On the anti-Wick symbol as a Gelfand-Shilov generalized function

TL;DR

This paper demonstrates that the anti-Wick symbol of certain operators can be rigorously defined within the Gelfand-Shilov generalized function framework, expanding the understanding of symbol representations beyond tempered distributions.

Contribution

It proves that anti-Wick symbols can be characterized as Gelfand-Shilov generalized functions, even when they are not tempered distributions, using specific space embeddings.

Findings

Anti-Wick symbols can be defined as Gelfand-Shilov generalized functions.

The work establishes embeddings between Schwartz, Gelfand-Shilov, and Gevrey spaces.

Results extend the class of symbols representable in generalized function frameworks.

Abstract

The purpose of this article is to prove that the anti-Wick symbol of an operator mapping $ {\cal S}(\R^n)$ into ${\cal S}'(\R^n)$, which is generally not a tempered distribution, can still be defined as a Gelfand-Shilov generalized function. This result relies on test function spaces embeddings involving the Schwartz and Gelfand-Shilov spaces. An additional embedding concerning Schwartz and Gevrey spaces is also given.

Equations

$$K_A(x,y)={\int }_{{\mathbb{R}}^n}{e}^{2i\pi (x-y)\cdot \xi }{\sigma }^{Weyl}(A)\left(\frac{x+y}{2},\xi \right)d\xi ,$$

$${\sigma }^{Weyl}(A)(x,\xi )={\int }_{{\mathbb{R}}^n}{e}^{-2i\pi t\cdot \xi }{K}_A\left(x+\frac{t}{2},x-\frac{t}{2}\right)dt.$$

$$<Af,g>=(2\pi {)}^{-n}{\int }_{{\mathbb{R}}^{2n}}F(X)<f,{\Psi }_X><{\Psi }_X,g>dX$$

$${\Psi }_X={\tau }_X{\Psi }_0,{\Psi }_0(u)=2^{\frac{n}{4}}{e}^{-\pi |u{|}^2},{\tau }_{(x,\xi )}f(u)=f(u-x)e^{2i\pi \left(u-\frac{x}{2}\right)\cdot \xi }(X=(x,\xi )\in {\mathbb{R}}^{2n},u\in {\mathbb{R}}^n)$$

$${\sigma }^{Weyl}(A)=2^n{\sigma }^{AW}(A)\ast e^{-2\pi \Gamma }\text{where}\Gamma (X)=|X{|}^2$$

$${\sigma }^{Weyl}(A)=e^{\frac{1}{8\pi }\Delta }{\sigma }^{AW}(A)$$

$$<T(A),e^{\frac{1}{8\pi }\Delta }\phi >=<{\sigma }^{Weyl}(A),\phi >,\phi \in \mathcal{S}({\mathbb{R}}^n).$$

$${\sigma }^{Weyl}(A)=e^{\frac{1}{8\pi }\Delta }{T}_A.$$

$$|x^{\alpha }{\partial }_x^{\beta }\phi (x)|\le A^{|\alpha |+|\beta |}(\alpha !{)}^{\lambda }(\beta !{)}^{\mu }.$$

$$e^{\phi (x)}|u(x+iy)|\le K{e}^{\psi (y)},$$

$$\phi (x)=\frac{\lambda }{2}\sum _{j=1}^n{\left|\frac{x_j}{A}\right|}^{\frac{1}{\lambda }}\text{and}\psi (y)=2(1-\mu )\sum _{j=1}^n|A{y}_j{|}^{\frac{1}{1-\mu }},$$

$$\underset{k\ge 0}{\sup }\frac{x^k}{k!}\ge \frac{1}{2}{e}^{\frac{x}{2}}.$$

$$\frac{1}{2^{n\lambda }}{e}^{\phi (x)}\le \underset{\alpha }{\sup }\frac{|x^{\alpha }|}{A^{\alpha }(\alpha !{)}^{\lambda }}.$$

$${\left(\frac{x^k}{k!}\right)}^{\nu }\le \frac{1}{2^{k\nu }}{e}^{2\nu x}$$

$$\sum _{k=0}^{\infty }{\left(\frac{x^k}{k!}\right)}^{\nu }\le C{e}^{2\nu x}\text{with}C=\sum _{k=0}^{\infty }\frac{1}{2^{k\nu }}.$$

$$\sum _{\beta }\left|{\partial }_x^{\beta }u(x)\frac{(iy{)}^{\beta }}{\beta !}\right|\le C^n{e}^{\psi (y)}.$$

$${\int }_{{\mathbf{C}}^n}{e}^{-2\pi |\mathrm{Im}z{|}^2}(1+|\mathrm{Re}z|{)}^m|\phi (z)|dz<\infty .$$

$$e^{\psi (y)}=e^{2\pi |y{|}^2}g(y)$$

$$\hat{u}(\xi )={\int }_{{\mathbb{R}}^n}u(x)e^{-2i\pi x\cdot \xi }dx.$$

$$\hat{u}(\xi )={\int }_{{\mathbb{R}}^n}u(x)e^{-2i\pi x\cdot \xi }dx={\int }_{{\mathbb{R}}^n}u(x+iy)e^{-2i\pi (x+iy)\cdot \xi }dx,$$

$$e^{-2\pi (|y{|}^2+y\cdot \xi )}\hat{u}(\xi )={\int }_{{\mathbb{R}}^n}u(x+iy)e^{-2\pi (|y{|}^2+ix\cdot \xi )}dx.$$

$$2^{-\frac{n}{2}}{e}^{\frac{\pi }{2}|\xi {|}^2}\hat{u}(\xi )={\int }_{{\mathbb{C}}^n}u(x+iy)e^{-2\pi (|y{|}^2+ix\cdot \xi )}dxdy.$$

$$\kappa (\xi )=2^{\frac{n}{2}}{\int }_{{\mathbb{C}}^n}u(x+iy)e^{-2\pi (|y{|}^2+ix\cdot \xi )}dxdy.$$

$$e^{-\frac{\pi }{2}|\xi {|}^2}\kappa (\xi )=\hat{u}(\xi ).$$

$$\Phi (x)={\int }_{{\mathbb{R}}^n}\kappa (\xi )e^{2i\pi x\cdot \xi }d\xi$$

$$e^{\frac{1}{8\pi }\Delta }\Phi =u.$$

$$\left|\frac{d^m}{d{x}^m}{e}^{-\frac{x^2}{2}}\right|\le \sqrt{2}(2\pi )\frac{1}{4}\sqrt{m!}(m+1{)}^{\frac{1}{4}},$$

$$f_m(x)=\frac{d^m}{d{x}^m}{e}^{-\frac{x^2}{2}}=(-1{)}^m{e}^{-\frac{x^2}{2}}{H}_m(x)$$

$$|f_m(x){|}^2\le 2{\int }_{\mathbb{R}}|f_m(t)||f_m^{\prime }(t)|dt\le 2\Vert f_m\Vert \Vert f_{m+1}\Vert$$

$$\Vert f_m{\Vert }^2={\int }_{\mathbb{R}}{e}^{-x^2}|H_m(x){|}^{2}dx\le {\int }_{\mathbb{R}}{e}^{-\frac{x^2}{2}}|H_m(x){|}^{2}dx=\sqrt{2\pi }m!.$$

Full text

Keywords: Anti-Wick symbol, Gelfand-Shilov generalized functions, pseudodifferential calculus, Gevrey spaces, test function spaces embeddings.

MSC 2010: 47G30,46F05

1 Statement of the result.

For any operator $A$ mapping ${\cal S}(\mathbb{R}^{n})$ into ${\cal S}^{\prime}(\mathbb{R}^{n})$, it is well known that the Weyl symbol $\sigma^{Weyl}(A)$ and the distribution kernel $K_{A}$ are tempered distributions on $\mathbb{R}^{2n}$ satisfying (with the notations of [8]):

[TABLE]

or equivalently:

[TABLE]

(The above spaces ${\cal S}(\mathbb{R}^{n})$ and ${\cal S}^{\prime}(\mathbb{R}^{n})$ refer respectively to the standard Schwartz space and its topological dual).

The two identities (1.1) and (1.2) are naturally understood in the sense of tempered distributions. Recall that these equalities can be regarded as a basis of Weyl pseudodifferential calculus [7].

Besides, an operator $A$ can also be given under the form:

[TABLE]

where the $\Psi_{X}$ are the coherent states:

[TABLE]

and where $F$ is, for instance, a bounded continuous function on $\mathbb{R}^{2n}$, called anti-Wick symbol of $A$. This symbol can be denoted $\sigma^{AW}(A)$. For that purpose, see, e.g., [2, 3, 4, 8].

In this case, it is well known that:

[TABLE]

(where $\ast$ is the standard convolution product) or equivalently:

[TABLE]

(where $\Delta$ denotes the Laplacian operator).

Our aim here is to prove that it is always possible to define the anti-Wick symbol of any operator $A$ mapping ${\cal S}(\mathbb{R}^{n})$ into ${\cal S}^{\prime}(\mathbb{R}^{n})$. This symbol is not a tempered distribution, but a Gelfand-Shilov generalized function ([5, 6]).

First, one can a priori define a linear form $T(A)$ on the space $e^{\frac{1}{8\pi}\Delta}{\cal S}(\mathbb{R}^{n})$ by:

[TABLE]

One then has:

[TABLE]

As a consequence and by analogy with (1.5), one can consider that $T(A)$ is by definition the anti-Wick symbol of $A$.

Next, Theorem 1.1 below proves that $T(A)$ defined above is actually also a Gelfand-Shilov generalized function.

Before stating Theorem 1.1, let us recall here the definition of the space $S(\lambda,\mu)(\mathbb{R}^{n})$ of these test functions used by Gelfand and Shilov [5, 6]. See also, $e.g.$ [9, 10, 11] for applications and [13] for related spaces.

The space $S(\lambda,\mu)(\mathbb{R}^{n})$ refers to the space of functions $\varphi\in C^{\infty}(\mathbb{R}^{n})$ such that, there exists a constant $A>0$ satisfying for all multi-indices $\alpha$ and $\beta$, for all $x\in\mathbb{R}^{n}$:

[TABLE]

The following Theorem is proved in Section 2.

Theorem 1.1**.**

If $\lambda>0$ and $0<\mu<\frac{1}{2}$ then the space $S(\lambda,\mu)(\mathbb{R}^{n})$ is continuously embedded in the space $e^{\frac{1}{8\pi}\Delta}{\cal S}(\mathbb{R}^{n})$.

As a consequence, one obtains that the anti-Wick symbol $T(A)$ of any operator $A$ mapping ${\cal S}(\mathbb{R}^{n})$ into ${\cal S}^{\prime}(\mathbb{R}^{n})$, is well defined (by restriction) as a continuous linear form on $S(\lambda,\mu)(\mathbb{R}^{2n})$ for any $\lambda>0$ and any $\mu<\frac{1}{2}$. That is, $T(A)$ is a Gelfand-Shilov generalized function.

Let us also mention the following fact as a complementary result concerning anti-Wick symbols. In [1], one provides conditions written in terms of the action of an operator $A$ on coherent states to ensure that the anti-Wick symbol of $A$ is a bounded continuous function on $\mathbb{R}^{2n}$. Also note that this latter result is actually to be compared with Unterberger result [14] giving a similar necessary and sufficient condition in order that the Weyl symbol of $A$ is a $C^{\infty}$ function on $\mathbb{R}^{2n}$, being bounded together with all of its derivatives.

Theorem 1.1 is proved in Section 2 and a related test function spaces embedding concerning Gevrey spaces is given in Section 3.

2 Proof of Theorem 1.1.

The Proposition below follows from Proposition 6.1.8 of Nicola-Rodino [12] but we give a proof for the sake of the reader convenience.

Proposition 2.1**.**

Suppose that $\lambda>0$ and $0<\mu<1$. Then, any function in $S(\lambda,\mu)(\mathbb{R}^{n})$ extends to a holomorphic function $u$ on $\mathbb{C}^{n}$ satisfying for some constant $K>0$:

[TABLE]

with

[TABLE]

where the constant $A>0$ is the constant in (1.8).

Proof of Proposition 2.1. For any $x>0$, one has:

[TABLE]

If $\lambda>0$, this yields for all $x\in\mathbb{R}^{n}$:

[TABLE]

If $\nu>0$ and $x>0$, one has:

[TABLE]

implying

[TABLE]

Therefore, for all $u\in S(\lambda,\mu)(\mathbb{R}^{n})$ with $\mu<1$ and $\nu=1-\mu$:

[TABLE]

Thus inequality (2.1) with (2.2) is valid and the proof is then complete.

$\Box$

Next, we denote by $E(\mathbb{C}^{n})$ the space of holomorphic functions $\varphi$ on $\mathbb{C}^{n}$ satisfying for all $m\geq 0$:

[TABLE]

Proposition 2.2**.**

If $\lambda>0$ and $\mu<\frac{1}{2}$ then any element of $S(\lambda,\mu)(\mathbb{R}^{n})$ can be extended to a holomorphic function belonging to $E(\mathbb{C}^{n})$. This defines a continuous embedding of $S(\lambda,\mu)(\mathbb{R}^{n})$ into $E(\mathbb{C}^{n})$.

Proof of Proposition 2.2. One notices for the function $\psi$ defined in (2.2) that:

[TABLE]

where the function $g$ belongs to $L^{1}(\mathbb{R}^{n})$ if $\mu<\frac{1}{2}$. Besides, the function $e^{-\varphi(x)}$ is rapidly decreasing if $\lambda>0$. Then, Proposition 2.2 follows from Proposition 2.1.

$\Box$

The following Fourier transform is used in the sequel:

[TABLE]

Proposition 2.3**.**

For any $u\in E(\mathbb{C}^{n})$, there is a function $\Phi\in{\cal S}(\mathbb{R}^{n})$ such that $e^{\frac{1}{8\pi}\Delta}\Phi$ is equal to $u$ restricted to $\mathbb{R}^{n}$.

Proof of Proposition 2.3. Let $u$ be any function belonging to $E(\mathbb{C}^{n})$. Then, one has:

[TABLE]

for any $\xi\in\mathbb{R}^{n}$. This implies:

[TABLE]

Therefore, one deduces integrating with respect to $y$:

[TABLE]

Set:

[TABLE]

This function $\psi$ belongs to ${\cal S}(\mathbb{R}^{n})$ and verifies:

[TABLE]

Thus, one observes that the function $\Phi$ defined by:

[TABLE]

is indeed satisfying:

[TABLE]

$\Box$

Proof of Theorem 1.1. It is a direct consequence of Propositions 2.2 et 2.3 when replacing $n$ by $2n$.

$\Box$

3 An additional embedding.

We give here a supplementary embedding involving Schwartz and Gevrey spaces.

We begin with the following Lemma.

Lemma 3.1**.**

One has:

[TABLE]

for any $m\geq 0$ and all $x\in\mathbb{R}$:

Proof of Lemma 3.1. Set:

[TABLE]

where the $H_{m}$ are the Hermite functions.

One checks that:

[TABLE]

where the above norms are the $L^{2}(\mathbb{R})$ norms. Besides, one has:

[TABLE]

The proof of the Lemma thus follows.

$\Box$

The Gevrey type space $G^{s}$ considered here is defined as the space of smooth functions $f$ such that, there exist $K>0$ and $C>0$ satisfying for all multi-indices $\alpha\in\mathbb{N}^{n}$ and all $x\in\mathbb{R}^{n}$, $|\partial_{x}^{\alpha}f(x)|\leq KC^{|\alpha|}(|\alpha|!)^{s}$.

We then obtain the next result.

Proposition 3.2**.**

If the function $f\in{\cal S}(\mathbb{R}^{n})$ then the function $e^{\frac{1}{8\pi}\Delta}f$ belongs to the Gevrey space $G^{\frac{1}{2}}(\mathbb{R}^{n})$.

Proof of Proposition 3.2. Recall that:

[TABLE]

for any $f\in{\cal S}(\mathbb{R}^{n})$. According to Lemma 3.1 above, there exists $C>0$ satisfying:

[TABLE]

for any multi-indice $\alpha\in\mathbb{N}^{n}$ and all $x\in\mathbb{R}^{n}$.

This proves Proposition 3.2.

$\Box$