}\in {\operatorname{Sh}}_\rho ^{(\omega )}({\mathbf R^{2d}})$. Then the following is true:

1.  if $z=x+i\xi$, $x,\xi \in {\mathbf R^{d}}$, then $$\label{Eq:RelAssignDiag}
    | \mathsf S_{\mathfrak V} {\mathfrak a}(z ,z) -{\mathfrak a}(\sqrt 2\, x,-\sqrt 2\, \xi )|
    \lesssim
    \omega (\sqrt 2\, \overline z)
    {\langle z\rangle}^{-2\rho}\text ;$$

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

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

\[Cor:ElliptEquiv\] Let ${\mathfrak a}$ be as in Theorem \[Thm:ElliptEquiv\]. Then the following is true:

1.  if $\rho _0\in [0,2\rho )$, then ${\mathfrak a}\in {\operatorname{Sh}}_\rho ^{(\omega )}({\mathbf R^{2d}})$ is weakly elliptic of order $\rho _0$, if and only if $\mathsf S_{\mathfrak V}{\mathfrak a}\in \wideparen A _{{\operatorname{Sh}},\rho}^{(\omega )}({\mathbf C^{2d}})$ is weakly elliptic of order $\rho _0$;

2.  ${\mathfrak a}\in {\operatorname{Sh}}_\rho ^{(\omega )}({\mathbf R^{2d}})$ is elliptic if and only if $\mathsf S_{\mathfrak V}{\mathfrak a}\in \wideparen A _{{\operatorname{Sh}},\rho}^{(\omega )}({\mathbf C^{2d}})$ is elliptic.

For the proof of Theorem \[Thm:ElliptEquiv\] we need the following proposition, related to Propositions \[Prop:WickToAntiWick\] and \[Prop:AsymptoticExpansion1\].

\[Prop:WickShubinTaylorExp\] Let $N\ge 0$ be an integer, $\rho \ge 0$, $\omega \in {\mathscr P}_{{\operatorname{Sh}},\rho}({\mathbf R^{2d}})\simeq
{\mathscr P}_{{\operatorname{Sh}},\rho}({\mathbf C^{d}})$, $\omega _k(x,\xi )=\omega (x,\xi ){\langle (x,\xi )\rangle}^{-2\rho k}$ and ${\mathfrak a}\in {\operatorname{Sh}}_\rho ^{(\omega )}({\mathbf R^{2d}})$. Then for some ${\mathfrak c}_N\in {\operatorname{Sh}}_\rho ^{(\omega _{N+1})}({\mathbf R^{2d}})$ and constants $\{ c_{\alpha} \}_{|\alpha| \le 2N}$ with $c_{0}=1$, it holds $$\label{Eq:WickShubinTaylorExp}
\mathsf S_{\mathfrak V}{\mathfrak a}(2^{-\frac 12}z ,2^{-\frac 12}z)
=
\sum _{k=0}^N{\mathfrak a}_k(x,-\xi ) + {\mathfrak c}_N(x,-\xi ),
\qquad
{\mathfrak a}_k = \sum _{|\alpha |=2k} c_{\alpha} \partial ^\alpha {\mathfrak a}.$$

Let $\psi$ be as in Proposition \[Prop:SBaTaRel\]. If we put $z=w$, then and Taylor's formula give $$\begin{gathered}
\label{Eq:TaylorNonWick}
(2\pi )^{d} \mathsf S_{\mathfrak V}{\mathfrak a}(2^{-\frac 12}z ,2^{-\frac 12}z)
=
(2\pi )^{\frac {3d}2} {\mathcal{T}}_\psi {\mathfrak a}(x,-\xi ,0,0)
\\[1ex]
=
2^d\iint _{{\mathbf R^{2d}}} {\mathfrak a}(t+x,\tau -\xi )e^{-(|t|^2+|\tau |^2)}\, dtd\tau
=
\sum _{k=0}^{2N+1} {\mathfrak b}_k(x,-\xi ) +{\mathfrak c}(x,-\xi )\end{gathered}$$ where $$\begin{aligned}
{\mathfrak b}_k(x,\xi ) &= \frac {2^d}{k!}\iint _{{\mathbf R^{2d}}}
{\langle {\mathfrak a}^{(k)}(x,\xi);(t,\tau ),\dots ,(t,\tau )\rangle}
e^{-(|t|^2+|\tau |^2)}\, dtd\tau 
\intertext{and}
{\mathfrak c}(x,\xi )&= 
\frac 1{(2N+1)!}\int _0^1 (1-\theta )^{2N+1} {\mathfrak c}_\theta (x,\xi )\, d\theta ,
\intertext{with}
{\mathfrak c}_\theta (x,\xi ) &= 2^d\iint _{{\mathbf R^{2d}}}
{\langle {\mathfrak a}^{(2N+2)}(x+\theta t,\xi +\theta \tau  );(t,\tau ),\dots ,(t,\tau )\rangle}
e^{-(|t|^2+|\tau |^2)}\, dtd\tau .\end{aligned}$$

If $k$ is odd, then $$(t,\tau )\mapsto {\langle {\mathfrak a}^{(k)}(x,\xi);(t,\tau ),\dots ,(t,\tau )\rangle}
e^{-(|t|^2+|\tau |^2)}$$ is odd which implies that the integral is zero. Hence ${\mathfrak b}_k(x,\xi )=0$ when $k$ is odd. For $k=0$ we observe that the integral for ${\mathfrak b}_0$ becomes $$2^d\iint _{{\mathbf R^{2d}}} e^{-(|t|^2+|\tau |^2)}\, dtd\tau
=
(2\pi )^d,$$ and it follows from these relations that $$(2\pi )^{-d}\sum _{k=0}^{2N+1}{\mathfrak b}_k = \sum _{k=0}^N{\mathfrak a}_k,$$ with ${\mathfrak a}_k$ as in and $c_{0}=1$. Hence the result follows if we prove that the last term in satisfies ${\mathfrak c}_N \in {\operatorname{Sh}}_\rho ^{(\omega _{N+1})}({\mathbf R^{2d}})$.

For $\theta \in [0,1]$ and $\alpha \in {\mathbf N^{2d}}$ we have $$\begin{gathered}
|\partial ^\alpha {\mathfrak c}_\theta (x,\xi )|
\lesssim
\iint _{{\mathbf R^{2d}}} |\partial^\alpha {\mathfrak a}^{(2N+2)}(x+\theta t,\xi +\theta \tau  )|
{\langle (t,\tau )\rangle}^{2N+2} e^{-(|t|^2+|\tau |^2)}\, dtd\tau 
\\[1ex]
\lesssim
\iint _{{\mathbf R^{2d}}} \omega (x+\theta t,\xi +\theta \tau  )
{\langle (x+\theta t,\xi +\theta \tau  )\rangle}^{-(2N+2+|\alpha |)\rho}
{\langle (t,\tau )\rangle}^{2N+2}
e^{-(|t|^2+|\tau |^2)}\, dtd\tau 
\\[1ex]
\lesssim
\omega (x,\xi)
{\langle (x,\xi )\rangle}^{-(2N+2+|\alpha |)\rho}
\iint _{{\mathbf R^{2d}}}
{\langle (t,\tau )\rangle}^{N_0}
e^{-(|t|^2+|\tau |^2)}\, dtd\tau 
\\[1ex]
\asymp
\omega (x,\xi)
{\langle (x,\xi )\rangle}^{-(2N+2+|\alpha |)\rho}\end{gathered}$$ for some $N_0>0$. In the last inequality we have used the fact that $\omega$ is polynomially moderate.

This implies $$|\partial ^\alpha {\mathfrak c}(x,\xi )|\lesssim \int _0^1 |\partial ^\alpha {\mathfrak c}_\theta (x,\xi )|
\, d\theta
\lesssim
\omega (x,\xi)
{\langle (x,\xi )\rangle}^{-(2N+2+|\alpha |)\rho},$$ which shows that ${\mathfrak c}, {\mathfrak c}_N \in {\operatorname{Sh}}_\rho ^{(\omega _{N+1})}({\mathbf R^{2d}})$.

\[Proof of Theorem \[Thm:ElliptEquiv\]\] Let $\psi$ be as in Proposition \[Prop:SBaTaRel\] and $N=0$ in Proposition \[Prop:WickShubinTaylorExp\]. Then $$\tag*{(\ref{Eq:RelAssignDiag})$'$}
| \mathsf S_{\mathfrak V}{\mathfrak a}(2^{-\frac 12}z ,2^{-\frac 12}z) -{\mathfrak a}(x,-\xi )|
\lesssim
\omega (x,-\xi)
{\langle (x,-\xi )\rangle}^{-2\rho},$$ and (1) follows.

Suppose $\rho _0\in [0,2\rho)$. Then it follows from the latter inequality that $$| \mathsf S_{\mathfrak V}{\mathfrak a}(z ,z)|
\gtrsim {\langle z\rangle}^{-\rho _0}\omega (\sqrt 2\, \overline z),
\qquad |z|\ge R$$ for some $R>0$, if and only if $$|{\mathfrak a}(x,\xi )|\gtrsim {\langle (x,\xi )\rangle}^{-\rho _0}\omega (x,\xi ), \qquad |z|\ge R$$ for some $R>0$, and the asserted equivalence in (2) follows.

Shubin hypoellipticity in a Wick setting
----------------------------------------

\[Def:ShubinWickHypoelliptic\] Let $\rho >0$, $\rho _0\ge 0$, $\omega \in {\mathscr P}_{{\operatorname{Sh}},\rho}({\mathbf C^{d}})$ and $a\in \wideparen {\mathcal A}_{{\operatorname{Sh}},\rho}^{(\omega )}({\mathbf C^{2d}})$. Then $a$ is called *hypoelliptic* (in the *Shubin-Wick sense* in $\wideparen {\mathcal A}_{{\operatorname{Sh}},\rho}^{(\omega )}({\mathbf C^{2d}})$) of order $\rho _0$, if there is an $R>0$ such that for every $\alpha ,\beta \in {\mathbf N^{d}}$, it holds $$\begin{aligned}
{2}
|\partial _z^\alpha \overline \partial _w^\beta  a(z,z)|
&\lesssim
|a(z,z)|{\langle z\rangle}^{-\rho |\alpha +\beta |}, &
\qquad |z| &\ge R \text .
\label{Eq:ShubinWickHypoelliptic1}
\intertext{and}
|a(z,z)| &\gtrsim \omega _0(\sqrt 2 \overline z)
{\langle z\rangle}^{-\rho _0}, &
\qquad |z| &\ge R.
\label{Eq:ShubinWickHypoelliptic2}\end{aligned}$$

According to Definition \[Def:Hypoelliptic\], if $\omega$, $\rho$ and $\rho _0$ are as in the definition, then ${\mathfrak a}\in {\operatorname{Sh}}_\rho ^{(\omega )}({\mathbf R^{2d}})$ is hypoelliptic of order $\rho _0$ means that there is an $R>0$ such that for every $\alpha  \in {\mathbf N^{2d}}$, it holds $$\begin{aligned}
{2}
|\partial ^\alpha {\mathfrak a}(x,\xi )|
&\lesssim
|{\mathfrak a}(x,\xi )|{\langle (x,\xi )\rangle}^{-\rho |\alpha |}, &
\qquad |(x,\xi )| &\ge R \text .
\label{Eq:ShubinWickHypoellipticReal1}
\intertext{and}
|{\mathfrak a}(x,\xi )| &\gtrsim \omega (x,\xi )
{\langle (x,\xi )\rangle}^{-\rho _0}, &
\qquad |(x,\xi )| &\ge R.
\label{Eq:ShubinWickHypoellipticReal2}\end{aligned}$$

Similar to Theorem \[Thm:ElliptEquiv\] we have the following.

\[Thm:HypoelliptShubinEquiv\] Let $\rho >0$, $\rho _0\ge 0$, $\omega \in {\mathscr P}_{{\operatorname{Sh}},\rho}({\mathbf R^{2d}})\simeq
{\mathscr P}_{{\operatorname{Sh}},\rho}({\mathbf C^{d}})$, ${\mathfrak a}\in {\operatorname{Sh}}_\rho ^{(\omega )}({\mathbf R^{2d}})$ and $a=\mathsf S_{\mathfrak V}{\mathfrak a}$. Then ${\mathfrak a}$ is hypoelliptic of order $\rho _0$ in ${\operatorname{Sh}}_\rho ^{(\omega )}({\mathbf R^{2d}})$, if and only if $a$ is *hypoelliptic* of order $\rho _0$ in $\wideparen {\mathcal A}_{{\operatorname{Sh}},\rho}^{(\omega )}({\mathbf C^{2d}})$.

Suppose that ${\mathfrak a}\in {\operatorname{Sh}}_\rho ^{(\omega )}({\mathbf R^{2d}})$ 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 and are fulfilled. Then Proposition \[Prop:WickShubinTaylorExp\] gives for $z=x+i\xi$ with $|z|\ge R$ where $R > 0$ is sufficiently large $$\begin{gathered}
|a(2^{-\frac 12}z,2^{-\frac 12}z)|
\gtrsim
|{\mathfrak a}(x,-\xi )|-
\sum _{k=1}^N\sum _{|\alpha |=2k}
(|\partial ^\alpha {\mathfrak a}(x,-\xi )| +
|{\mathfrak c}(x,-\xi )| )
\\[1ex]
\gtrsim
|{\mathfrak a}(x,-\xi )| -|{\mathfrak a}(x,-\xi )|{\langle (x,-\xi )\rangle}^{-2\rho}-
\omega (x,-\xi ){\langle (x,-\xi )\rangle}^{-\rho (2N+2)}
\\[1ex]
\gtrsim
|{\mathfrak a}(x,-\xi )| -|{\mathfrak a}(x,-\xi )|{\langle (x,-\xi )\rangle}^{-2\rho}
\\[1ex]
\gtrsim 
|{\mathfrak a}(x,-\xi )| \gtrsim \omega (x,-\xi ){\langle (x,-\xi )\rangle}^{-\rho _0},\end{gathered}$$ and follows. In particular it follows from the previous estimates that $$\label{Eq:IneqHypoShubinHypoWick}
|a(2^{-\frac 12}z,2^{-\frac 12}z)|
\gtrsim
|{\mathfrak a}(x,-\xi )|,
\qquad |z|\ge R.$$

For fixed $\alpha ,\beta \in {\mathbf N^{d}}$, let $\Omega _k$ be the set of all $(\gamma ,\delta )\in {\mathbf N^{2d}}\times {\mathbf N^{2d}}$ such that $|\gamma |=2k$ and $|\delta |=|\alpha +\beta |$. By Proposition \[Prop:WickShubinTaylorExp\] and we have for some $R$ large enough and $|z|\ge R$, $$\begin{gathered}
|(\partial _z^\alpha \overline \partial _w^\beta a)(2^{-\frac 12}z,2^{-\frac 12}z)|
\lesssim
\sum _{k=0}^N\sum _{(\gamma ,\delta )\in \Omega _k}
(|\partial ^{\gamma +\delta}{\mathfrak a}(x,-\xi )| +
|\partial ^\delta {\mathfrak c}(x,-\xi )| )
\\[1ex]
\lesssim
\sum _{k=0}^N\sum _{(\gamma ,\delta )\in \Omega _k}
\big ( |{\mathfrak a}(x,-\xi )|{\langle (x,-\xi )\rangle}^{-\rho (2k+|\alpha +\beta |)} +
\omega (x,-\xi ){\langle (x,-\xi )\rangle}^{-\rho (2N+|\alpha +\beta |)} \big )
\\[1ex]
\asymp
|{\mathfrak a}(x,-\xi )|{\langle (x,-\xi )\rangle}^{-\rho |\alpha +\beta |} +
\omega (x,-\xi ){\langle (x,-\xi )\rangle}^{-\rho (2N+|\alpha +\beta |)}
\\[1ex]
\lesssim
|{\mathfrak a}(x,-\xi )|{\langle (x,-\xi )\rangle}^{-\rho |\alpha +\beta |} +
|{\mathfrak a}(x,-\xi )|{\langle (x,-\xi )\rangle}^{\rho _0-\rho (2N+|\alpha +\beta |)}
\\[1ex]
\asymp
|{\mathfrak a}(x,-\xi )|{\langle (x,-\xi )\rangle}^{-\rho |\alpha +\beta |}
\lesssim
|a(2^{-\frac 12}z,2^{-\frac 12}z)|{\langle (x,-\xi )\rangle}^{-\rho |\alpha +\beta |},\end{gathered}$$ which implies that holds.

This shows that $a$ is hypoelliptic of order $\rho _0$ in $\wideparen {\mathcal A}_{{\operatorname{Sh}},\rho}^{(\omega )}({\mathbf C^{2d}})$ when ${\mathfrak a}$ is hypoelliptic of order $\rho _0$ in ${\operatorname{Sh}}_\rho ^{(\omega )}({\mathbf R^{2d}})$.

Suppose instead that $a$ is hypoelliptic of order $\rho _0$ in $\wideparen {\mathcal A}_{{\operatorname{Sh}},\rho}^{(\omega )}({\mathbf C^{2d}})$. By using Proposition \[Prop:AsymptoticExpansion1\], and instead of Proposition \[Prop:WickShubinTaylorExp\], similar computations as in the first part of the proof shows that and hold for some $R>0$. This shows that ${\mathfrak a}$ is hypoelliptic of order $\rho _0$ in ${\operatorname{Sh}}_\rho ^{(\omega )}({\mathbf R^{2d}})$ when $a$ is hypoelliptic of order $\rho _0$ in $\wideparen {\mathcal A}_{{\operatorname{Sh}},\rho}^{(\omega )}({\mathbf C^{2d}})$, and the result follows.

Ellipticity in the case of polynomial symbols
---------------------------------------------

Next we discuss ellipticity for polynomial symbols, i.e. $$\begin{aligned}
{2}
{\mathfrak a}(x,\xi ) &= \sum _{|\alpha +\beta | \le N}c(\alpha ,\beta )
x^\alpha \xi ^\beta , &
\quad
x,\xi &\in {\mathbf R^{d}},
\label{Eq:RealPolSymbol}
\intertext{and}
a(z,w) &= \sum _{|\alpha +\beta | \le N}c(\alpha ,\beta )z^\alpha {\overline w}^\beta ,
&
\quad
z,w &\in {\mathbf C^{d}}.
\label{Eq:ComplexPolSymbol}
\intertext{The corresponding principal symbols are}
{\mathfrak a}_p (x,\xi ) &= \sum _{|\alpha +\beta | = N}c(\alpha ,\beta )x^\alpha \xi ^\beta , &
\quad
x,\xi &\in {\mathbf R^{d}},
\label{Eq:RealPrincipalPolSymbol}
\intertext{and}
a_p (z,w ) &= \sum _{|\alpha +\beta | = N}c(\alpha ,\beta )
z^\alpha {\overline w} ^\beta , &
\quad
z,w &\in {\mathbf C^{d}},
\label{Eq:ComplexPrincipalPolSymbol}\end{aligned}$$ respectively.

First we relate polynomials on ${\mathbf R^{2d}}$ to Shubin classes.

\[Prop:DiffOpShubin\] Let ${\mathfrak a}$ and ${\mathfrak a}_p$ be as in and for some $c(\alpha ,\beta )\in \mathbf C$, $\alpha ,\beta \in {\mathbf N^{d}}$ and $N\ge 0$, and let $\omega _{N}(x,\xi )={\langle (x,\xi )\rangle}^{N}$, $x,\xi \in {\mathbf R^{d}}$. Then the following is true:

1.  ${\mathfrak a}\in {\operatorname{Sh}}_1^{(\omega _{N})}({\mathbf R^{2d}})$;

2.  ${\mathfrak a}$ is elliptic with respect to $\omega _{N}$, if and only if ${\mathfrak a}_p(x,\xi )\neq 0$ when $(x,\xi )\neq 0$.

The result can be considered folklore. In order to be self-contained we present the arguments.

First we prove (1). Let $t=\max (|x_1|,\dots ,|x_d|,|\xi _1|,\dots ,|\xi _d|)$ when $x=(x_1,\dots ,x_d)\in {\mathbf R^{d}}$ and $\xi =(\xi _1,\dots ,\xi _d)\in {\mathbf R^{d}}$. Then $$|{\mathfrak a}(x,\xi )| \le  \sum _{|\alpha +\beta | \le N_0}|c(\alpha ,\beta )| t^{|\alpha +\beta |}
\lesssim  1+t^{N}
\le {\langle (x,\xi )\rangle}^{N},$$ which gives the desired bound for $|{\mathfrak a}(x,\xi)|$. Since the degree of a polynomial is lowered by at least one for every differentiation we get $$|\partial^\alpha {\mathfrak a}(x,\xi )| \lesssim {\langle (x,\xi )\rangle}^{N-|\alpha |}$$ for every $\alpha \in {\mathbf N^{2d}}$, which gives (1).

In order to prove (2) we let ${\mathfrak a}_p$ be as in . First suppose that ${\mathfrak a}_p(x,\xi )\neq 0$ when $(x,\xi )\neq (0,0)$, and let $g$ be the continuous function on ${\mathbf R^{2d}}\setminus 0$ given by $$g(x,\xi ) =\frac {|{\mathfrak a}_p(x,\xi )|}{|(x,\xi )|^{N}},\qquad (x,\xi )\neq (0,0).$$ Since $g$ is continuous and positive, and the sphere $$\mathbf S^{2d-1}={\{ \, (x,\xi )\in {\mathbf R^{2d}}\, ;\, |x|^2+|\xi |^2 = 1\, \} }$$ is compact, it follows that there are constants $c_1,c_2>0$ such that $$c_1\le g(x,\xi )\le c_2,\qquad (x,\xi )\in \mathbf S^{2d-1}.$$ By homogeneity it now follows $$c_1|(x,\xi )|^{N} \le |{\mathfrak a}_p(x,\xi )| \le c_2|(x,\xi )|^{N},\qquad x,\xi \in {\mathbf R^{d}}.$$

Hence, if $${\mathfrak b}(x,\xi )= {\mathfrak a}(x,\xi )-{\mathfrak a}_p(x,\xi )
= \sum _{|\alpha +\beta | \le N-1}c(\alpha ,\beta )x^\alpha \xi ^\beta ,$$ then the first part of the proof implies that for some constants $C>0$ and $R>0$ we have $$|{\mathfrak a}(x,\xi )| \ge |{\mathfrak a}_p(x,\xi )| -|{\mathfrak b}(x,\xi )| \ge c_1|(x,\xi )|^{N_0} - C{\langle (x,\xi )\rangle}^{N-1}
\gtrsim {\langle (x,\xi )\rangle}^{N}$$ when $|(x,\xi )|\ge R$. Hence ${\mathfrak a}$ is elliptic with respect to $\omega _{N_0}$.

Suppose instead ${\mathfrak a}_p(x_0,\xi _0)=0$ for some $(x_0,\xi _0)\neq (0,0)$. For any $(x,\xi )=(tx_0,t\xi _0)$ we have $$\begin{gathered}
|{\mathfrak a}(x,\xi )| \le |{\mathfrak a}_p(x,\xi )| +|{\mathfrak b}(x,\xi )|
=
|t^N{\mathfrak a}_p(x_0,\xi _0)| +|{\mathfrak b}(x,\xi )|
\\[1ex]
=
|{\mathfrak b}(x,\xi )| \lesssim {\langle (x,\xi )\rangle}^{N-1},\end{gathered}$$ giving that $|{\mathfrak a}(x,\xi )|\gtrsim {\langle (x,\xi )\rangle}^{N}$, $|(x,\xi )|\ge R$, cannot hold for any $R>0$.

By Theorems \[Thm:ElliptEquiv\], \[Thm:HypoelliptShubinEquiv\] and Proposition \[Prop:DiffOpShubin\] we get the following. The details are left for the reader.

\[Prop:DiffOpBargmannShubin\] Let $a$ and $a_p$ be as in and for some $c(\alpha ,\beta )\in \mathbf C$, $\alpha ,\beta \in {\mathbf N^{d}}$ and $N\ge 0$, and let $\omega _{N}(x,\xi )={\langle (x,\xi )\rangle}^{N}$, $x,\xi \in {\mathbf R^{d}}$. Then the following is true:

1.  $a\in {\mathcal A}_{{\operatorname{Sh}},1}^{(\omega _{N})}({\mathbf C^{2d}})$;

2.  $a$ is elliptic in ${\mathcal A}_{{\operatorname{Sh}},1}^{(\omega _{N})}({\mathbf C^{2d}})$ if and only if $a_p(z,z)\neq 0$ when $z\neq 0$.

\[Rem:Elliptic\] Let ${\mathfrak a}$, ${\mathfrak a}_p$, $a$ and $a_p$ be as in –. Then it follows from Propositions \[Prop:DiffOpShubin\] and Proposition \[Prop:DiffOpBargmannShubin\] that ${\mathfrak a}$ is elliptic, if and only if ${\mathfrak a}_p$ is elliptic, and that $a$ is elliptic, if and only if $a_p$ is elliptic.

We have now the following.

\[Thm:Ellipticity\] Let ${\mathfrak a}\in {\operatorname{Sh}}_1^{(\omega _{N})}({\mathbf R^{2d}})$ and ${\mathfrak a}_p$ be as in and for some $c(\alpha ,\beta )\in \mathbf C$, $\alpha ,\beta \in {\mathbf N^{d}}$ and $N\ge 0$. Then the following is true:

1.  the principal symbol $a_p(z,w)$ of $\mathsf S_{\mathfrak V}{\mathfrak a}$ is given by $$\label{Eq:ComplexPrincipalSymb}
    a_p(z,w) = 2^{-\frac N2}
    \sum _{|\alpha +\beta |= N}
    c(\alpha ,\beta )i^{|\beta |}
    (z+\overline w)^\alpha (z-\overline w)^\beta 
    \text ;$$

2.  ${\mathfrak a}$ is elliptic in ${\operatorname{Sh}}_1^{(\omega _{N})}({\mathbf R^{2d}})$ if and only if $a_p$ is elliptic in ${\mathcal A}_{{\operatorname{Sh}},1}^{(\omega _{N})}({\mathbf C^{2d}})$;

3.  ${\mathfrak a}_p(x,\xi )> 0$ for every $(x,\xi )\neq (0,0)$, if and only if $a_p(z,z)> 0$ for every $z\neq 0$.

Let $z=x+i\xi$, $x,\xi \in {\mathbf R^{d}}$, i. e. $x=\frac 12(z+\overline z)$ and $\xi =\frac 1{2i}(z-\overline z)$. By Theorem \[Thm:ElliptEquiv\] we get $$\label{Eq:PrincipalSymbolShubinWickIdent}
a_p(2^{-\frac 12}z,2^{-\frac 12}z) = {\mathfrak a}_p(x,-\xi ).$$ This implies $$\begin{gathered}
a_p(z,z) = 2^\frac N2\sum _{|\alpha +\beta | = N}c(\alpha ,\beta )x^\alpha (-\xi )^\beta
\\[1ex]
= 2^\frac N2\sum _{|\alpha +\beta | = N}c(\alpha ,\beta )2^{-|\alpha |}
(z+\overline z)^\alpha (2i)^{-|\beta |}(-(z-\overline z) )^\beta ,\end{gathered}$$ which gives $$\tag*{(\ref{Eq:ComplexPrincipalSymb})$'$}
a_p(z,z) = 2^{-\frac N2}
\sum _{|\alpha +\beta |= N}
c(\alpha ,\beta )i^{|\beta |}
(z+\overline z)^\alpha (z-\overline z)^\beta .$$ The formula now follows from $'$ and analytic continuation, using the fact that $a_p(z,w)$ is analytic in $z$ and conjugate analytic in $w$.

The assertion (2) follows by a combination of Corollary \[Cor:ElliptEquiv\], Propositions \[Prop:DiffOpShubin\] and \[Prop:DiffOpBargmannShubin\], and the assertion (3) is a direct consequence of .

A necessary condition for polynomially bounded Wick symbols {#sec6}
===========================================================

In @Fo [Section 2.7] Folland shows that polynomial symbols for pseudo-differential operators correspond to polynomial Wick and anti-Wick symbols. Thus partial differential operators with polynomial coefficients corresponds to polynomial Wick symbols.

Here we show that a Wick symbol that is polynomially bounded must be a polynomial. This gives a characterization of Wick symbols corresponding to polynomial symbols for pseudo-differential operators.

Cauchy's integral formula implies that an entire function which is polynomially bounded must be a polynomial:

\[Prop:PolynomialSymbol\] Let $F\in A({\mathbf C^{d}})$ have Maclaurin series $$F(z) = \sum _{\alpha \in {\mathbf N^{d}}} c(\alpha )e_\alpha (z),\quad z\in {\mathbf C^{d}}.$$ Suppose that for some $j\in \{ 1,\dots ,d\}$, $C>0$, $N \ge 0$, and an open neighbourhood $I\subseteq \mathbf C$ of the origin we have $$|F(z)|\le C {\langle z_j\rangle}^N,\quad z_j\in \mathbf C,$$ provided $z_k\in I$, $k\in \{ 1,\dots ,d\} \setminus \{ j\}$. Then $c(\alpha ) = 0$ when $\alpha _j > N$.

By interchanging the variables, we may assume that $j=d$. Let $R\ge 1$ and ${\varepsilon}>0$ be chosen such that $$D_{\varepsilon}\equiv {\{ \, z_0\in \mathbf C\, ;\, |z_0|\le {\varepsilon}\, \} } \subseteq I.$$ Take $\alpha \in {\mathbf N^{d}}$ such that $\alpha _d>N$, let $\beta =(\alpha _1+1,\dots ,\alpha _d+1)\in {\mathbf N^{d}}$ and $\gamma _{\varepsilon}\subseteq \mathbf C$ be the boundary circle of $D_{\varepsilon}$. Then Cauchy's integral formula gives $$\begin{gathered}
\frac {|c(\alpha )|}{\alpha !^{\frac 12}}
=
\left |
\frac { \partial^\alpha F (0) }{\alpha !}
\right |
= (2\pi )^{-d}
\left |
\idotsint _{\gamma_{\varepsilon}^{d-1}}
\left (
\int _{|z_d|=R}
\frac { F(z)}{z^\beta}\, dz_d
\right )
\, dz_1\cdots dz_{d-1}
\right | 
\\[1ex]
\le
(2\pi )^{-d}
\idotsint _{\gamma_{\varepsilon}^{d-1}}
\left (
\int _{|z_d|=R}
\frac {| F(z)|}{|z^\beta |}\, |dz_d |
\right )
\, |dz_1|\cdots |dz_{d-1}|
\\[1ex]
\lesssim
R^{-\alpha _d} {\langle R\rangle}^N{\varepsilon}^{-(\alpha _1+\dots +\alpha _{d-1})}
\to 0 \end{gathered}$$ as $R\to \infty$.

Let $a\in \wideparen A({\mathbf C^{2d}})$ and suppose $$\label{Eq:GlobPolEst}
|a(z,w)|\lesssim {\langle (z,w)\rangle} ^N$$ for some $N\ge 0$. Then $a$ is a polynomial in $z\in {\mathbf C^{d}}$ and $\overline w\in {\mathbf C^{d}}$ of degree at most $N$.

By Proposition \[Prop:PolynomialSymbol\] it follows that $a$ is a polynomial of degree at most $2dN$. We need to prove that the degree is at most $N$. In order to do this we may assume that $a$ has degree at least one.

For some integer $M\ge 1$ we have $$a(z,w) = a_M(z,w) +a_{M-1}(z,w),$$ where $$\begin{aligned}
a_M(z,w)
&=
\sum _{|\alpha +\beta |=M}c(\alpha ,\beta )z^\alpha \overline w^\beta
\intertext{is non-trivial and}
a_{M-1}(z,w)
&=
\sum _{|\alpha +\beta |\le M-1}c(\alpha ,\beta )z^\alpha \overline w^\beta .\end{aligned}$$ Since $a_M$ is non-trivial, there are $z_0,w_0\in {\mathbf C^{d}}$ such that $|z_0|^2+|w_0|^2=1$ and $|a_M(z_0,w_0)|=c_0\neq 0$. By homogeneity we get $$|a_M(tz_0,tw_0)| = c_0|t|^M,\qquad t \in {\mathbf R}.$$ In the same way we get $$|a_{M-1}(tz_0,tw_0)| \le C(1+|t|)^{M-1},\qquad t\in {\mathbf R}$$ for some constant $C$ which is independent of $t$.

Suppose contrary to the assumption that $M>N$. For $t\in {\mathbf R}$ with $|t|\ge 1$ we have $$\begin{gathered}
\left |
\frac {a(tz_0,tw_0)}{{\langle (tz_0,tw_0)\rangle}^N}
\right |
\gtrsim
|t|^{-N}\left (|a_M(tz_0,tw_0)| - |a_{M-1}(tz_0,tw_0)|    \right )
\\[1ex]
\ge
|t|^{-N}\left (c_0|t|^M - C(1+|t|)^{M-1} \right ) \to \infty
\quad \text{as}\quad
|t|\to \infty .\end{gathered}$$ This contradicts , and the hence our assumption that $M>N$ must be false.

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{'source': 'arXiv_out/2011'}