# Non-isotropic translation and modulation invariant Hilbert spaces

Joachim Toft  Department of Mathematics, Linnæus University, Växjö, Sweden
<EMAIL_ADDRESS>and  P. K. Ratnakumar  Harish-Chandra Research Institute
(HBNI), Chhatnag Road, Jhunsi, Allahabad, 211019, Uttarpradesh, India
<EMAIL_ADDRESS>
###### Abstract.

Let $\mathcal{H}$ be a Hilbert space of distributions on $\mathbf{R}^{d}$
which contains at least one non-zero element in
$\mathscr{D}^{\prime}(\mathbf{R}^{d})$. If there is a constant $C_{0}>0$ such
that

$\|e^{i\langle\,\cdot\,,\xi\rangle}f(\,\cdot\,-x)\|_{\mathcal{H}}\leq
C_{0}\|f\|_{\mathcal{H}},\qquad f\in\mathcal{H},\ x,\xi\in\mathbf{R}^{d},$

then we prove that $\mathcal{H}=L^{2}(\mathbf{R}^{d})$, with equivalent norms.

###### Key words and phrases:

modulation spaces, Feichtinger’s minimization principle

###### 1991 Mathematics Subject Classification:

46C15, 46C05, 42B35

##  0\. Introduction

In [15] it is proved that a suitable non-trivial Hilbert space
$\mathcal{H}\subseteq\mathscr{S}^{\prime}(\mathbf{R}^{d})$, which is norm
preserved under translations

$\displaystyle\big{(}\,y\mapsto f(y)\,\big{)}$
$\displaystyle\mapsto\big{(}\,y\mapsto f(y-x)\,\big{)}$ and modulations
$\displaystyle\big{(}\,y\mapsto f(y)\,\big{)}$
$\displaystyle\mapsto\big{(}\,y\mapsto e^{i\langle y,\xi\rangle}f(y)\,\big{)}$

is equal to $L^{2}(\mathbf{R}^{d})$. (See [9] or Section 1 for notations.) It
is here also proved that the norms between $\mathcal{H}$ and
$L^{2}(\mathbf{R}^{d})$ only differs by a multiplicative constant, i. e. for
some constant $C>0$ one has

$\|f\|_{\mathcal{H}}=C\|f\|_{L^{2}(\mathbf{R}^{d})},\qquad
f\in\mathcal{H}=L^{2}(\mathbf{R}^{d}).$ (0.1)

This property is analogous to Feichtinger’s minimization property, which shows
that the Feichtinger algebra $S_{0}(\mathbf{R}^{d})$, which is the same as the
modulation space $M^{1,1}(\mathbf{R}^{d})$, is the smallest non-trivial Banach
space of tempered distributions which is norm invariant under translations and
modulations. (See e. g. [7] for general facts about modulation spaces.)

The condition on non-triviality in [15] is simply that $\mathcal{H}$ should
contain at least one non-trivial element in $M^{1,1}(\mathbf{R}^{d})$. Hence
the main result in [15] can be formulated as follows.

######  Theorem 0.1.

Let $\mathcal{H}$ be a Hilbert space which is continuously embedded in
$\mathscr{S}^{\prime}(\mathbf{R}^{d})$, and norm preserved under translations
and modulations. If $\mathcal{H}$ contains a non-zero element in
$M^{1,1}(\mathbf{R}^{d})$, then $\mathcal{H}=L^{2}(\mathbf{R}^{d})$, and (0.1)
holds for some constant $C>0$ which is independent of
$f\in\mathcal{H}=L^{2}(\mathbf{R}^{d})$.

An alternative approach, using the Bargmann transform, to reach similar
properties is given by Bais, Pinlodi and Venku Naidu in [1]. In fact, by using
their result [1, Theorem 3.1] in combination with some well-known arguments in
the distribution theory, one obtains the following improvement of Theorem 0.1.

######  Theorem 0.2.

Let $\mathcal{H}$ be a Hilbert space which is continuously embedded in
$\mathscr{S}^{\prime}(\mathbf{R}^{d})$, and norm preserved under translations
and modulations. If $\mathcal{H}$ is non-trivial, then
$\mathcal{H}=L^{2}(\mathbf{R}^{d})$, and (0.1) holds for some constant $C>0$
which is independent of $f\in\mathcal{H}=L^{2}(\mathbf{R}^{d})$.

We observe that the condition

$|(f,\phi)|\leq C\|f\|_{\mathcal{H}},\qquad f\in\mathcal{H},\
\phi(x)=e^{-\frac{1}{2}|x|^{2}},$ (0.2)

for some constant $C>0$ in the hypothesis in [1, Theorem 3.1] is absent in
Theorem 0.2. Hence Theorem 0.2 is slightly more general than [1, Theorem 3.1].

Here we remark that we may relax the hypothesis in Theorem 0.2 by assuming
that $\mathcal{H}$ is continuously embedded in the space
$\mathscr{D}^{\prime}(\mathbf{R}^{d})$ instead of the smaller space
$\mathscr{S}^{\prime}(\mathbf{R}^{d})$. In fact, let $\mathcal{B}$ be a
translation invariant Banach space which is continuously embedded in
$\mathscr{D}^{\prime}(\mathbf{R}^{d})$ and satisfies

$\|f(\,\cdot\,-x)\|_{\mathcal{B}}\leq C\|f\|_{\mathcal{B}},$

for some constant $C\geq 1$ which is independent of $f\in\mathcal{B}$ and
$x\in\mathbf{R}^{d}$. Then it follows by some standard arguments that
$\mathcal{B}$ is continuously embedded in
$\mathscr{S}^{\prime}(\mathbf{R}^{d})$ (see e. g. [15, Proposition 1.5]).

In the paper we investigate properties on weaker forms of translation and
modulation invariant Hilbert spaces compared to [1, 15]. More precisely we
show that except for the norm identity (0.1), Theorem 0.2 still holds true
after the condition that $\mathcal{H}$ is norm preserved under translations
and modulations, is relaxed into the weaker condition

$\|f(\,\cdot\,-x)e^{i\langle\,\cdot\,,\xi\rangle}\|_{\mathcal{H}}\leq
C_{0}\|f\|_{\mathcal{H}},\qquad f\in\mathcal{H},\ x,\xi\in\mathbf{R}^{d}.$
(0.3)

In our extension, the identity (0.1) should be replaced by the norm
equivalence

$C_{0}^{-1}C\|f\|_{L^{2}(\mathbf{R}^{d})}\leq\|f\|_{\mathcal{H}}\leq
C_{0}C\|f\|_{L^{2}(\mathbf{R}^{d})},\qquad
f\in\mathcal{H}=L^{2}(\mathbf{R}^{d}).$ (0.4)

More precisely our extension of Theorem 0.2 is the following.

######  Theorem 0.3.

Let $\mathcal{H}$ be a Hilbert space which is continuously embedded in
$\mathscr{S}^{\prime}(\mathbf{R}^{d})$, and such that (0.3) holds true for
some constant $C_{0}\geq 1$ which is independent of $f\in\mathcal{H}$ and
$x,\xi\in\mathbf{R}^{d}$. If $\mathcal{H}$ is non-trivial, then
$\mathcal{H}=L^{2}(\mathbf{R}^{d})$, and (0.4) holds for some constant $C>0$
which is independent of $f\in\mathcal{H}=L^{2}(\mathbf{R}^{d})$.

In Section 2 we also present some improvements of this result, where it is
assumed that $\mathcal{H}$ in Theorem 0.3 is embedded in suitable larger
(ultra-)distribution spaces which contain
$\mathscr{S}^{\prime}(\mathbf{R}^{d})$ (see Theorem 2.1). These investigations
are based on some general properties of translation and modulation invariant
quasi-Banach spaces (see Proposition 2.2).

## Acknowledgement

The authors are grateful to HRI and the infosys foundations for support for
the first author during his visit at HRI in December 2023, making this
collaboration possible. The first author is also grateful for the excellent
hospitality during his visit at HRI in December 2023. The first author was
supported by Vetenskapsrådet (Swedish Science Council) within the project
2019-04890.

##  1\. Proof of the main result

In this section we prove our main result Theorem 0.3. First we show how the
condition (0.2) in [1, Theorem 3.1] can be removed, which leads to Theorem
0.2. Then we prove a lemma, which in combination with Theorem 0.2, essentially
leads to Theorem 0.3.

First we have the following lemma.

######  Lemma 1.1.

Suppose that $\mathcal{H}$ satisfies the hypothesis in Theorem 0.3. Then there
is a constant $C>0$ such that (0.2) holds.

Lemma 1.1 follows by straight-forward continuity arguments. In order to assist
the reader here we present the arguments.

###### Proof.

Let $H_{d}=|x|^{2}-\Delta_{d}$ be the $d$-dimensional harmonic oscillator.
Then $\mathscr{S}^{\prime}(\mathbf{R}^{d})$ is the inductive limit of the
increasing Hilbert spaces

$\mathcal{H}_{j}\equiv H_{d}^{j}(L^{2}(\mathbf{R}^{d})),\qquad
j\in\mathbf{N},$

with norms and scalar products given by

$\|f\|_{\mathcal{H}_{j}}\equiv\|H_{d}^{-j}f\|_{L^{2}(\mathbf{R}^{d})}\quad\text{and}\quad(f,g)_{\mathcal{H}_{j}}\equiv(H_{d}^{-j}f,H_{d}^{-j}g)_{L^{2}(\mathbf{R}^{d})},\quad
f,g\in\mathcal{H}_{j}.$

(See e. g. [3, 13].) The fact that $\mathcal{H}$ is continuously embedded in
$\mathscr{S}^{\prime}(\mathbf{R}^{d})$, implies that $\mathcal{H}$ is
continuously embedded in $\mathcal{H}_{j}$, provided $j$ is chosen
sufficiently large. For such $j$ we have

$|(f,\phi)|=|(f,\phi)_{L^{2}}|\lesssim\|f\|_{\mathcal{H}_{j}}\|H_{d}^{j}\phi\|_{L^{2}}\leq
C\|f\|_{\mathcal{H}},$

for some constant $C>0$, and the result follows. ∎

Theorem 0.2 now follows from [1, Theorem 3.1] and Lemma 1.1.

######  Lemma 1.2.

Suppose that $\mathcal{H}$ satisfies the hypothesis in Theorem 0.3. Then there
is a norm $\|\,\cdot\,\|$ on $\mathcal{H}$ with the following properties:

  1. (1)

$\|\,\cdot\,\|$ is equivalent to $\|\,\cdot\,\|_{\mathcal{H}}$;

  2. (2)

$\|\,\cdot\,\|$ is a Hilbert norm;

  3. (3)

$\|e^{i\langle\,\cdot\,,\xi\rangle}f(\,\cdot\,-x)\|=\|f\|$ for every
$f\in\mathcal{H}$ and $x,\xi\in\mathbf{R}^{d}$.

###### Proof.

Let $Q_{R}=[-R,R]^{2d}$ be the cube in $\mathbf{R}^{2d}$ with center at origin
and side length $2R$, $R\geq 1$. Then the volume of $Q_{R}$ is given by
$|Q_{R}|=(2R)^{2d}$. We define the norm $\|\,\cdot\,\|_{[R]}$ by the formula

$\|f\|_{[R]}^{2}=\frac{1}{|Q_{R}|}\iint_{Q_{R}}\|e^{i\langle\,\cdot\,,\xi\rangle}f(\,\cdot\,-x)\|_{\mathcal{H}}^{2}\,dxd\xi,\qquad
f\in\mathcal{H}.$

By the definition it follows that $\|\,\cdot\,\|_{[R]}$ is equivalent to
$\|\,\cdot\,\|_{\mathcal{H}}$, and (0.3) gives

$C_{0}^{-1}\|f\|_{\mathcal{H}}\leq\|f\|_{[R]}\leq
C_{0}\|f\|_{\mathcal{H}},\qquad f\in\mathcal{H}.$ (1.1)

Here the constant $C$ is the same as above. Furthermore, $\|\,\cdot\,\|_{[R]}$
is a norm which arises from the scalar product

$(f,g)_{[R]}=\frac{1}{|Q_{R}|}\iint_{Q_{R}}({e^{i\langle\,\cdot\,,\xi\rangle}f(\,\cdot\,-x)},{e^{i\langle\,\cdot\,,\xi\rangle}g(\,\cdot\,-x)})_{\mathcal{H}}\,dxd\xi,\quad
f,g\in\mathcal{H}.$

Now let $\mho_{0}$ be a dense countable subset of $\mathcal{H}$ containing
$0$, and let

$\mho=\\{\,f-g,f+g,f-ig,f+ig\,;\,f,g\in\mho_{0}\,\\}.$

Then $\mho$ is countable, and closed under additions, subtractions and
multiplications with $\pm i$. Let $\mho=\\{f_{j}\\}_{j=1}^{\infty}$ be a
counting of $\mho$. Since

$C_{0}^{-1}\|f_{1}\|_{\mathcal{H}}\leq\|f_{1}\|_{[m]}\leq
C_{0}\|f_{1}\|_{\mathcal{H}},$

for every $m\in\mathbf{N}$, there is a subsequence
$\\{m_{k}\\}_{k=1}^{\infty}$ of $\mathbf{N}$ such that

$\|f_{1}\|\equiv\lim_{k\to\infty}\|f_{1}\|_{[m_{k}]}$

exists. By Cantor’s diagonalization principle, there is a subsequence
$\\{n_{k}\\}_{k=1}^{\infty}$ of $\\{m_{k}\\}_{k=1}^{\infty}$ such that

$\|f_{j}\|\equiv\lim_{k\to\infty}\|f_{j}\|_{[n_{k}]}$

exists for every $f_{j}\in\mho$, and by (1.1) we get

$C_{0}^{-1}\|f\|_{\mathcal{H}}\leq\|f\|\leq C_{0}\|f\|_{\mathcal{H}},$ (1.2)

when $f\in\mho$.

Next suppose that $f\in\mathcal{H}$ is arbitrary, and let
$\\{f_{0,j}\\}_{j=1}^{\infty}\subseteq\mho_{0}$, be chosen such that

$\lim_{j\to\infty}\|f-f_{0,j}\|_{\mathcal{H}}=0.$

Then

$\|f_{0,j}-f_{0,k}\|\leq C_{0}\|f_{0,j}-f_{0,k}\|_{\mathcal{H}}\to
0,\quad\text{as}\quad j,k\to\infty.$

Since $\left|\|f_{0,j}\|-\|f_{0,k}\|\right|\leq\|f_{0,j}-f_{0,k}\|$, it
follows that $\\{\|f_{0,j}\|\\}_{j=1}^{\infty}$ is a Cauchy sequence in
$\mathbf{R}$. Hence

$\|f\|\equiv\lim_{j\to\infty}\|f_{0,j}\|$

exists and is independent of the chosen particular sequence
$\\{f_{0,j}\\}_{j=1}^{\infty}$. Since (1.2) holds for any $f_{0,j}\in\mho$, it
follows from the recent estimates and limit properties (1.2) extends to any
$f\in\mathcal{H}$. This gives (1).

For $f,g\in\mathcal{H}$, their scalar product $(f,g)_{[n_{k}]}$ can be
evaluated by

$(f,g)_{[n_{k}]}=\frac{1}{4}\left(\|f+g\|_{[n_{k}]}^{2}-\|f-g\|_{[n_{k}]}^{2}+i\|f+ig\|_{[n_{k}]}^{2}-i\|f-ig\|_{[n_{k}]}^{2}.\right)$

By letting $k$ tends to $\infty$, it follows that $\|\,\cdot\,\|$ is a Hilbert
norm with scalar product

$(f,g)\equiv\frac{1}{4}\left(\|f+g\|^{2}-\|f-g\|^{2}+i\|f+ig\|^{2}-i\|f-ig\|^{2}\right),$

giving that $\|\,\cdot\,\|$ fulfills (2).

It remains to prove that (3) holds. By repetition it suffices to prove

$\|e^{i\langle\,\cdot\,,\xi\rangle}f(\,\cdot\,-x)\|=\|f\|$ (1.3)

when $x=0$ and $\xi=\xi_{j}e_{j}$ or when $\xi=0$ and $x=x_{j}e_{j}$ for some
$x_{j},\xi_{j}\in\mathbf{R}$. Here $e_{j}$ denotes the vector of order $j$ in
the standard basis of $\mathbf{R}^{d}$. Then we only prove (1.3) for $\xi=0$
and $x=x_{j}e_{j}$. The other cases follow by similar arguments and are left
for the reader.

Let $R$ be chosen such that $R>|x_{j}|$. Then $Q_{R}$ and $-x_{j}e_{j}+Q_{R}$
intersects. We have

$\displaystyle\|f(\,\cdot\,-x_{j}e_{j})\|_{[R]}^{2}$
$\displaystyle=\frac{1}{|Q_{R}|}\iint_{Q_{R}}\|e^{i\langle\,\cdot\,,\eta\rangle}f(\,\cdot\,-(x_{j}e_{j}+y))\|_{\mathcal{H}}^{2}\,dyd\eta$
$\displaystyle=\frac{1}{|Q_{R}|}\iint_{-x_{j}e_{j}+Q_{R}}\|e^{i\langle\,\cdot\,,\eta\rangle}f(\,\cdot\,-y)\|_{\mathcal{H}}^{2}\,dyd\eta$
$\displaystyle=\frac{1}{|Q_{R}|}\iint_{Q_{R}}\|e^{i\langle\,\cdot\,,\eta\rangle}f(\,\cdot\,-y)\|_{\mathcal{H}}^{2}\,dyd\eta+E_{R}(f)$
$\displaystyle=\|f\|_{[R]}^{2}+E_{R}(f),$ (1.4)

where

$E_{R}(f)=\frac{1}{|Q_{R}|}\iint_{-x_{j}e_{j}+Q_{R}}\|e^{i\langle\,\cdot\,,\eta\rangle}f(\,\cdot\,-y)\|_{\mathcal{H}}^{2}\,dyd\eta\\\
-\frac{1}{|Q_{R}|}\iint_{Q_{R}}\|e^{i\langle\,\cdot\,,\eta\rangle}f(\,\cdot\,-y)\|_{\mathcal{H}}^{2}\,dyd\eta.$

If

$\Delta_{R}=\big{(}-x_{j}e_{j}+Q_{R})\setminus
Q_{R}\big{)}\bigcup\big{(}Q_{R}\setminus-x_{j}e_{j}+Q_{R}\big{)},$

then it follows that $|\Delta_{R}|=2|x_{j}|(2R)^{2d-1}$ and that

$|E_{R}(f)|\leq\frac{1}{|Q_{R}|}\iint_{\Delta_{R}}\|e^{i\langle\,\cdot\,,\eta\rangle}f(\,\cdot\,-y)\|_{\mathcal{H}}^{2}\,dyd\eta.$

This gives

$\displaystyle|E_{R}(f)|$
$\displaystyle\leq\frac{1}{|Q_{R}|}\iint_{\Delta_{R}}\|e^{i\langle\,\cdot\,,\eta\rangle}f(\,\cdot\,-y)\|_{\mathcal{H}}^{2}\,dyd\eta$
$\displaystyle\leq\frac{C_{0}}{|Q_{R}|}\iint_{\Delta_{R}}\|f\|_{\mathcal{H}}^{2}\,dyd\eta$
$\displaystyle=\frac{C_{0}|\Delta_{R}|\|f\|_{\mathcal{H}}^{2}}{|Q_{R}|}=\frac{C_{0}|x_{j}|\|f\|_{\mathcal{H}}^{2}}{R},$

which tends to zero as $R$ turns to infinity.

By letting $R=n_{k}$ in the previous analysis, (1.4) gives

$\displaystyle\|f(\,\cdot\,-x_{j}e_{j})\|$
$\displaystyle=\lim_{k\to\infty}\|f(\,\cdot\,-x_{j}e_{j})\|_{[n_{k}]}^{2}$
$\displaystyle=\lim_{k\to\infty}\left(\frac{1}{|Q_{n_{k}}|}\iint_{Q_{n_{k}}}\|e^{i\langle\,\cdot\,,\eta\rangle}f(\,\cdot\,-y)\|_{\mathcal{H}}^{2}\,dyd\eta+E_{n_{k}}(f)\right)$
$\displaystyle=\|f\|+0,$

which gives (3) and thereby the result. ∎

###### Proof of Theorem 0.3.

By Lemma 1.2, we may replace the norm for $\mathcal{H}$ by an equivalent norm
which is invariant under translations and modulations. The result now follows
from Theorem 0.2. ∎

##  2\. Extensions and variations

In this section we slightly improve Theorem 0.3, and show that the result is
still true when $\mathcal{H}$ is continuously embedded in certain distribution
spaces and ultra-distribution spaces, containing
$\mathscr{S}^{\prime}(\mathbf{R}^{d})$. In the first part we recall the
definition of the Pilipović ultra-distribution space
$\mathcal{H}_{\flat}^{\prime}(\mathbf{R}^{d})$ (also denoted by
$\mathcal{H}_{\flat_{1}}^{\prime}(\mathbf{R}^{d})$ in [13]). Thereafter we
extend Theorem 0.3.

We recall that the Pilipović space $\mathcal{H}_{\flat}(\mathbf{R}^{d})$
consists of all Hermite function expansions

$f(x)=\sum_{\alpha\in\mathbf{N}^{d}}c(f,\alpha)h_{\alpha}(x),$ (2.1)

where the Hermite coefficients $c(f,\alpha)$ should satisfy

$|c(f,\alpha)|\lesssim h^{|\alpha|}\alpha!^{-\frac{1}{2}},$

for some $h>0$ (see [13]). The distribution space
$\mathcal{H}_{\flat}^{\prime}(\mathbf{R}^{d})$ can be identified with the set
of all formal expansions in (2.1) such that

$|c(f,\alpha)|\lesssim h^{|\alpha|}\alpha!^{\frac{1}{2}},$

for every $h>0$. The distribution action between
$\mathcal{H}_{\flat}(\mathbf{R}^{d})$ and
$\mathcal{H}_{\flat}^{\prime}(\mathbf{R}^{d})$ is then given by

$(f,\phi)_{L^{2}}=\sum_{\alpha\in\mathbf{N}^{d}}c(f,\alpha)\overline{c(\phi,\alpha)}.$
(2.2)

We have now the following extension of Theorem 0.3.

######  Theorem 2.1.

Let $\mathcal{H}$ be a Hilbert space which is continuously embedded in
$\mathscr{D}^{\prime}(\mathbf{R}^{d})$ or in
$\mathcal{H}_{\flat}^{\prime}(\mathbf{R}^{d})$, and such that (0.3) holds true
for some constant $C_{0}\geq 1$ which is independent of $f\in\mathcal{H}$ and
$x,\xi\in\mathbf{R}^{d}$. If $\mathcal{H}$ is non-trivial, then
$\mathcal{H}=L^{2}(\mathbf{R}^{d})$, and (0.4) holds for some constant $C>0$
which is independent of $f\in\mathcal{H}=L^{2}(\mathbf{R}^{d})$.

The proof of Theorem 2.1 follows by a combination of [15, Proposition 1.5],
Theorem 0.3, and Proposition 2.2 below. First we need some preparations.

Let $\mathcal{B}$ be a vector space. A _quasi-norm_ on $\mathcal{B}$ is a map
from $\mathcal{B}$ to $\mathbf{R}_{+}\cup\\{0\\}$ such that

$\displaystyle\|\alpha f\|_{\mathcal{B}}$
$\displaystyle=|\alpha|\cdot\|f\|_{\mathcal{B}},$ $\displaystyle\qquad f$
$\displaystyle\in\mathcal{B},\ \alpha\in\mathbf{C}$ and
$\displaystyle\|f+g\|_{\mathcal{B}}^{p}$
$\displaystyle\leq\|f\|_{\mathcal{B}}^{p}+\|g\|_{\mathcal{B}}^{p},$
$\displaystyle\qquad f,g$ $\displaystyle\in\mathcal{B}.$

If the topology on $\mathcal{B}$ is defined through this quasi-norm, then
$\mathcal{B}$ is called a quasi-normed space. A complete quasi-normed space is
called a _quasi-Banach space_.

Next we recall some facts concerning the Gelfand-Shilov space
$\Sigma_{1}(\mathbf{R}^{d})$ and its distribution space
$\Sigma_{1}^{\prime}(\mathbf{R}^{d})$, given in e. g. [10, 13]. We recall that
$\Sigma_{1}(\mathbf{R}^{d})$ consists of all $f\in C^{\infty}(\mathbf{R}^{d})$
such that for every $h>0$, there is a constant $C_{h}>0$ such that

$|x^{\alpha}\partial^{\beta}f(x)|\leq C_{h}h^{|\alpha+\beta|}\alpha!\beta!\,.$

The smallest choice of $C_{h}$ defines a semi-norm on
$\Sigma_{1}(\mathbf{R}^{d})$, and by defining a (project limit) topology from
these semi-norms, it follows that $\Sigma_{1}(\mathbf{R}^{d})$ is a Fréchet
space.

There are several characterizations of $\Sigma_{1}(\mathbf{R}^{d})$ and its
(strong) dual or distribution space $\Sigma_{1}^{\prime}(\mathbf{R}^{d})$. For
example, $\Sigma_{1}(\mathbf{R}^{d})$ is the set of all expansions in (2.1)
such that

$|c(f,\alpha)|\leq C_{r}e^{-r|\alpha|^{\frac{1}{2}}}$

for every $r>0$. The smallest $C_{r}$ defines a semi-norm for
$\Sigma_{1}(\mathbf{R}^{d})$, and the (project limit) topology in this setting
agrees with the topology for $\Sigma_{1}(\mathbf{R}^{d})$. We may then
identify $\Sigma_{1}^{\prime}(\mathbf{R}^{d})$ with all expansions in (2.1)
such that

$|c(f,\alpha)|\leq Ce^{r_{0}|\alpha|^{\frac{1}{2}}},$

for some constants $C>0$ and $r_{0}>0$, with distribution action given by
(2.2) when $f\in\Sigma_{1}^{\prime}(\mathbf{R}^{d})$ and
$\phi\in\Sigma_{1}(\mathbf{R}^{d})$.

In the same way we may identify $\mathscr{S}(\mathbf{R}^{d})$ and
$\mathscr{S}^{\prime}(\mathbf{R}^{d})$ as the sets of all expansions in (2.1)
such that

$|c(f,\alpha)|\leq C_{r}(1+|\alpha|)^{-r}$

for every $r>0$, and

$|c(f,\alpha)|\leq C(1+|\alpha|)^{r_{0}}$

for some $C>0$ and $r>0$, respectively (see e. g. [11]). The distribution
action is given by (2.2) when $f\in\mathscr{S}^{\prime}(\mathbf{R}^{d})$ and
$\phi\in\mathscr{S}(\mathbf{R}^{d})$.

From these identifications it is evident that

$\mathcal{H}_{\flat}(\mathbf{R}^{d})\subseteq\Sigma_{1}(\mathbf{R}^{d})\subseteq\mathscr{S}(\mathbf{R}^{d})\subseteq\mathscr{S}^{\prime}(\mathbf{R}^{d})\subseteq\Sigma_{1}^{\prime}(\mathbf{R}^{d})\subseteq\mathcal{H}_{\flat}^{\prime}(\mathbf{R}^{d}),$
(2.3)

with dense and continuous embeddings.

In [13], the images of the spaces (2.3) under the Bargmann transform are
presented. Let $A(\mathbf{C}^{d})$ be the set of entire functions on
$\mathbf{C}^{d}$, and let

$\displaystyle\mathcal{A}_{0,\infty}^{\prime}(\mathbf{C}^{d})$
$\displaystyle\equiv\bigcup_{r\geq 0}\\{\,F\in
A(\mathbf{C}^{d})\,;\,|F(z)|\leq Ce^{\frac{1}{2}|z|^{2}}(1+|z|)^{r}\ \text{for
some}\ C,r>0\,\\}$ $\displaystyle\mathcal{A}_{0,1}^{\prime}(\mathbf{C}^{d})$
$\displaystyle\equiv\bigcup_{r\geq 0}\\{\,F\in
A(\mathbf{C}^{d})\,;\,|F(z)|\leq Ce^{\frac{1}{2}|z|^{2}+r|z|}\ \text{for
some}\ C,r>0\,\\}$ and
$\displaystyle\mathcal{A}_{\flat}^{\prime}(\mathbf{C}^{d})$
$\displaystyle\equiv A(\mathbf{C}^{d}).$

We equip $\mathcal{A}_{0,\infty}^{\prime}(\mathbf{C}^{d})$ and
$\mathcal{A}_{0,1}^{\prime}(\mathbf{C}^{d})$ with inductive limit topologies
through the semi-norms

$\displaystyle\|F\|_{\mathcal{A}_{0,\infty}^{\prime}\,,r}$
$\displaystyle\equiv\|F\cdot
e^{-\frac{1}{2}|\,\cdot\,|^{2}}(1+|\,\cdot\,|)^{-r}\|_{L^{\infty}},$
$\displaystyle\quad r$ $\displaystyle>0,$ and
$\displaystyle\|F\|_{\mathcal{A}_{0,1}^{\prime}\,,r}$
$\displaystyle\equiv\|F\cdot
e^{-(\frac{1}{2}|z|^{2}+r|\,\cdot\,|)}\|_{L^{\infty}},$ $\displaystyle\quad r$
$\displaystyle>0,$ respectively. We also equipp
$\mathcal{A}_{\flat}^{\prime}(\mathbf{C}^{d})=A(\mathbf{C}^{d})$ with the
projective limit topology, given by the semi-norms
$\displaystyle\|F\|_{\mathcal{A}_{\flat}^{\prime}\,,r}$
$\displaystyle\equiv\|F\|_{B_{r}(0)},$ $\displaystyle\quad r$
$\displaystyle>0.$

In [13] it is proved that the Bargmann mappings

$\displaystyle\mathfrak{V}_{d}$ $\displaystyle:$
$\displaystyle\,\mathscr{S}^{\prime}(\mathbf{R}^{d})$
$\displaystyle\to\mathcal{A}_{0,\infty}^{\prime}(\mathbf{C}^{d}),$ (2.4)
$\displaystyle\mathfrak{V}_{d}$ $\displaystyle:$
$\displaystyle\Sigma_{1}^{\prime}(\mathbf{R}^{d})$
$\displaystyle\to\mathcal{A}_{0,1}^{\prime}(\mathbf{C}^{d})$ (2.5) and
$\displaystyle\mathfrak{V}_{d}$ $\displaystyle:$
$\displaystyle\mathcal{H}_{\flat}^{\prime}(\mathbf{R}^{d})$
$\displaystyle\to\mathcal{A}_{\flat}^{\prime}(\mathbf{C}^{d})$ (2.6)

are homeomorphisms. Finally we recall that in terms of the Weyl map

$(W_{w}F)(z)\equiv e^{-\frac{1}{2}|w|^{2}+(z,w)}F(z-w),\quad F\in
A(\mathbf{C}^{d}),\ z,w\in\mathbf{C}^{d},$ (2.7)

the Bargmann transform transfer translations and modulations as

$\displaystyle(\mathfrak{V}_{d}(e^{i\langle\,\cdot\,,\xi\rangle}f(\,\cdot\,-x)))(z)$
$\displaystyle=e^{\frac{i}{2}\langle
x,\xi\rangle}(W_{\overline{w}/\sqrt{2}}(\mathfrak{V}_{d}f))(z),$ (2.8)
$\displaystyle z$ $\displaystyle\in\mathbf{C}^{d},\
w=x+i\xi\in\mathbf{C}^{d}.$

######  Proposition 2.2.

Let $\mathcal{B}$ be a quasi-Banach space which is continuously embedded in
$\mathcal{H}_{\flat}^{\prime}(\mathbf{R}^{d})$, and such that for some
constant $C>0$ and function $v\in
L^{\infty}_{loc}(\mathbf{R}^{2d};\mathbf{R}_{+})$, it holds

$\|e^{i\langle\,\cdot\,,\xi\rangle}f(\,\cdot\,-x)\|_{\mathcal{B}}\leq
Cv(x,\xi)\|f\|_{\mathcal{B}},\quad x,\xi\in\mathbf{R}^{d}.$ (2.9)

Then the following is true:

  1. (1)

$\mathcal{B}$ is continuously embedded in
$\Sigma_{1}^{\prime}(\mathbf{R}^{d})$;

  2. (2)

if in addition $v(x,\xi)\leq C_{0}(1+|x|+|\xi|)^{N}$, for some constants
$C_{0}>0$ and $N>0$, then $\mathcal{B}$ is continuously embedded in
$\mathscr{S}^{\prime}(\mathbf{R}^{d})$.

###### Proof.

We may assume that $C\geq 1$. Let

$v_{0}(x,\xi)\equiv\sup_{f\in\mathcal{B}\setminus
0}\left(\frac{\|e^{i\langle\,\cdot\,,\xi\rangle}f(\,\cdot\,-x)\|_{\mathcal{B}}}{\|f\|_{\mathcal{B}}}\right).$

Then $v_{0}\leq Cv$, and $v_{0}$ is submultiplicative, i. e.,

$v_{0}(x+y,\xi+\eta)\leq v_{0}(x,\xi)v_{0}(y,\eta),\qquad
x,y,\xi,\eta\in\mathbf{R}^{d}.$ (2.10)

Furthermore, (2.9) gives

$\|e^{i\langle\,\cdot\,,\xi\rangle}f(\,\cdot\,-x)\|_{\mathcal{B}}\leq
v_{0}(x,\xi)\|f\|_{\mathcal{B}},\quad x,\xi\in\mathbf{R}^{d}.$

Hence we may assume that (2.9) holds with $v_{0}$ in place of $v$, and that
$C=1$. We also observe that (2.10) gives

$v_{0}(x)\leq Ce^{r(|x|+|\xi|)},\quad x,\xi\in\mathbf{R}^{d},$ (2.11)

for some constants $C>0$ and $r>0$ (see [8]).

Suppose that $f\in\mathcal{B}$ and let $F=\mathfrak{V}_{d}f$. By the
assumptions, (2.8) and that (2.6) is a homeomorphism, it follows that for some
$C>0$ we have

$|F(z-w)e^{-\frac{1}{2}|w|^{2}+\operatorname{Re}(z,w)}|v_{0}(\sqrt{2}\overline{w})^{-1}\leq
C,\qquad z\in B_{r}(0),\ w\in\mathbf{C}^{d},$

which implies

$\sup_{z\in\mathbf{C}^{d}}\left(|F(z)|e^{-\frac{1}{2}|z|^{2}}v_{0}(-\sqrt{2}\overline{z})^{-1}\right)<\infty.$
(2.12)

By combining the latter estimate with (2.11) shows that
$F\in\mathcal{A}_{0,1}^{\prime}(\mathbf{C}^{d})$. The asserted embedding in
(1) now follows from the homeomorphic property of (2.5).

Next suppose that $v$ satisfies the estimates in (2). Then (2.12) shows that
$F\in\mathcal{A}_{0,\infty}^{\prime}(\mathbf{C}^{d})$. The asserted embedding
in (2) now follows from the homeomorphic property of (2.4), giving the result.
∎

######  Remark 2.3.

By the definition of modulation spaces and their mapping properties under the
Bargmann transform, it follows from the estimate (2.12) that $\mathcal{B}$ in
Proposition 2.2 (1) is continuously embedded in the modulation space
$M^{\infty}_{(\omega)}(\mathbf{R}^{d})$, with
$\omega(x,\xi)=v_{0}(-x-i\xi)^{-1}$, $x,\xi\in\mathbf{R}^{d}$. (See e. g.
[13].)

###### Proof of Theorem 2.1.

If $\mathcal{H}$ is continuously embedded in
$\mathscr{D}^{\prime}(\mathbf{R}^{d})$ or in
$\mathcal{H}_{\flat}(\mathbf{R}^{d})$, then [15, Proposition 1.5] and
Proposition 2.2 (2) show that $\mathcal{H}$ is continuously embedded in
$\mathscr{S}^{\prime}(\mathbf{R}^{d})$. The result now follows from Theorem
0.3. ∎

## References

  * [1] S. R. Bais, P. Mohan, D. V. Naidu _A characterization of translation and modulation invariant Hilbert space of tempered distributions_ , Arch. Math. 122 (2024), 429–436.
  * [2] V. Bargmann _On a Hilbert space of analytic functions and an associated integral transform_ , Comm. Pure Appl. Math., 14 (1961), 187–214.
  * [3] V. Bargmann, _On a Hilbert space of analytic functions and an associated integral transform. Part II. A family of related function spaces. Application to distribution theory_ , Comm. Pure Appl. Math. 20 (1967), 1–101.
  * [4] H. G. Feichtinger _Modulation spaces on locally compact abelian groups. Technical report_ , University of Vienna, Vienna, 1983; also in: M. Krishna, R. Radha, S. Thangavelu (Eds) Wavelets and their applications, Allied Publishers Private Limited, NewDelhi Mumbai Kolkata Chennai Nagpur Ahmedabad Bangalore Hyderbad Lucknow, 2003, pp. 99–140.
  * [5] H. G. Feichtinger _Modulation spaces: Looking back and ahead_ , Sampl. Theory Signal Image Process. 5 (2006), 109–140.
  * [6] H. Feichtinger, F. Luef, E. Cordero _Banach Gelfand triples for Gabor analysis_ , in: L. Rodino, M. W. Wong (Eds), Pseudo-Differential Operators, Quantization and Signals, Lecture notes in math. 1949, Springer, Berlin Heidelberg, 2008, pp. 1–33.
  * [7] K. Gröchenig  _Foundations of Time-Frequency Analysis_ ,  Birkhäuser, Boston, 2001.
  * [8] K. Gröchenig, _Weight functions in time-frequency analysis, in: L. Rodino, M. W. Wong (Eds.) Pseudodifferential Operators: Partial Differential Equations and Time-Frequency Analysis_, Fields Institute Comm., 52 (2007), pp. 343–366.
  * [9] L. Hörmander _The Analysis of Linear Partial Differential Operators_ , vol I, Springer-Verlag, Berlin Heidelberg NewYork Tokyo, 1983.
  * [10] S. Pilipović _Tempered ultradistributions_ , Boll. U.M.I. 7 (1988), 235–251.
  * [11] M. Reed and B. Simon _Methods of modern mathematical physics_ , Academic Press, London New York, 1979.
  * [12] S. Thangavelu _Lectures on Hermite and Laguerre expansions_ , Mathematical Notes 42, Princeton University Press, Princeton, N.J., 1993.
  * [13] J. Toft, _Images of function and distribution spaces under the Bargmann transform_ , J. Pseudo-Differ. Oper. Appl. 8 (2017), 83–139.
  * [14] J. Toft _Schatten properties, nuclearity and minimality of phase shift invariant spaces_ , Appl. Comput. Harmon. Anal. 46 (2019), 154–176.
  * [15] J. Toft, A. Gumber, R. Manna, P. K. Ratnakumar _Translation and modulation invariant Hilbert spaces_ , Monatsh. Math. 196 (2021), 389–398.