Paley–Wiener properties for spaces of entire functions

Abstract

We deduce Paley–Wiener results in the Bargmann setting. At the same time we deduce characterisations of Pilipović spaces of low orders. In particular we improve the characterisation of the Gröchenig test function space $\mathcal {H}_{\flat _1}=\mathcal {S}_C$, deduced in Toft (J Pseudo-Differ Oper Appl 8:83–139, 2017).

1 Introduction

Paley–Wiener theorems characterize functions and distributions with certain restricted supports in terms of estimates of their Fourier–Laplace transforms. For example, let f be a distribution on $\mathbf {R}^{d}$ and let $B_{r_0}(0)\subseteq \mathbf {R}^{d}$ be the ball with center at origin and radius $r_0$. Then f is supported in $B_{r_0}(0)$ if and only if

$$\begin{aligned} |\widehat{f}(\zeta )| \lesssim \langle \zeta \rangle ^{N} e^{r_0 |{\text {Im}}(\zeta )|}, \quad \zeta \in \mathbf {C}^{d}, \end{aligned}$$

for some $N\ge 0$. Furthermore, f is supported in $B_{r_0}(0)$ and smooth, if and only if

$$\begin{aligned} |\widehat{f}(\zeta )| \lesssim \langle \zeta \rangle ^{-N} e^{r_0 |{\text {Im}}(\zeta )|}, \quad \zeta \in \mathbf {C}^{d}, \end{aligned}$$

for every $N\ge 0$.

A similar approach for ultra-regular functions of Gevrey types and corresponding ultra-distribution spaces can be done. In fact, let $s>1$, $\mathcal {D}_s'(\mathbf {R}^{d})$ be the set of all Gevrey distributions of order s and let $\mathcal {E}_s(\mathbf {R}^{d})$ be the set of all smooth functions with Gevrey regularity s. (See [6] and Sect. 2 for notations.) Then it can be proved that $f\in \mathcal {D}_s'(\mathbf {R}^{d})$ is supported in $B_{r_0}(0)$, if and only if

$$\begin{aligned} |\widehat{f}(\zeta )| \lesssim e^{r_0 |{\text {Im}}(\zeta )|+ r |\zeta |^{\frac{1}{s}}}, \quad \zeta \in \mathbf {C}^{d}, \end{aligned}$$

for every $r>0$. Furthermore $f\in \mathcal {E}_s(\mathbf {R}^{d})$ is supported in $B_{r_0}(0)$, if and only if

$$\begin{aligned} |\widehat{f}(\zeta )| \lesssim e^{r_0 |{\text {Im}}(\zeta )|- r |\zeta |^{\frac{1}{s}}}, \quad \zeta \in \mathbf {C}^{d}, \end{aligned}$$

for some $r>0$.

We observe that s in the latter result can not be pushed to be smaller, because if $s\le 1$, it does not make any sense to discuss compact support properties of $\mathcal {D}_s'(\mathbf {R}^{d})$ and $\mathcal {E}_s(\mathbf {R}^{d})$.

In the paper we consider analogous Paley–Wiener properties after the Fourier–Laplace transform above is replaced by the reproducing kernel $\Pi _A$ for the Bargmann transform, and the image spaces are replaced by suitable subspaces of entire functions on $\mathbf {C}^{d}$. These subspaces were considered in [4, 12] and are given by

$$\begin{aligned} \mathcal {A}_{\flat _\sigma } (\mathbf {C}^{d})&= \bigcup _{r>0} \mathcal {A}_{r,\flat _\sigma } (\mathbf {C}^{d}),&\qquad \mathcal {A}_{0,\flat _\sigma } (\mathbf {C}^{d})&= \bigcap _{r>0} \mathcal {A}_{r,\flat _\sigma } (\mathbf {C}^{d}),\nonumber \\ \mathcal {A}_{s} (\mathbf {C}^{d})&= \bigcup _{r>0} \mathcal {A}_{r,s} (\mathbf {C}^{d}),&\qquad \mathcal {A}_{0,s} (\mathbf {C}^{d})&= \bigcap _{r>0} \mathcal {A}_{r,s} (\mathbf {C}^{d}),\nonumber \end{aligned}$$

(1.1)

when $\sigma >0$ and $0<s<\frac{1}{2}$, where $\mathcal {A}_{r,\flat _\sigma } (\mathbf {C}^{d})$ and $\mathcal {A}_{r,s} (\mathbf {C}^{d})$ are the sets of all entire F on $\mathbf {C}^{d}$ such that

$$\begin{aligned} |F(z)|\lesssim e^{r|z|^{\frac{2\sigma }{\sigma +1}}} \quad \text {respective}\quad |F(z)|\lesssim e^{r(\log \langle z\rangle )^{\frac{1}{1-2s}}}. \end{aligned}$$

The spaces in (1.1) appear naturally when considering the Bargmann transform images of extended classes of Fourier invariant Gelfand–Shilov spaces, called Pilipović spaces (see [4, 12]).

If (z, w) is the scalar product of $z,w\in \mathbf {C}^{d}$, then the reproducing kernel of the Bargmann transform is given by

$$\begin{aligned} (\Pi _AF)(z) = \pi ^{-d}\langle F,e^{(z,\, \cdot \, )-|\, \cdot \, |^2}\rangle \end{aligned}$$

when F is a suitable function or (ultra-)distribution. If

$$\begin{aligned} z&\mapsto F(z) e^{R|z|-|z|^2} \in L^1(\mathbf {C}^{d}), \quad R>0 \end{aligned}$$

(1.2)

holds and $d\lambda (w)$ is the Lebesgue measure on $\mathbf {C}^{d}$, then

$$\begin{aligned} (\Pi _AF)(z) = \pi ^{-d}\int _{\mathbf {C}^{d}}F(w)e^{(z,w)-|w|^2}\, d\lambda (w). \end{aligned}$$

A recent Paley–Wiener result with respect to the transform $\Pi _A$ and image spaces (1.1) is given in [12], where it is proved that if $L^\infty _c(\mathbf {C}^{d}) = \mathscr {E}'(\mathbf {C}^{d})\cap L^\infty (\mathbf {C}^{d})$, then

$$\begin{aligned} \Pi _A(\mathscr {E}'(\mathbf {C}^{d})) = \Pi _A(L^\infty _c (\mathbf {C}^{d})) = \mathcal {A}_{\flat _1}(\mathbf {C}^{d}). \end{aligned}$$

(1.3)

Evidently, $L^\infty _c (\mathbf {C}^{d})\subseteq \mathscr {E}'(\mathbf {C}^{d})$, and the gap between these spaces are rather large. It might therefore be somewhat surprising that the first equality holds in (1.3).

In Sect. 3 we improve (1.3) in different ways. Firstly we show that we may replace $L^\infty _c(\mathbf {C}^{d})$ in (1.3) with the smaller space $L^\infty _{A,c}(\mathbf {C}^{d})$ given by

$$\begin{aligned} L^\infty _{A,c}(\mathbf {C}^{d}) = \bigcup L^\infty _{A,c}(K), \end{aligned}$$

where $L^\infty _{A,c}(K)$ is the set of all $F\cdot \chi _K$, where F is analytic in a neighbourhood of the compact set K and $\chi _K$ is the characteristic function of K. Secondly, we may replace $\mathscr {E}'(\mathbf {C}^{d})$ with the set $\mathcal {E}_s'(\mathbf {C}^{d})$ of all compactly supported Gevrey distributions of order $s>1$. Summing up we improve (1.3) into

In Sect. 3 we also deduce various kinds of related mapping properties when $\mathcal {A}_{\flat _1}(\mathbf {C}^{d})$ in (1.3) is replaced by any of the spaces in (1.1). More precisely, let $\chi \in L^\infty _c(\mathbf {C}^{d})$ be non-negative, radial symmetric in each complex variable $z_j$ and bounded from below by a positive constant near origin. Then we prove

$$\begin{aligned} \Pi _A(\mathcal {A}_{0,\flat _{\sigma _0}}' (\mathbf {C}^{d})\cdot \chi )&= \mathcal {A}_{\flat _{\sigma }} (\mathbf {C}^{d}), \quad \sigma \in ({\textstyle {\frac{1}{2}}},1),\ \sigma _0 = \frac{\sigma }{2\sigma -1},\\ \Pi _A(\mathcal {A}_{\flat _{\sigma _0}} (\mathbf {C}^{d})\cdot \chi )&= \mathcal {A}_{\flat _{\sigma }} (\mathbf {C}^{d}), \quad \sigma \in (0,{\textstyle {\frac{1}{2}}}),\ \sigma _0 = \frac{\sigma }{1-2\sigma },\\ \Pi _A(\mathcal {A}_{s} (\mathbf {C}^{d})\cdot \chi )&= \mathcal {A}_{s} (\mathbf {C}^{d}),\quad s \in [0,{\textstyle {\frac{1}{2}}}), \end{aligned}$$

and similarly for $\sigma =\frac{1}{2}$ and when $\mathcal {A}_{\flat _\sigma }$ and $\mathcal {A}_s$ are replaced by $\mathcal {A}_{0,\flat _\sigma }$ and $\mathcal {A}_{0,s}$, respectively. (Cf. Theorems 3.2–3.8 and Propositions 3.9–3.11.)

Finally, in Sect. 4 we use the results in Sect. 3 to deduce characterizations of Pilipović spaces of small orders. We remark that some continuations of the present analysis, involving larger spaces of analytic functions or formal power series expansions, can be found in [13].

2 Preliminaries

In this section we recall some basic facts. We start by discussing Pilipović spaces and some of their properties. Then we recall some facts on modulation spaces. Finally we recall the Bargmann transform and some of its mapping properties, and introduce suitable classes of entire functions on $\mathbf {C}^{d}$.

2.1 The Pilipović spaces

In order to define Pilipović spaces we recall the definition of Hermite functions. We recall that the Hermite function of order $\alpha \in \mathbf {N}^{d}$ is defined by

$$\begin{aligned} h_\alpha (x) = \pi ^{-\frac{d}{4}}(-1)^{|\alpha |} (2^{|\alpha |}\alpha !)^{-\frac{1}{2}}e^{\frac{|x|^2}{2}} (\partial ^\alpha e^{-|x|^2}). \end{aligned}$$

It follows that

$$\begin{aligned} h_{\alpha }(x)= \pi ^{-\frac{d}{4}}(2^{|\alpha |}\alpha ! )^{-\frac{1}{2}} e^{-\frac{|x|^2}{2}}p_{\alpha }(x), \end{aligned}$$

for some polynomial $p_\alpha$ on $\mathbf {R}^{d}$, which is called the Hermite polynomial of order $\alpha$. The Hermite functions are eigenfunctions to the Fourier transform, and to the Harmonic oscillator $H_d\equiv |x|^2-\Delta$ which acts on functions and (ultra-)distributions defined on $\mathbf {R}^{d}$. More precisely, we have

$$\begin{aligned} H_dh_\alpha = (2|\alpha |+d)h_\alpha ,\qquad H_d\equiv |x|^2-\Delta . \end{aligned}$$

It is well-known that the set of Hermite functions is a basis for $\mathscr {S}(\mathbf {R}^{d})$ and an orthonormal basis for $L^2(\mathbf {R}^{d})$ (cf. [10]). In particular, if $f\in L^2(\mathbf {R}^{d})$, then

$$\begin{aligned} \Vert f\Vert _{L^2(\mathbf {R}^{d})}^2 = \sum _{\alpha \in \mathbf {N}^{d}}|c_h(f,\alpha )|^2, \end{aligned}$$

where

$$\begin{aligned} f(x) = \sum _{\alpha \in \mathbf {N}^{d}}c_h(f,\alpha )h_\alpha \end{aligned}$$

(2.1)

is the Hermite series expansion of f, and

$$\begin{aligned} c_h(f,\alpha ) = (f,h_\alpha )_{L^2(\mathbf {R}^{d})} \end{aligned}$$

(2.2)

is the Hermite coefficient of f of order $\alpha \in \mathbf {R}^{d}$.

In order to define the full scale of Pilipović spaces, their order s should belong to the extended set

$$\begin{aligned} \mathbf {R}_\flat = \mathbf {R}_+\cup \{ \, \flat _\sigma \, ;\, \sigma \in \mathbf {R}_+\, \} , \end{aligned}$$

of $\mathbf {R}_+$, with extended inequality relations as

$$\begin{aligned} s_1<\flat _\sigma<s_2 \quad \text {and}\quad \flat _{\sigma _1}<\flat _{\sigma _2} \end{aligned}$$

when $s_1<\frac{1}{2}\le s_2$ and $\sigma _1<\sigma _2$. (Cf. [12].)

For such s and $r\in \mathbf{R} _+^d$ we set

$$\begin{aligned} \vartheta _{r,s}(\alpha )&\equiv {\left\{ \begin{array}{ll} e^{-(\frac{1}{r_1}\cdot \alpha _1^{\frac{1}{2s}} +\cdots + \frac{1}{r_d}\cdot \alpha _d^{\frac{1}{2s}})}, &{} s\in \mathbf {R}_+ \backslash \{ \frac{1}{2}\},\\ r^{\alpha }\alpha !^{-\frac{1}{2\sigma }}, &{} s = \flat _\sigma ,\\ r^{\alpha }, &{} s =\frac{1}{2} , \qquad \alpha \in \mathbf {N}^{d} \end{array}\right. } \end{aligned}$$

(2.3)

and

$$\begin{aligned} \vartheta _{r,s}'(\alpha )&\equiv {\left\{ \begin{array}{ll} e^{(\frac{1}{r_1}\cdot \alpha _1^{\frac{1}{2s}} +\cdots + \frac{1}{r_d}\cdot \alpha _d^{\frac{1}{2s}})}, &{} s \in \mathbf {R}_+\backslash \{ \frac{1}{2}\},\\ r^\alpha \alpha !^{\frac{1}{2\sigma }}, &{} s=\flat _\sigma ,\\ r^ \alpha , &{} s = \frac{1}{2} , \qquad \alpha \in \mathbf {N}^{d}. \end{array}\right. } \end{aligned}$$

(2.4)

Definition 2.1

Let $s\in \overline{\mathbf {R}_\flat } = \mathbf {R}_\flat \cup \{ 0\}$, and let $\vartheta _{r,s}$ and $\vartheta _{r,s}'$ be as in (2.3) and (2.4).

• (1) $\mathcal {H}_0(\mathbf {R}^{d})$ consists of all finite Hermite series expansions (2.1), and $\mathcal {H}_0'(\mathbf {R}^{d})$ consists of all formal Hermite series expansions in (2.1);

• (2) if $s\in \mathbf {R}_\flat$, then $\mathcal {H}_s(\mathbf {R}^{d})$ ($\mathcal {H}_{0,s}(\mathbf {R}^{d})$) consists of all $f\in L^2(\mathbf {R}^{d})$ such that

  $$\begin{aligned} |c_h(f,h_\alpha )| \lesssim \vartheta _{r,s}(\alpha ) \end{aligned}$$

  holds true for some $r\in \mathbf {R}^{d}_+$ (for every $r\in \mathbf {R}^{d}_+$);

• (3) if $s\in \mathbf {R}_\flat$, then $\mathcal {H}_s'(\mathbf {R}^{d})$ ($\mathcal {H}_{0,s}'(\mathbf {R}^{d})$) consists of all formal Hermite series expansions in (2.1) such that

  $$\begin{aligned} |c_h(f,h_\alpha )| \lesssim \vartheta _{r,s}'(\alpha ) \end{aligned}$$

  holds true for every $r\in \mathbf {R}^{d}_+$ (for some $r\in \mathbf {R}^{d}_+$).

The spaces $\mathcal {H}_s(\mathbf {R}^{d})$ and $\mathcal {H}_{0,s}(\mathbf {R}^{d})$ are called Pilipović spaces of Roumieu respectively Beurling type of order s, and $\mathcal {H}_s'(\mathbf {R}^{d})$ and $\mathcal {H}_{0,s}'(\mathbf {R}^{d})$ are called Pilipović distribution spaces of Roumieu respectively Beurling type of order s.

Remark 2.2

Let $\mathscr {S}_s(\mathbf {R}^{d})$ and $\Sigma _s(\mathbf {R}^{d})$ be the Fourier invariant Gelfand–Shilov spaces of order $s\in \mathbf {R}_+$ and of Rourmeu and Beurling types respectively (see [12] for notations). Then it is proved in [8, 9] that

$$\begin{aligned} \mathcal {H}_{0,s}(\mathbf {R}^{d})&=\Sigma _s(\mathbf {R}^{d})\ne \{ 0\} , \quad s> \frac{1}{2},\\ \mathcal {H}_{0,s}(\mathbf {R}^{d})&\ne \Sigma _s(\mathbf {R}^{d}) = \{ 0\} , \quad s\le \frac{1}{2},\\ \mathcal {H}_s(\mathbf {R}^{d})&=\mathcal {S}_s(\mathbf {R}^{d})\ne \{ 0\} , \quad s\ge \frac{1}{2} \end{aligned}$$

and

$$\begin{aligned} \mathcal {H}_s(\mathbf {R}^{d})&\ne \mathcal {S}_s(\mathbf {R}^{d})= \{ 0\} , \quad s< \frac{1}{2}. \end{aligned}$$

In Proposition 2.3 below we give further characterisations of Pilipović spaces.

Next we recall the topologies for Pilipović spaces. Let $s\in \mathbf {R}_\flat$, $r>0$, and let $\Vert f\Vert _{\mathcal {H}_{s;r}}$ and $\Vert f\Vert _{\mathcal {H}_{s;r}'}$ be given by

$$\begin{aligned} \Vert f\Vert _{\mathcal {H}_{s;r}}&\equiv \sup _{\alpha \in \mathbf {N}^{d}} |c_h(f,\alpha )\vartheta _{r,s}'(\alpha )| , \quad s\in \mathbf {R}_\flat , \end{aligned}$$

(2.5)

and

$$\begin{aligned} \Vert f\Vert _{\mathcal {H}_{s;r}'}&\equiv \sup _{\alpha \in \mathbf {N}^{d}} |c_h(f,\alpha )\vartheta _{r,s}(\alpha )| , \quad s\in \mathbf {R}_\flat . \end{aligned}$$

(2.6)

when f is a formal expansion in (2.1). Then $\mathcal {H}_{s;r}(\mathbf {R}^{d})$ consists of all expansions (2.1) such that $\Vert f\Vert _{\mathcal {H}_{s;r}}$ is finite, and $\mathcal {H}_{s;r}'(\mathbf {R}^{d})$ consists of all expansions (2.1) such that $\Vert f\Vert _{\mathcal {H}_{s;r}'}$ is finite. It follows that both $\mathcal {H}_{s;r}(\mathbf {R}^{d})$ and $\mathcal {H}_{s;r}'(\mathbf {R}^{d})$ are Banach spaces under the norms $f\mapsto \Vert f\Vert _{\mathcal {H}_{s;r}}$ and $f\mapsto \Vert f\Vert _{\mathcal {H}_{s;r}'}$, respectively.

We let the topologies of $\mathcal {H}_s(\mathbf {R}^{d})$ and $\mathcal {H}_{0,s}(\mathbf {R}^{d})$ be the inductive respectively projective limit topology of $\mathcal {H}_{s;r}(\mathbf {R}^{d})$ with respect to $r>0$. In the same way, the topologies of $\mathcal {H}_s'(\mathbf {R}^{d})$ and $\mathcal {H}_{0,s}'(\mathbf {R}^{d})$ are the projective respectively inductive limit topology of $\mathcal {H}_{s;r}'(\mathbf {R}^{d})$ with respect to $r>0$. It follows that all the spaces in Definition 2.1 are complete, and that $\mathcal {H}_{0,s}(\mathbf {R}^{d})$ and $\mathcal {H}_s'(\mathbf {R}^{d})$ are Fréchet spaces with semi-norms $f\mapsto \Vert f\Vert _{\mathcal {H}_{s;r}}$ and $f\mapsto \Vert f\Vert _{\mathcal {H}_{s;r}'}$, respectively.

The following characterisations for Pilipović spaces can be found in [12]. The proof is therefore omitted.

Proposition 2.3

Let $s\in \mathbf {R}_+\cup \{ 0\}$ and let $f\in \mathcal {H}_0'(\mathbf {R}^{d})$. Then $f\in \mathcal {H}_{0,s}(\mathbf {R}^{d})\ (f\in \mathcal {H}_s(\mathbf {R}^{d}))$, if and only if $f\in C^\infty (\mathbf {R}^{d})$ and satisfies

$$\begin{aligned} \mathop {\mathrm{sup}}\limits _{N\in \mathrm{N}}\left( \frac{\Vert H_d^N f\Vert _{L^\infty } }{h^N N!^{2s}}\right) <\infty \end{aligned}$$

for every $h>0\ ($for some $h>0)$.

From now on we let

$$\begin{aligned} \phi (x)=\pi ^{-\frac{d}{4}}e^{-\frac{|x|^2}{2}}. \end{aligned}$$

(2.7)

2.2 Spaces of entire functions and the Bargmann transform

Let $\Omega \subseteq \mathbf {C}^{d}$ be open. Then $A(\Omega )$ is the set of all analytic functions in $\Omega$. If instead $\Omega \subseteq \mathbf {C}^{d}$ is closed, then $A(\Omega )$ is the set of all functions which are analytic in an open neighbourhood of $\Omega$. In particular, if $z_0\in \mathbf {C}^{d}$ is fixed, then $A(\{z_0\} )$ is the set of all complex-valued functions which are defined and analytic near $z_0$.

We shall now consider the Bargmann transform which is defined by the formula

$$\begin{aligned} (\mathfrak {V}_df)(z) =\pi ^{-\frac{d}{4}}\int _{\mathbf {R}^{d}}\exp \Big ( -\frac{1}{2}(\langle z,z\rangle +|y|^2)+2^{\frac{1}{2}}\langle z,y\rangle \Big )f(y)\, dy, \end{aligned}$$

when $f\in L^2(\mathbf {R}^{d})$ (cf. [1]). We note that for all $f\in L^2(\mathbf {R}^{d})$, then the Bargmann transform $\mathfrak {V}_df$ of f is the entire function on $\mathbf {C}^{d}$, given by

$$\begin{aligned} (\mathfrak {V}_df)(z) =\int _{\mathbf {R}^{d}}\mathfrak {A}_d(z,y)f(y)\, dy, \end{aligned}$$

or

$$\begin{aligned} (\mathfrak {V}_df)(z) =\langle f,\mathfrak {A}_d(z,\, \cdot \, )\rangle , \end{aligned}$$

(2.8)

where the Bargmann kernel $\mathfrak {A}_d$ is given by

$$\begin{aligned} \mathfrak {A}_d(z,y)=\pi ^{-\frac{d}{4}} \exp \left( -\frac{1}{2}(\langle z,z\rangle +|y|^2)+2^{\frac{1}{2}}\langle z,y\rangle \right) . \end{aligned}$$

Here

$$\begin{aligned} \langle z,w\rangle= & {} \sum _{j=1}^dz_jw_j,\quad \text {when} \ z=(z_1,\dots ,z_d) \in \mathbf {C}^{d}\ \text {and} \ w=(w_1,\dots ,w_d)\in \mathbf {C}^{d}, \end{aligned}$$

and otherwise $\langle \, \cdot \, ,\, \cdot \, \rangle$ denotes the duality between test function spaces and their corresponding duals. We note that the right-hand side in (2.8) makes sense when $f\in \mathcal {S}_{1/2}'(\mathbf {R}^{d})$ and defines an element in $A(\mathbf {C}^{d})$, since $y\mapsto \mathfrak {A}_d(z,y)$ can be interpreted as an element in $\mathcal {S}_{1/2} (\mathbf {R}^{d})$ with values in $A(\mathbf {C}^{d})$.

It was proved in [1] that $f\mapsto \mathfrak {V}_df$ is a bijective and isometric map from $L^2(\mathbf {R}^{d})$ to the Hilbert space $A^2(\mathbf {C}^{d}) \equiv B^2(\mathbf {C}^{d})\cap A(\mathbf {C}^{d})$, where $B^2(\mathbf {C}^{d})$ consists of all measurable functions F on $\mathbf {C}^{d}$ such that

$$\begin{aligned} \Vert F\Vert _{B^2}\equiv \left( \int _{\mathbf {C}^{d}}|F(z)|^2d\mu (z) \right) ^{\frac{1}{2}}<\infty . \end{aligned}$$

(2.9)

Here $d\mu (z)=\pi ^{-d} e^{-|z|^2}\, d\lambda (z)$, where $d\lambda (z)$ is the Lebesgue measure on $\mathbf {C}^{d}$. We recall that $A^2(\mathbf {C}^{d})$ and $B^2(\mathbf {C}^{d})$ are Hilbert spaces, where the scalar products are given by

$$\begin{aligned} (F,G)_{B^2}\equiv \int _{\mathbf {C}^{d}} F(z)\overline{G(z)}\, d\mu (z), \quad F,G\in B^2(\mathbf {C}^{d}). \end{aligned}$$

(2.10)

If $F,G\in A^2(\mathbf {C}^{d})$, then we set $\Vert F\Vert _{A^2}=\Vert F\Vert _{B^2}$ and $(F,G)_{A^2}=(F,G)_{B^2}$.

Furthermore, Bargmann showed that there is a convenient reproducing formula on $A^2(\mathbf {C}^{d})$. More precisely, let

$$\begin{aligned} (\Pi _AF)(z) \equiv \int _{\mathbf {C}^{d}}F(w)e^{(z,w)}\, d\mu (w), \end{aligned}$$

(2.11)

when $z\mapsto F(z)e^{R|z|-|z|^2}$ belongs to $L^1(\mathbf {C}^{d})$ for every $R\ge 0$. Here

$$\begin{aligned} (z,w)= & {} \sum _{j=1}^dz_j\overline{w_j},\quad \text {when} \ z=(z_1,\dots ,z_d) \in \mathbf {C}^{d}\ \text {and} \ w=(w_1,\dots ,w_d)\in \mathbf {C}^{d}, \end{aligned}$$

is the scalar product of $z\in \mathbf {C}^{d}$ and $w\in \mathbf {C}^{d}$. Then it is proved in [1, 2] that $\Pi _A$ is the orthogonal projection of $B^2(\mathbf {C}^{d})$ onto $A^2(\mathbf {C}^{d})$. In particular, $\Pi _AF =F$ when $F\in A^2(\mathbf {C}^{d})$.

In [1] it is also proved that

$$\begin{aligned} \mathfrak {V}_dh_\alpha = e_\alpha ,\quad \text {where}\ e_\alpha (z)\equiv \frac{z^\alpha }{\sqrt{\alpha !}},\ z\in \mathbf {C}^{d} . \end{aligned}$$

(2.12)

In particular, the Bargmann transform maps the orthonormal basis $\{ h_\alpha \}_{\alpha \in \mathbf {N}^{d}}$ in $L^2(\mathbf {R}^{d})$ bijectively into the orthonormal basis $\{ e_\alpha \}_{\alpha \in \mathbf {N}^{d}}$ of monomials in $A^2(\mathbf {C}^{d})$. Hence, there is a natural way to identify formal Hermite series expansion by formal power series expansions

$$\begin{aligned} F(z) = \sum _{\alpha \in \mathbf {N}^{d}}c(F,\alpha )e_\alpha (z), \end{aligned}$$

(2.13)

by letting the series (2.1) be mapped into

$$\begin{aligned} \sum _{\alpha \in \mathbf {N}^{d}}c_h(f,\alpha )e_\alpha (z). \end{aligned}$$

(2.14)

It follows that if $f,g\in L^2(\mathbf {R}^{d})$ and $F,G\in A^2(\mathbf {C}^{d})$, then

$$\begin{aligned} (f,g)_{L^2(\mathbf {R}^{d})}&= \sum _{\alpha \in \mathbf {N}^{d}}c_h(f,\alpha ) \overline{c_h(g,\alpha )},\nonumber \\ (F,G)_{A^2(\mathbf {C}^{d})}&= \sum _{\alpha \in \mathbf {N}^{d}}c(F,\alpha ) \overline{c(G,\alpha )}. \end{aligned}$$

(2.15)

Here and in what follows, $(\, \cdot \, ,\, \cdot \, )_{L^2(\mathbf {R}^{d})}$ and $(\, \cdot \, ,\, \cdot \, )_{A^2(\mathbf {C}^{d})}$ denote the scalar products in $L^2(\mathbf {R}^{d})$ and $A^2(\mathbf {C}^{d})$, respectively. Furthermore,

$$\begin{aligned} c_h(f,\alpha ) = c(F,\alpha ) \quad \text {when}\ F=\mathfrak {V}_df. \end{aligned}$$

(2.16)

We now recall the following spaces of power series expansions given in [12].

Definition 2.4

Let $s\in \overline{\mathbf {R}_\flat } = \mathbf {R}_\flat \cup \{ 0\}$, and let $\vartheta _{r,s}$ and $\vartheta _{r,s}'$ be as in (2.3) and (2.4).

• (1) $\mathcal {A}_0(\mathbf {C}^{d})$ consists of all analytic polynomials on $\mathbf {C}^{d}$, and $\mathcal {A}_0'(\mathbf {C}^{d})$ consists of all formal power series expansions on $\mathbf {C}^{d}$ in (2.13);

• (2) if $s\in \mathbf {R}_\flat$, then $\mathcal {A}_s(\mathbf {C}^{d})$ ($\mathcal {A}_{0,s}(\mathbf {C}^{d})$) consists of all $F\in L^2(\mathbf {C}^{d})$ such that

  $$\begin{aligned} |c(F,h_\alpha )| \lesssim \vartheta _{r,s}(\alpha ) \end{aligned}$$

  holds true for some $r>0$ (for every $r>0$);

• (3) if $s\in \mathbf {R}_\flat$, then $\mathcal {A}_s'(\mathbf {C}^{d})$ ($\mathcal {A}_{0,s}'(\mathbf {C}^{d})$) consists of all formal power series expansions in (2.13) such that

  $$\begin{aligned} |c(F,h_\alpha )| \lesssim \vartheta _{r,s}'(\alpha ) \end{aligned}$$

  holds true for every $r>0$ (for some $r>0$).

Let $f\in \mathcal {H}_0'(\mathbf {R}^{d})$ with formal Hermite series expansion (2.1). Then the Bargmann transform $\mathfrak {V}_df$ of f is defined to be the formal power series expansion (2.14). It follows that $\mathfrak {V}_d$ agrees with the earlier definition when acting on $L^2(\mathbf {R}^{d})$, that $\mathfrak {V}_d$ is linear and bijective from $\mathcal {H}_0'(\mathbf {R}^{d})$ to $\mathcal {A}_0'(\mathbf {C}^{d})$, and restricts to bijections from the spaces

$$\begin{aligned} \mathcal {H}_{0,s}(\mathbf {R}^{d}),\quad \mathcal {H}_s(\mathbf {R}^{d}),\quad \mathcal {H}_s'(\mathbf {R}^{d}) \quad \text {and}\quad \mathcal {H}_{0,s}'(\mathbf {R}^{d}) \end{aligned}$$

(2.17)

to

$$\begin{aligned} \mathcal {A}_{0,s}(\mathbf {C}^{d}),\quad \mathcal {A}_s(\mathbf {C}^{d}),\quad \mathcal {A}_s'(\mathbf {C}^{d}) \quad \text {and}\quad \mathcal {A}_{0,s}'(\mathbf {C}^{d}) \end{aligned}$$

(2.18)

respectively, when $s\in \mathbf {R}_\flat$. We also let the topologies of the spaces in (2.18) be inherited from the spaces in (2.17).

If $s\in \overline{\mathbf {R}_\flat }$, $f\in \mathcal {H}_s(\mathbf {R}^{d})$, $g\in \mathcal {H}_s'(\mathbf {R}^{d})$, $F \in \mathcal {A}_s(\mathbf {C}^{d})$ and $G \in \mathcal {A}_s'(\mathbf {C}^{d})$, then $(f,g)_{L^2(\mathbf {R}^{d})}$ and $(F,G)_{A^2(\mathbf {C}^{d})}$ are defined by the formula (2.15). It follows that (2.16) holds for such choices of f and F. Furthermore, the duals of $\mathcal {H}_s(\mathbf {R}^{d})$ and $\mathcal {A}_s(\mathbf {C}^{d})$ can be identified with $\mathcal {H}_s'(\mathbf {R}^{d})$ and $\mathcal {A}_s'(\mathbf {C}^{d})$, respectively, through the forms in (2.15). The same holds true with

$$\begin{aligned}&\mathcal {H}_{0,s}, \quad \mathcal {H}_{0,s}' ,\quad \mathcal {A}_{0,s},\quad \text {and}\quad \mathcal {A}_{0,s}' \end{aligned}$$

in place of

$$\begin{aligned}&\mathcal {H}_s, \quad \mathcal {H}_s' , \quad \mathcal {A}_s, \quad \text {and}\quad \mathcal {A}_s', \end{aligned}$$

respectively, at each occurrence.

In order to identify the spaces of power series expansions above with spaces of analytic functions, we let

$$\begin{aligned} M_{1,r,s}(z)&= {\left\{ \begin{array}{ll} r_1(\log \langle z_1\rangle )^{\frac{1}{1-2s}} + \cdots + r_d(\log \langle z_d\rangle )^{\frac{1}{1-2s}}, &{} s<\frac{1}{2} ,\\ r_1|z_1|^{\frac{2\sigma }{\sigma +1}}+\cdots +r_d|z_d|^{\frac{2\sigma }{\sigma +1}}, &{} s=\flat _\sigma ,\ \sigma>0,\\ \frac{|z|^2}{2} -(r_1|z_1|^{\frac{1}{s}}+\cdots +r_d|z_d|^{\frac{1}{s}}), &{} s\ge \frac{1}{2} , \end{array}\right. }\nonumber \\ M_{1,r,s}^0(z)&= {\left\{ \begin{array}{ll} M_{1,r,s}(z), &{} s\ne \frac{1}{2}, \\ r_1|z_1|^2+\cdots +r_d|z_d|^2, &{} s=\frac{1}{2}, \end{array}\right. }\nonumber \\ M_{2,r,s}(z)&= {\left\{ \begin{array}{ll} r_1|z_1|^{\frac{2\sigma }{\sigma -1}}+\cdots +r_d|z_d|^{\frac{2\sigma }{\sigma -1}}, &{} s=\flat _\sigma ,\ \sigma >1,\\ \frac{|z|^2}{2} +(r_1|z_1|^{\frac{1}{s}}+\cdots +r_d|z_d|^{\frac{1}{s}}), &{} s\ge \frac{1}{2} , \end{array}\right. }\nonumber \\ M_{2,r,s}^0(z)&= {\left\{ \begin{array}{ll} M_{2,r,s}(z), &{} s\ne \frac{1}{2},\\ r_1|z_1|^2+\cdots +r_d|z_d|^2, &{} s=\frac{1}{2}, \end{array}\right. } \end{aligned}$$

(2.19)

when $r\in \mathbf {R}^{d}_+$ and $z\in \mathbf {C}^{d}$. For conveniency we set $M_r=M_{1,\flat _1,r}$.

By [12] we have the following. The proof is therefore omitted.

Proposition 2.5

Let $M_{1,r,s}$, $M_{1,r,s}^0$, $M_{2,r,s}$ and $M_{2,r,s}^0$ be as in (2.19) when $s\in \overline{\mathbf {R}_\flat }$ and $r\in \mathbf {R}^{d}_+$. Then

$$\begin{aligned} \mathcal {A}_{0,s}(\mathbf {C}^{d})&= \{ \, F\in A(\mathbf {C}^{d})\, ;\, Fe^{-M_{1,r,s}^0}\in L^\infty (\mathbf {C}^{d})\ \text {for every}\ r\in \mathbf {R}^{d}_+\, \} ,&s>0,\\ \mathcal {A}_s(\mathbf {C}^{d})&= \{ \, F\in A(\mathbf {C}^{d})\, ;\, Fe^{-M_{1,r,s}}\in L^\infty (\mathbf {C}^{d})\ \text {for some}\ r\in \mathbf {R}^{d}_+\, \} ,&s\ge 0,\\ \mathcal {A}_s'(\mathbf {C}^{d})&= \{ \, F\in A(\mathbf {C}^{d})\, ;\, Fe^{-M_{2,r,s}}\in L^\infty (\mathbf {C}^{d})\ \text {for every}\ r\in \mathbf {R}^{d}_+\, \} ,&s> \flat _1,\\ \mathcal {A}_{0,s}'(\mathbf {C}^{d})&= \{ \, F\in A(\mathbf {C}^{d})\, ;\, Fe^{-M_{2,r,s}^0}\in L^\infty (\mathbf {C}^{d})\ \text {for some}\ r\in \mathbf {R}^{d}_+\, \} ,&s> \flat _1,\\ \mathcal {A}_{\flat _1}'(\mathbf {C}^{d})&= A(\mathbf {C}^{d}) \quad \text {and}\quad \mathcal {A}_{0,\flat _1}'(\mathbf {C}^{d}) = A(\{ 0\}).&\end{aligned}$$

Next we recall the link between the Bargmann transform and the short-time Fourier transform $f\mapsto V_\phi f$ with window function $\phi$ given by (2.7), defined by

$$\begin{aligned} V_\phi f (x,\xi )\equiv \langle f,\overline{\phi (\, \cdot \, -x)}e^{-i\langle \, \cdot \, ,\xi \rangle }\rangle . \end{aligned}$$

Let S be the dilation operator given by

$$\begin{aligned} (SF)(x,\xi ) = F(2^{-\frac{1}{2}}x,-2^{-\frac{1}{2}}\xi ), \end{aligned}$$

(2.20)

when $F\in L^1_{loc}(\mathbf {R}^{2d})$. Then it follows by straight-forward computations that

$$\begin{aligned} (\mathfrak {V}_d f)(z)= & {} (\mathfrak {V}_df)(x+i\xi ) = (2\pi )^{\frac{d}{2}}e^{\frac{1}{2}(|x|^2+|\xi |^2)}e^{-i\langle x,\xi \rangle } V_\phi f(2^{\frac{1}{2}}x,-2^{\frac{1}{2}}\xi )\nonumber \\= & {} (2\pi )^{\frac{d}{2}}e^{\frac{1}{2}(|x|^2+|\xi |^2)}e^{-i\langle x,\xi \rangle }(S^{-1}(V_\phi f))(x,\xi ), \end{aligned}$$

(2.21)

or equivalently,

$$\begin{aligned} V_\phi f(x,\xi )= & {} (2\pi )^{-\frac{d}{2}} e^{-\frac{1}{4}(|x|^2+|\xi |^2)}e^{-i\langle x,\xi \rangle /2}(\mathfrak {V}_df) (2^{-\frac{1}{2}}x,-2^{-\frac{1}{2}}\xi )\nonumber \\= & {} (2\pi )^{-\frac{d}{2}}e^{-i\langle x,\xi \rangle /2}S(e^{-\frac{|\, \cdot \, |^2}{2}}(\mathfrak {V}_df))(x,\xi ) \end{aligned}$$

(2.22)

We observe that (2.21) and (2.22) can be formulated as

$$\begin{aligned} \mathfrak {V}_d = U_{\mathfrak {V}}\circ V_\phi ,\quad \text {and}\quad U_{\mathfrak {V}}^{-1} \circ \mathfrak {V}_d = V_\phi , \end{aligned}$$

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

$$\begin{aligned} (U_{\mathfrak {V}}F)(x,\xi ) = (2\pi )^{\frac{d}{2}} e^{\frac{1}{2}(|x|^2+|\xi |^2)}e^{-i\langle x,\xi \rangle } F(2^{\frac{1}{2}}x,-2^{\frac{1}{2}}\xi ) . \end{aligned}$$

(2.23)

Let $D_{d,r}(z_0)$ be the polydisc

$$\begin{aligned} \{ \, z=(z_1,\dots ,z_d)\in \mathbf {C}^{d}\, ;\, |z_j-z_{0,j}|< r_j,\ j=1,\dots ,d\, \} , \end{aligned}$$

with center and radii given by

$$\begin{aligned} z_0 =(z_{0,1},\dots ,z_{0,d}) \in \mathbf {C}^{d} \quad \text {and}\quad r=(r_1,\dots ,r_d)\in (0,\infty )^d. \end{aligned}$$

Then

$$\begin{aligned} A(\mathbf {C}^{d}) = \bigcap _{r\in (0,\infty )^d} A(D_{d,r}(z)),\qquad A(\{0\}) =\bigcup _{r\in (0,\infty )^d} A(D_{d,r}(z_0)). \end{aligned}$$

2.3 Hilbert spaces of power series expansions and analytic functions

The spaces in Definition 2.4 can also be described by related unions and intersections of Hilbert spaces of analytic functions and power series expansions as follows. (See also [12].)

Let $\vartheta$ be a weight on $\mathbf {N}^{d}$, $\omega$ be a weight on $\mathbf {C}^{d}$, and let

$$\begin{aligned} \Vert F\Vert _{\mathcal {A}_{[\vartheta ]}^2(\mathbf {C}^{d})} \equiv \left( \sum _{\alpha \in \mathbf {N}^{d}} |c(F,\alpha )\vartheta (\alpha )|^2 \right) ^{\frac{1}{2}} \end{aligned}$$

(2.24)

when $F \in \mathcal {A}_0'(\mathbf {C}^{d})$ is given by (2.13), and

$$\begin{aligned} \Vert F\Vert _{A^2_{(\omega )}(\mathbf {C}^{d}) } \equiv \left( \int _{\mathbf {C}^{d}}|F(z)\omega (2^{\frac{1}{2}}\overline{z})|^2\, d\mu (z) \right) ^{\frac{1}{2}} \end{aligned}$$

(2.25)

when $F\in A(\mathbf {C}^{d})$. We let $\mathcal {A}_{[\vartheta ]}^2(\mathbf {C}^{d})$ be the set of all $F\in \mathcal {A}_0'(\mathbf {C}^{d})$ such that $\Vert F\Vert _{\mathcal {A}_{[\vartheta ]}^2}$ is finite, and $A^2_{(\omega )}(\mathbf {C}^{d})$ be the set of all $F\in A(\mathbf {C}^{d})$ such that $\Vert F\Vert _{A^2_{(\omega )}}$ is finite. It follows that these spaces are Hilbert spaces under these norms.

If $\vartheta$ and $\omega$ are related to each others as

$$\begin{aligned} \vartheta (\alpha ) = \left( \frac{1}{\alpha !} \int _{\mathbf {R}_+^d} \omega _0(r)^2r^\alpha \, dr \right) ^{\frac{1}{2}} \end{aligned}$$

(2.26)

and

$$\begin{aligned} \omega (z) = e^{\frac{|z|^2}{2}}\omega _0(|z_1|^2,\dots ,|z_d|^2), \end{aligned}$$

(2.27)

for some suitable weight $\omega _0$ on $\mathbf {R}_+^d$, then the following multi-dimensional version of [7, Theorem (4.1)] shows that $\mathcal {A}_{[\vartheta ]}^2(\mathbf {C}^{d}) = A^2_{(\omega )}(\mathbf {C}^{d})$ with equal norms. Here we identify entire functions with their power series expansions at origin. Consequently, the Bargmann transform is bijective and isometric from $\mathcal {H}_{[\vartheta ]}^2(\mathbf {R}^{d})$ to $A^2_{(\omega )}(\mathbf {C}^{d})$ for such choices of $\vartheta$ and $\omega$. We omit the proof since the result is an immediate consequence of [12, Theorem 3.5].

Theorem 2.6

Let $\alpha \in \mathbf {N}^{d}$ and $\omega _0$ be a positive measurable function on $\mathbf {R}_+^d$. Also let $\vartheta$ and $\omega$ be weights on $\mathbf {N}^{d}$ and $\mathbf {C}^{d}$, respectively, related to each others by (2.26) and (2.27), and such that

$$\begin{aligned} \frac{r^{|\alpha |}}{(\alpha !)^{\frac{1}{2}}}\lesssim \vartheta (\alpha ),\quad \alpha \in \mathbf {N}^{d}, \end{aligned}$$

(2.28)

holds for every $r>0$. Then $\mathcal {A}^2_{[\vartheta ]}(\mathbf {C}^{d})=A^2_{(\omega )}(\mathbf {C}^{d})$ with equality in norms.

In our situation, the involved weights should satisfy a split condition. In one dimension, (2.26), (2.27) and (2.28) take the forms

and

Lemma 2.7

Let $\omega _{0,j}$ be weights on $\overline{\mathbf {R}_+}$, $\omega _j$ be weights on $\mathbf {C}$ and $\vartheta _j$ be weights on $\mathbf {N}$ such that (2.26)$'$–(2.28)$'$ hold, $j=1,\dots ,d$, and set $\omega _0(z)\equiv \prod _{j=1}^d \omega _{0,j}(z_j)$, $z\equiv (z_1, \ldots , z_d) \in \mathbf {C}^{d}$. If $\vartheta$ and $\omega$ are given by (2.26), and (2.27), then

$$\begin{aligned} \vartheta (\alpha ) = \prod _{j=1}^d \vartheta _j(\alpha _j),\quad \alpha = (\alpha _1, \dots , \alpha _d) \in \mathbf {N}^{d} \end{aligned}$$

(2.29)

and

$$\begin{aligned} \omega (z) = \prod _{j=1}^d \omega _j(z_j), \quad z = (z_1,\dots ,z_d)\in \mathbf {C}^{d}. \end{aligned}$$

(2.30)

Proof

By [12, Theorem 3.5] and its proof, it follows that $\omega _{0,j}\cdot r^{\alpha _j} \in L^1(\mathbf {R}_+)$ for all $j \in \{1, \ldots , d\}$ and $\alpha _j \in \mathbf {N}$. Hence, Fubini’s theorem gives

$$\begin{aligned} \vartheta (\alpha )= & {} \left( \frac{1}{\alpha !} \int _{\mathbf {R}^d_+} \omega _0(r)^2 r^{\alpha }\, dr \right) ^{\frac{1}{2}} = \left( \frac{1}{\alpha !}\prod _{j=1}^d \int _0^{\infty } \omega _{0,j}(r_j)^2 r_j^{\alpha _j}\, dr_j \right) ^{\frac{1}{2}}\\= & {} \prod _{j=1}^d \vartheta _{j}(\alpha _j), \end{aligned}$$

and (2.29) follows. The assertion (2.30) follows from the definitions. $\square$

2.4 A test function space introduced by Gröchenig

In this section we recall some comparison results deduced in [12], between a test function space, $\mathcal {S}_C(\mathbf {R}^{d})$, introduced by Gröchenig in [5] to handle modulation spaces with elements in spaces of ultra-distributions.

The definition of $\mathcal {S}_C(\mathbf {R}^{d})$ is given as follows.

Definition 2.8

Let $\phi (x)=\pi ^{-\frac{d}{4}}e^{-\frac{|x|^2}{2}}$. Then $\mathcal {S}_C(\mathbf {R}^{d})$ and $\mathcal {S}_G(\mathbf {R}^{d})$ consist of all $f\in \mathscr {S}'(\mathbf {R}^{d})$ such that $f=V_\phi ^*F$, for some $F\in L^\infty (\mathbf {R}^{d})\cap \mathscr {E}'(\mathbf {R}^{d})$ and $F\in \mathscr {E}'(\mathbf {R}^{d})$, respectively.

It follows that $f\in \mathcal {S}_C(\mathbf {R}^{d})$, if and only if

$$\begin{aligned} f(x) = (2\pi )^{-\frac{d}{2}}\iint _{\mathbf {R}^{2d}} F(y,\eta )e^{-\frac{1}{2}|x-y|^2} e^{i\langle x,\eta \rangle }\, dyd\eta , \end{aligned}$$

(2.31)

for some $F\in L^\infty (\mathbf {R}^{d})\cap \mathscr {E}'(\mathbf {R}^{d})$.

Remark 2.9

By the identity $(V_\phi h,F)=(h,V_\phi ^*F)$ and the fact that the map $(f,\phi )\mapsto V_\phi f$ is continuous from $\mathscr {S}(\mathbf {R}^{d})\times \mathscr {S}(\mathbf {R}^{d})$ to $\mathscr {S}(\mathbf {R}^{2d})$, it follows that $f=V_\phi ^*F$ is uniquely defined as an element in $\mathscr {S}'(\mathbf {R}^{d})$ when $F\in \mathscr {S}'(\mathbf {R}^{2d})$ (cf. [3]). In particular, the space $\mathcal {S}_G(\mathbf {R}^{d})$ in Definition 2.8 is well-defined, and it is evident that $\mathcal {S}_C(\mathbf {R}^{d}) \subseteq \mathcal {S}_G(\mathbf {R}^{d})$.

The following is a restatement of [12, Lemma 4.9]. The result is essential when deducing the characterizations of Pilipović spaces in Sect. 4.

Lemma 2.10

Let $F\in L^\infty (\mathbf {C}^{d})\cup \mathscr {E}'(\mathbf {C}^{d})$. Then the Bargmann transform of $f=V_\phi ^*F$ is given by $\Pi _AF_0$, where

$$\begin{aligned} F_0(x+i\xi ) = (2\pi ^3)^{\frac{d}{4}}F(\sqrt{2} x,-\sqrt{2} \xi )e^{\frac{1}{2}(|x|^2+|\xi |^2)} e^{-i\langle x,\xi \rangle }. \end{aligned}$$

(2.32)

Moreover, the images of $\mathcal {S}_C(\mathbf {R}^{d})$ and $\mathcal {S}_G(\mathbf {R}^{d})$ under the Bargmann transform are given by

$$\begin{aligned} \{ \, \Pi _AF\, ;\, F\in L^\infty (\mathbf {C}^{d})\cap \mathscr {E}'(\mathbf {C}^{d})\, \} \quad \text {and}\quad \{ \, \Pi _AF\, ;\, F\in \mathscr {E}'(\mathbf {C}^{d})\, \} , \end{aligned}$$

(2.33)

respectively.

The next results follow from [12, Theorem 4.10]. The proof is therefore omitted.

Proposition 2.11

It holds $\mathcal {S}_C(\mathbf {R}^{d}) = \mathcal {S}_G(\mathbf {R}^{d}) = \mathcal {H}_{\flat _1}(\mathbf {R}^{d})$.

Due to the image properties for the spaces in Proposition 2.11 under the Bargmann transform, the next result is equivalent with the previous one.

Proposition 2.12

The sets in (2.33) are equal to $\mathcal {A}_{\flat _1}(\mathbf {C}^{d})$.

In the next section we extend Propositions 2.11 and 2.12 by proving that the conclusions in Proposition 2.12 hold for suitable smaller and larger sets than those in (2.33). We also deduce similar identifications for other Pilipović spaces and their Bargmann images.

3 Paley–Wiener properties for Bargmann–Pilipović spaces

In this section we consider spaces of compactly supported functions with interiors in $\mathcal {A}_s(\mathbf {C}^{d})$ or in $\mathcal {A}_s'(\mathbf {C}^{d})$. We show that the images of such functions under the reproducing kernel $\Pi _A$ are equal to $\mathcal {A}_s(\mathbf {C}^{d})$, for some other choice of $s\le \flat _1$. In the first part we state the main results given in Theorems 3.2–3.4. They are straight-forward consequences of Propositions 3.9, where more detailed information concerning involved constants are given. Thereafter we deduce results which are needed for their proofs. Depending of the choice of s, there are several different situations for characterizing $\mathcal {A}_s(\mathbf {C}^{d})$. This gives rise to a quite large flora of main results, where each one takes care of one situation.

First we note that if $A^2_{(\omega )}=\mathcal {A}_{[\vartheta ]}^2$, then a split of the variables in the weight $\omega$ in $A^2_{(\omega )}$ induce a split of the variables in $\vartheta$ in $\mathcal {A}_{[\vartheta ]}^2$. (See Lemma 2.7.)

In order to present the main results, it is suitable to make the following definition.

Definition 3.1

Let $t_1,t_2\in \mathbf {R}^{d}_+$ be such that $t_1\le t_2$. Then the function $\chi \in L^\infty (\mathbf {C}^{d})$ is called positive, bounded and radial symmetric with respect to $t_1$ and $t_2$, if the following conditions are fulfilled:

• $\chi \in L^\infty (\mathbf {C}^{d})\cap \mathscr {E}' (\overline{D_{t_2}(0)})$ is non-negative;

• $\chi (z_1,\dots ,z_d)=\chi _0(|z_1|,\dots ,|z_d|)$ for some function $\chi _0$;

• $\chi \ge c$ on $D_{t_1}(0)$ for some constant $c>0$.

The set of positive, bounded and radial symmetric functions with respect to $t_1$ and $t_2$ is denoted by $\mathcal {R}^\infty _{t_1,t_2}(\mathbf {C}^{d})$, and $\mathcal {R}^\infty (\mathbf {C}^{d})$ is defined by

$$\begin{aligned} \mathcal {R}^\infty (\mathbf {C}^{d}) \equiv \bigcup _{t_1\le t_2\in \mathbf {R}^{d}_+} \mathcal {R}^\infty _{t_1,t_2}(\mathbf {C}^{d}). \end{aligned}$$

3.1 Main results

We begin with characterizing the largest spaces in our investigations, which appears when $s=\flat _1$, and then proceed with spaces of decreasing order. First we recall that elements in $\mathcal {A}_s(\mathbf {C}^{d})$ and $\mathcal {A}_{0,s}(\mathbf {C}^{d})$ fulfill conditions of the forms

$$\begin{aligned} |F(z)| \lesssim e^{r_1|z_1|^{\frac{2\sigma }{\sigma +1}}+\cdots +r_d|z_d|^{\frac{2\sigma }{\sigma +1}}} \end{aligned}$$

(3.1)

when $s=\flat _\sigma$ and

$$\begin{aligned} |F(z)| \lesssim e^{r_1 (\log \langle z_1\rangle )^{\frac{1}{1-2s}} +\cdots + r_d (\log \langle z_d\rangle )^{\frac{1}{1-2s}}} \end{aligned}$$

(3.2)

when $s\in [0,\frac{1}{2})$.

Theorem 3.2

Let $F\in A(\mathbf {C}^{d})$, $\sigma =1$ and $s>1$. Then the following conditions are equivalent:

• (1) $F\in \mathcal {A}_{\flat _1}(\mathbf {C}^{d})$;

• (2) (3.1) holds for some $r\in \mathbf {R}^{d}_+$;

• (3) For some $r_0\in \mathbf {R}^{d}_+$ and every $r\in \mathbf {R}^{d}_+$ with $r_0\le r$ and every $\chi \in \mathcal {R}_{r_0,r}^\infty (\mathbf {C}^{d})$, there exists $F_0\in A(\overline{D_r(0)})$ such that $F = \Pi _A(F_0\cdot \chi )$;

• (4) For some $r_0\in \mathbf {R}^{d}_+$ and every $r\in \mathbf {R}^{d}_+$ with $r_0<r$ and some $\chi \in \mathcal {R}_{r_0,r}^\infty (\mathbf {C}^{d})$, there exists $F_0\in A(\overline{D_r(0)})$ such that $F = \Pi _A(F_0\cdot \chi )$;

• (5) There exists $F_0\in \mathscr {E}'(\mathbf {C}^{d})\cap L^\infty (\mathbf {C}^{d})$ such that $F = \Pi _AF_0$;

• (6) There exists $F_0\in \mathcal {E}_s'(\mathbf {C}^{d})$ such that $F = \Pi _AF_0$.

Remark 3.3

Since

$$\begin{aligned} \mathscr {E}'(\mathbf {C}^{d})\cap L^\infty (\mathbf {C}^{d}) \subseteq \mathscr {E}'(\mathbf {C}^{d}) \subseteq \mathcal {E}_s'(\mathbf {C}^{d}), \end{aligned}$$

Theorem 3.2 still holds true after $\mathcal {E}_s'$ has been replaced by $\mathscr {E}'$ in (6).

Theorem 3.4

Let $F\in A(\mathbf {C}^{d})$, $\sigma =1$ and $\chi \in \mathcal {R}^\infty (\mathbf {C}^{d})$. Then the following is true:

• (i) The following conditions are equivalent:

  • (1) $F\in \mathcal {A}_{0,\flat _1}(\mathbf {C}^{d})$;

  • (2) (3.1) holds for every $r\in \mathbf {R}^{d}_+$;

  • (3) There exists $F_0\in A(\mathbf {C}^{d})$ such that $F = \Pi _A(F_0\cdot \chi )$;

• (ii) The map $F\mapsto \Pi _A(F\cdot \chi )$ from $A(\mathbf {C}^{d})$ to $\mathcal {A}_{0,\flat _1}(\mathbf {C}^{d})$ is a homeomorphism.

The next result deals with the case when $s=\flat _\sigma$ with $\sigma \in (\frac{1}{2},1)$.

Theorem 3.5

Let $F\in A(\mathbf {C}^{d})$, $\chi \in \mathcal {R}^\infty (\mathbf {C}^{d})$, $\sigma \in (\frac{1}{2},1)$ and let

$$\begin{aligned} \sigma _0 = \frac{\sigma }{2\sigma -1}. \end{aligned}$$

Then the following is true:

• (i) The following conditions are equivalent:

  • (1) $F\in \mathcal {A}_{\flat _\sigma }(\mathbf {C}^{d})$ ($F\in \mathcal {A}_{0,\flat _\sigma }(\mathbf {C}^{d})$);

  • (2) (3.1) holds for some (for every) $r\in \mathbf {R}^{d}_+$;

  • (3) There exists $F_0\in \mathcal {A}_{0,\flat _{\sigma _0}}'(\mathbf {C}^{d})$ ($F_0\in \mathcal {A}_{\flat _{\sigma _0}}'(\mathbf {C}^{d})$) such that $F = \Pi _A(F_0\cdot \chi )$;

• (ii) The mappings $F\mapsto \Pi _A(F\cdot \chi )$ from $\mathcal {A}_{0,\flat _{\sigma _0}}'(\mathbf {C}^{d})$ to $\mathcal {A}_{\flat _{\sigma }}(\mathbf {C}^{d})$ and from $\mathcal {A}_{\flat _{\sigma _0}}'(\mathbf {C}^{d})$ to $\mathcal {A}_{0,\flat _{\sigma }}(\mathbf {C}^{d})$ are homeomorphisms.

The next result deals with the case when $s=\flat _\sigma$ with $\sigma =\frac{1}{2}$.

Theorem 3.6

Let $F\in A(\mathbf {C}^{d})$, $\sigma =\frac{1}{2}$ and $\chi \in \mathcal {R}^\infty (\mathbf {C}^{d})$. Then the following is true:

• (i) The following conditions are equivalent:

  • (1) $F\in \mathcal {A}_{\flat _{\sigma }}(\mathbf {C}^{d})$ ($F\in \mathcal {A}_{0,\flat _{\sigma }}(\mathbf {C}^{d})$);

  • (2) (3.1) holds for some (for every) $r\in \mathbf {R}^{d}_+$;

  • (3) There exists $F_0\in \mathcal {A}_{0,1/2}'(\mathbf {C}^{d})$ ($F_0\in \mathcal {A}_{0,1/2}(\mathbf {C}^{d})$) such that $F = \Pi _A(F_0\cdot \chi )$;

• (ii) The mappings $F\mapsto \Pi _A(F\cdot \chi )$ from $\mathcal {A}_{0,1/2}'(\mathbf {C}^{d})$ to $\mathcal {A}_{\flat _{1/2}}(\mathbf {C}^{d})$ and from $\mathcal {A}_{0,1/2}(\mathbf {C}^{d})$ to $\mathcal {A}_{0,\flat _{1/2}}(\mathbf {C}^{d})$ are homeomorphisms.

The next result deals with the case when $s=\flat _\sigma$ with $\sigma \in (0,\frac{1}{2})$.

Theorem 3.7

Let $F\in A(\mathbf {C}^{d})$, $\chi \in \mathcal {R}^\infty (\mathbf {C}^{d})$, $\sigma \in (0,\frac{1}{2})$ and let

$$\begin{aligned} \sigma _0 = \frac{\sigma }{1-2\sigma }. \end{aligned}$$

Then the following is true:

• (i) The following conditions are equivalent:

  • (1) $F\in \mathcal {A}_{\flat _\sigma }(\mathbf {C}^{d})$ ($F\in \mathcal {A}_{0,\flat _\sigma }(\mathbf {C}^{d})$);

  • (2) (3.1) holds for some (for every) $r\in \mathbf {R}^{d}_+$;

  • (3) There exists $F_0\in \mathcal {A}_{\flat _{\sigma _0}}(\mathbf {C}^{d})$ ($F_0\in \mathcal {A}_{0,\flat _{\sigma _0}}(\mathbf {C}^{d})$) such that $F = \Pi _A(F_0\cdot \chi )$;

• (ii) The mappings $F\mapsto \Pi _A(F\cdot \chi )$ from $\mathcal {A}_{\flat _{\sigma _0}}(\mathbf {C}^{d})$ to $\mathcal {A}_{\flat _{\sigma }}(\mathbf {C}^{d})$ and from $\mathcal {A}_{0,\flat _{\sigma _0}}(\mathbf {C}^{d})$ to $\mathcal {A}_{0,\flat _{\sigma }}(\mathbf {C}^{d})$ are homeomorphisms.

In the next result we consider the case when $s\in [0,\frac{1}{2})$ is real.

Theorem 3.8

Let $F\in A(\mathbf {C}^{d})$, $\chi \in \mathcal {R}^\infty (\mathbf {C}^{d})$, $s \in [0,\frac{1}{2})$. Then the following is true:

• (i) The following conditions are equivalent:

  • (1) $F\in \mathcal {A}_s(\mathbf {C}^{d})$ ($F\in \mathcal {A}_{0,s}(\mathbf {C}^{d})$);

  • (2) (3.2) holds for some (for every) $r\in \mathbf {R}^{d}_+$;

  • (3) There exists $F_0\in \mathcal {A}_{s}(\mathbf {C}^{d})$ ($F_0\in \mathcal {A}_{0,s}(\mathbf {C}^{d})$) such that $F = \Pi _A(F_0\cdot \chi )$;

• (ii) The mappings $F\mapsto \Pi _A(F\cdot \chi )$ from $\mathcal {A}_{s}(\mathbf {C}^{d})$ to $\mathcal {A}_{s}(\mathbf {C}^{d})$ and from $\mathcal {A}_{0,s}(\mathbf {C}^{d})$ to $\mathcal {A}_{0,s}(\mathbf {C}^{d})$ are homeomorphisms.

The previous theorems are essentially consequences of Propositions 3.9–3.11, where more detailed information about involved constants are given.

Proposition 3.9

Let $F\in A(\mathbf {C}^{d})$, $\sigma =1$, $s>1$ and $r\in \mathbf {R}^{d}_+$. Then the following conditions are equivalent:

• (1) For some $r_0\in \mathbf {R}^{d}_+$ such that $r_0<r$, (3.1) holds with $r_0$ in place of r;

• (2) For some $t_1\in \mathbf {R}^{d}_+$ such that $t_1<r$, every $t_2\in \mathbf {R}^{d}_+$ with $t_1\le t_2 <r$ and every $\chi \in \mathcal {R}_{t_1,t_2}^\infty (\mathbf {C}^{d})$, there exists $F_0\in A(\overline{D_r(0)})$ such that $F = \Pi _A(F_0\cdot \chi )$;

• (3) For some $F_0\in A(\overline{D_r(0)})$ it holds $F = \Pi _A(F_0\cdot \chi _{D_r(0)})$;

• (4) For some $F_0\in \mathscr {E}'(D_r(0))\cap L^\infty (\mathbf {C}^{d})$ it holds $F = \Pi _AF_0$;

• (5) For some $r_0\in \mathbf {R}^{d}_+$ such that $r_0<r$, there exists $F_0\in \mathcal {E}'_s(D_{r_0}(0))$ such that $F = \Pi _AF_0$.

Theorems 3.5–3.7 essentially follow from the following proposition.

Proposition 3.10

Let $\tau >\frac{1}{2}$, $r,t_1,t_2\in \mathbf {R}^{d}_+$ be such that $t_1\le t_2$, and let $\chi \in \mathcal {R}_{t_1,t_2}^\infty (\mathbf {C}^{d})$. Then the following is true:

• (1) Let $F\in A(\mathbf {C}^{d})$ be such that

  $$\begin{aligned} |F(z)| \lesssim e^{r_{0,1}|z_1|^{\frac{2}{2\tau +1}}+\cdots +r_{0,d}|z_d|^{\frac{2}{2\tau +1}}} \end{aligned}$$

  (3.3)

  holds for some $r_0\in \mathbf {R}^{d}_+$ such that $r_0<r$. Then for some $r_0\in \mathbf {R}^{d}_+$ such that $r_0<r$, there exists $F_0\in A(\mathbf {C}^{d})$ such that $F=\Pi _A(F_0\cdot \chi )$ and

  $$\begin{aligned} |F_0(z)| \lesssim e^{R_{0,1}|z_1|^{\frac{2}{2\tau -1}}+\cdots +R_{0,d}|z_d|^{\frac{2}{2\tau -1}} }, \end{aligned}$$

  (3.4)

  where

  $$\begin{aligned} R_0 = \frac{2\tau -1}{2} \left( \frac{2r_0}{2\tau +1} \right) ^{{{\frac{2\tau +1}{2\tau -1}}}} t_1^{-\frac{4}{2\tau -1}} ; \end{aligned}$$

  (3.5)

• (2) Let $r_0,R_0\in \mathbf {R}^{d}_+$ be such that $r_0<r$ and

  $$\begin{aligned} R_0 = \frac{2\tau -1}{2} \left( \frac{2r_0}{2\tau +1} \right) ^{{{\frac{2\tau +1}{2\tau -1}}}} t_2^{-\frac{4}{2\tau -1}}, \end{aligned}$$

  (3.6)

  $F_0\in A(\mathbf {C}^{d})$ be such that (3.4) hold and let $F=\Pi _A(F_0\cdot \chi )$. Then $F\in A(\mathbf {C}^{d})$ and satisfies (3.3) for some $r_0\in \mathbf {R}^{d}_+$ such that $r_0<r$.

Theorem 3.8 follows from the following two propositions, where the first one concerns the case when $s>0$ and the second one make a more detailed explanation of the case $s=0$, i. e. the case of analytic polynomials.

Proposition 3.11

Let $s\in (0,\frac{1}{2})$, $r\in \mathbf {R}^{d}_+$, and let $\chi \in \mathcal {R}^\infty (\mathbf {C}^{d})$. Then the following is true:

• (1) Suppose $F\in A(\mathbf {C}^{d})$ satisfies

  $$\begin{aligned} |F(z)| \lesssim e^{r_{0,1} (\log \langle z_1\rangle )^{\frac{1}{1-2s}} +\cdots + r_{0,d} (\log \langle z_d\rangle )^{\frac{1}{1-2s}}} \end{aligned}$$

  (3.7)

  for some $r_0\in \mathbf {R}^{d}_+$ such that $r_0<r$. Then there is an $F_0\in A(\mathbf {C}^{d})$ such that $F=\Pi _A(F_0\cdot \chi )$ and

  $$\begin{aligned} |F_0(z)| \lesssim e^{r_{0,1} (\log \langle z_1\rangle )^{\frac{1}{1-2s}} +\cdots + r_{0,d} (\log \langle z_d\rangle )^{\frac{1}{1-2s}}} \end{aligned}$$

  (3.8)

  for some $r_0\in \mathbf {R}^{d}_+$ such that $r_0<r$;

• (2) Suppose $F_0\in A(\mathbf {C}^{d})$ satisfies (3.8) for some $r_0\in \mathbf {R}^{d}_+$ such that $r_0<r$, and let $F=\Pi _A(F_0\cdot \chi )$. Then $F\in A(\mathbf {C}^{d})$ and satisfies (3.7) for some $r_0\in \mathbf {R}^{d}_+$ such that $r_0<r$.

Proposition 3.12

Let $\chi \in \mathcal {R}^\infty (\mathbf {C}^{d})$ and let $N\ge 0$ be an integer. Then the following is true:

• (1) Suppose $F\in A(\mathbf {C}^{d})$ is given by

  $$\begin{aligned} F(z) = \sum _{|\alpha |\le N} c(F,\alpha ) z^\alpha , \end{aligned}$$

  (3.9)

  where $\{c(F,\alpha ) \}_{|\alpha |\le N} \subseteq \mathbf {C}$. Then there is an $F_0\in A(\mathbf {C}^{d})$ such that $F=\Pi _A(F_0\cdot \chi )$ and

  $$\begin{aligned} F_0(z) = \sum _{|\alpha |\le N} c(F_0,\alpha ) z^\alpha , \end{aligned}$$

  (3.10)

  where $\{ c(F_0,\alpha ) \}_{|\alpha |\le N} \subseteq \mathbf {C}$ and satisfies $c(F_0,\alpha )=0$ when $c(F,\alpha )=0$;

• (2) Suppose $F_0\in A(\mathbf {C}^{d})$ satisfies (3.10) for some $\{c(F_0,\alpha ) \}_{|\alpha |\le N} \subseteq \mathbf {C}$, and let $F=\Pi _A(F_0\cdot \chi )$. Then $F\in A(\mathbf {C}^{d})$ and satisfies (3.9) for some $\{ c(F,\alpha ) \}_{|\alpha |\le N} \subseteq \mathbf {C}$ such that $c(F,\alpha )=0$ when $c(F_0,\alpha )=0$.

3.2 Preparing results and their proofs

For the proofs of Propositions 3.9–3.11 and thereby of Theorems 3.2–3.8 we need some preparatory results. Because the proof of Proposition 3.9 needs some room, we put parts of the statement in the following separate proposition. At the same time we slightly refine some parts concerning the image of compactly supported elements in $L^\infty$ under the map $\Pi _A$.

Proposition 3.13

Let $s>1$, $r_0,r\in \mathbf {R}^{d}_+$ be such that $r_0<r$ and suppose that either

$$\begin{aligned} F_0\in \mathcal {E}_s'(D_{r_0}(0)), \quad F_0\in \mathcal {E}_{0,s}'(D_{r_0}(0)), \quad F_0\in \mathscr {E}'(D_{r_0}(0)) \quad \text {or}\quad F_0\in L^\infty (\overline{D_{r}(0)}). \end{aligned}$$

Then $F=\Pi _AF_0 \in A(\mathbf {C}^{d})$ and satisfies

$$\begin{aligned} |F(z)| \lesssim e^{r_1|z_1|+\cdots +r_d|z_d|}. \end{aligned}$$

Proof

By the inclusions

$$\begin{aligned} \mathscr {E}\hookrightarrow \mathcal {E}_{0,s+\varepsilon }' \hookrightarrow \mathcal {E}_s' \hookrightarrow \mathcal {E}_{0,s}' \end{aligned}$$

when $\varepsilon >0$, it suffices to consider the case when $F_0\in \mathcal {E}_s'(D_{r_0}(0))$ or $F_0\in L^\infty (\overline{D_{r}(0)})$ hold.

Let $r_2=r$. First suppose that $F_0\in \mathcal {E}_s'(D_{r_0}(0))$ holds, choose $r_1\in \mathbf {R}^{d}_+$ such that $r_0<r_1<r_2$, $\Psi (y,\eta )=e^{-(|y|^2+|\eta |^2)}$ and let $\Phi _z(y,\eta )=e^{(z,y+i\eta )}$. By identifying $\mathbf {C}^{d}$ with $\mathbf {R}^{2d}$ and using the fact that $F_0\in \mathcal {E}_s'(D_{r_0}(0))$ we obtain

$$\begin{aligned} |\Pi _AF_0(z)| = \pi ^{-d}|\langle F_0,\Phi _z\Psi \rangle | \lesssim \sup _{\alpha \in \mathbf {N}^{2d}} \left( \frac{\Vert D^\alpha (\Phi _z\Psi )\Vert _{L^\infty (D_{r_1}(0))}}{h_1^{|\alpha |}\alpha !^s} \right) \end{aligned}$$

(3.11)

for every $h_1>0$. We also have $\Psi \in \mathcal {E}_{1/2}(\mathbf {R}^{2d}) \hookrightarrow \mathcal {E}_{0,s}(\mathbf {R}^{2d})$, which implies

$$\begin{aligned} \Vert D^\alpha \Psi \Vert _{L^\infty (D_{r_1}(0))} \lesssim h_2^{|\alpha |}\alpha !^s \end{aligned}$$

for every $h_2>0$. Furthermore,

$$\begin{aligned} |D^\alpha \Phi _z(y,\eta )| = |m_\alpha (z) e^{(z,y+i\eta )}| \le |m_\alpha (z) |e^{r_{1,1}|z_1|+\cdots +r_{1,d}|z_d|},\quad y+i\eta \in D_{r_1}(0), \end{aligned}$$

where

$$\begin{aligned} m_\alpha (z)= \prod _{j=1}^dz_j^{\alpha _j+\alpha _{d+j}}, \quad z\in \mathbf {C}^{d},\ \alpha \in \mathbf {N}^{2d}. \end{aligned}$$

By choosing $h_1=4$ and $h_2=1$ above, and letting $y+i\eta \in D_{r_1}(0)$, Leibnitz rule gives

$$\begin{aligned}&4^{-|\alpha |}\alpha !^{-s} |D^\alpha (\Phi _z\Psi )(y,\eta )|\nonumber \\&\quad \le 4^{-|\alpha |}\alpha !^{-s}\sum _{\gamma \le \alpha } {\alpha \atopwithdelims ()\gamma }|D^\gamma \Phi _z(y,\eta )| \, |D^{\alpha -\gamma }\Psi (y,\eta )|\nonumber \\&\quad \lesssim 4^{-|\alpha |}\alpha !^{-s} \left( \sum _{\gamma \le \alpha } {\alpha \atopwithdelims ()\gamma } |m_\gamma (z) |(\alpha -\gamma )!^s \right) e^{r_{1,1}|z_1|+\cdots +r_{1,d}|z_d|}\nonumber \\&\quad \lesssim 4^{-|\alpha |} \left( \sum _{\gamma \le \alpha } {\alpha \atopwithdelims ()\gamma } |m_\gamma (z) |\gamma !^{-s} \right) e^{r_{1,1}|z_1|+\cdots +r_{1,d}|z_d|}\nonumber \\&\quad \le \sup _{\gamma \le \alpha } \left( \prod _{j=1}^d\frac{|z_j|^{\gamma _j}}{\gamma _j!^s} \cdot \frac{|z_j|^{\gamma _{d+j}}}{\gamma _{d+j}!^s} \right) e^{r_{1,1}|z_1|+\cdots +r_{1,d}|z_d|}. \end{aligned}$$

(3.12)

In the last inequality we have used that the number of terms in the sums are bounded by $2^{|\alpha |}$, and that ${n \atopwithdelims ()k}\le 2^k$ when n, k are non-negative integers such that $k\le n$.

By combining (3.12) with the estimate

$$\begin{aligned} \frac{|z_j|^{\gamma _j}}{\gamma _j!^s} = \left( \frac{(|z_j|^{\frac{1}{s}})^{\gamma _j}}{\gamma _j!} \right) ^s \le e^{s|z_j|^{\frac{1}{s}}}, \end{aligned}$$

we get

$$\begin{aligned} \sup _{\alpha \in \mathbf {N}^{2d}} \left( \frac{\Vert D^\alpha (\Phi _z\Psi )\Vert _{L^\infty (D_{r_1}(0))}}{h_1^{|\alpha |}\alpha !^s} \right)\lesssim & {} e^{2s(|z_1|^{\frac{1}{s}}+\cdots +|z_d|^{\frac{1}{s}})} e^{r_{1,1}|z_1|+\cdots +r_{1,d}|z_d|}\\\lesssim & {} e^{r_{2,1}|z_1|+\cdots +r_{2,d}|z_d|}. \end{aligned}$$

In the last inequality we have used the fact that $r_1<r_2$ and $s>1$. From the latter estimate and (3.11) we obtain

$$\begin{aligned} |\Pi _AF_0(z)|\lesssim e^{r_{2,1}|z_1|+\cdots +r_{2,d}|z_d|}, \end{aligned}$$

and the result follows when $F_0\in \mathcal {E}_s'(D_{r_0}(0))$.

Suppose instead that $F_0\in L^\infty (Q)$ holds, where $Q=\overline{D_{r_2}(0)}\subseteq \mathbf {C}^{d}$, and let $Q_j=\overline{D_{r_{2,j}}(0)}\subseteq \mathbf {C}$. Then

$$\begin{aligned} |\Pi _AF_0(z)|\lesssim & {} \int _{Q} |F_0(z)||e^{(z,w)}|\, d\lambda (w)\\\le & {} \Vert F_0\Vert _{L^\infty }\prod _{j=1}^d \left( \int _{Q_j} e^{|z_j||w_j|}\, d\lambda (w_j) \right) \lesssim e^{r_{2,1}|z_1|+\cdots +r_{2,d}|z_d|}, \end{aligned}$$

and the result follows in this case as well. $\square$

In the next lemma we give options on compactly supported functions which are mapped on the basic monomials, $e_\alpha$ by the operator $\Pi _A$.

Lemma 3.14

Let $t_1,t_2\in \mathbf {R}^{d}_+$ be such that $t_1\le t_2$, $\chi \in \mathcal {R}^\infty _{t_1,t_2}(\mathbf {C}^{d})$, and let $\chi _0$ be such that $\chi _0(|z_1|,\dots ,|z_d|)=\chi (z_1,\dots ,z_d)$. If

$$\begin{aligned} F_{\alpha ,\chi }(z) = \varsigma _\alpha z^\alpha \chi (z), \end{aligned}$$

with

$$\begin{aligned} \varsigma _\alpha = 2^{-d}\alpha !^{\frac{1}{2}} \left( \int _{\Delta _{t_2}} \chi _0(u)e^{-|u|^2}u^{2\alpha }u_1\cdots u_d\, du \right) ^{-1},\quad \Delta _t = \{ \, u\in \mathbf {R}^{d}_+\, ;\, u\le t\, \} , \end{aligned}$$

(3.13)

then the following is true:

• (1) $\Pi _AF_{\alpha ,\chi }=e_\alpha$;

• (2) for some constant $C>0$ which only depends on $\Vert \chi \Vert _{L^\infty }$, c in Definition 3.1 and the dimension d, it holds

  $$\begin{aligned} C^{-1} \left( \prod _{j=1}^d t_{2,j}^{-2}(\alpha _j+1) \right) t_2^{-2\alpha } \alpha !^{\frac{1}{2}} \le \varsigma _\alpha \le C e^{|t_2|^2} \left( \prod _{j=1}^d t_{1,j}^{-2}(\alpha _j+1) \right) t_1^{-2\alpha }\alpha !^{\frac{1}{2}}. \end{aligned}$$

  (3.14)

Proof

By using polar coordinates in each complex variable when integrating we get

$$\begin{aligned} (\Pi _AF_{\alpha ,\chi })(z)= & {} \pi ^{-d}\varsigma _\alpha \int _{\mathbf {C}^{d}}w^\alpha \chi (w) e^{(z,w)-|w|^2}\, d\lambda (w)\nonumber \\= & {} \pi ^{-d}\varsigma _\alpha \int _{\Delta _{t_2}} I_\alpha (u,z)u^\alpha \chi _0(u)e^{-|u|^2}u_1\cdots u_d\, du, \end{aligned}$$

(3.15)

where

$$\begin{aligned} I_\alpha (u,z) = \int _{[0,2\pi )^d}e^{i\langle \alpha ,\theta \rangle } \left( \prod _{j=1}^d e^{z_ju_je^{-i\theta _j}} \right) \, d\theta = \prod _{j=1}^d I_{\alpha _j}(u_j,z_j) \end{aligned}$$

(3.16)

with

$$\begin{aligned} I_{\alpha _j}(u_j,z_j) = \int _0^{2\pi }e^{i\alpha _j\theta _j}e^{z_ju_je^{-i\theta _j}}\, d\theta _j. \end{aligned}$$

By Taylor expansions we get

$$\begin{aligned} I_{\alpha _j}(u_j,z_j)= & {} \int _0^{2\pi }e^{i\alpha _j\theta _j} \left( \sum _{k=0}^\infty \frac{z_j^ku_j^ke^{-ik\theta _j}}{k!} \right) \, d\theta _j\\= & {} \sum _{k=0}^\infty \left( \left( \int _0^{2\pi }e^{i(\alpha _j-k)\theta _j} \, d\theta _j \right) \frac{z_j^ku_j^k}{k!} \right) = \frac{2\pi z_j^{\alpha _j}u_j^{\alpha _j}}{\alpha _j!} \end{aligned}$$

By inserting this into (3.15) and (3.16) we get

$$\begin{aligned} (\Pi _AF_{\alpha ,\chi })(z)= & {} \pi ^{-d}\varsigma _\alpha \int _{\Delta _{t_2}} (2\pi )^d\frac{u^\alpha z^\alpha }{\alpha !} u^\alpha \chi _0(u)e^{-|u|^2}u_1\cdots u_d\, du\\= & {} \left( 2^d \varsigma _\alpha \alpha !^{-\frac{1}{2}} \int _{\Delta _{t_2}} u^{2\alpha } \chi _0(u)e^{-|u|^2}u_1\cdots u_d\, du \right) e_\alpha (z) = e_\alpha (z) \end{aligned}$$

and (1) follows.

Since $\chi$ is non-negative and fullfills $\chi \ge c$ on $D_{t_1}(0)$ we get

$$\begin{aligned} \varsigma _\alpha\lesssim & {} e^{|t_2|^2}\alpha !^{\frac{1}{2}} \left( \int _{\Delta _{t_1}}u^{2\alpha }u_1\cdots u_2\, du \right) ^{-1}\\= & {} e^{|t_2|^2}\alpha !^{\frac{1}{2}} \prod _{j=1}^d \left( \frac{t_{1,j}^{2\alpha _j+2}}{2\alpha _j+2} \right) ^{-1} \asymp e^{|t_2|^2}\alpha !^{\frac{1}{2}} \left( \prod _{j=1}^d (t_{1,j}^{-2}(\alpha _j+1)) \right) t_1^{-2\alpha } \end{aligned}$$

which gives the right inequality in (3.14). By the support properties of $\chi$ we also have

$$\begin{aligned} \varsigma _\alpha \gtrsim \alpha !^{\frac{1}{2}} \left( \int _{\Delta _{t_2}}t^{2\alpha }u_1\cdots u_2\, du \right) ^{-1}, \end{aligned}$$

and the left inequality in (3.14) follows by similar arguments, and (2) follows. $\square$

The next lemma shows that we may estimate entire functions by different Lebesgue norms. We omit the proof, since the result follows from [11, Theorem 3.2].

Lemma 3.15

Suppose $s,\tau \in \mathbf {R}$ and $r,r_0\in \mathbf {R}^{d}_+$ are such that

$$\begin{aligned} s< \frac{1}{2},\quad \tau >-\frac{1}{2} \quad \text {and}\quad r_0<r. \end{aligned}$$

Let $p,q\in [1,\infty ]$, $F\in A(\mathbf {C}^{d})$ and set

$$\begin{aligned} M_{1,r}(z) = r_1|z_1|^{\frac{2}{2\tau +1}}+\cdots +r_d|z_d|^{\frac{2}{2\tau +1}} \end{aligned}$$

and

$$\begin{aligned} M_{2,r}(z) = r_1(\log \langle z_1\rangle )^{\frac{1}{1-2s}} +\cdots + r_d(\log \langle z_d\rangle )^{\frac{1}{1-2s}}. \end{aligned}$$

Then

$$\begin{aligned} \Vert F \cdot e^{-M_{j,r} }\Vert _{L^p(\mathbf {C}^{d} )} \lesssim \Vert F \cdot e^{-M_{j,r_0} }\Vert _{L^q(\mathbf {C}^{d} )} . \end{aligned}$$

The next lemma relates Lebesgue estimates of entire functions with estimates on corresponding Taylor coefficients. Here we let the Gamma function on $\mathbf {R}^{d}_+$ be defined by

$$\begin{aligned} \Gamma _{ d}(x_1,\dots ,x_d) = \prod _{j=1}^d\Gamma (x_j), \end{aligned}$$

where $\Gamma$ is the Gamma function on $\mathbf {R}_+$.

Lemma 3.16

Let $\tau > -\frac{1}{2}$, $r\in \mathbf{R}_{+}^{d}$, $M_{1,r}$ be the same as in Lemma 3.15, $\omega (z)= e^{\frac{1}{2}|z|^2-M_{1,r}(z)}$, $z\in \mathbf {C}^{d}$, and let $\alpha _0 = (1,\dots ,1)\in \mathbf {N}^{d}$. Also let

$$\begin{aligned} \vartheta _{r}(\alpha ) = \left( (2\tau +1) (2r)^{-(2\tau +1)(\alpha +\alpha _0)} \left( \frac{\Gamma \left( (2\tau +1)(\alpha +\alpha _0) \right) }{\alpha !} \right) \right) ^{\frac{1}{2}} . \end{aligned}$$

If $F \in A (\mathbf {C}^{d})$ is given by (2.13), then

$$\begin{aligned} \Vert F \cdot e^{-M_{1,r}}\Vert _{L^2(\mathbf {C}^{d})} = \pi ^{\frac{d}{2}}\left( \sum _{\alpha \in \mathbf {N}^{d}} |c(F,\alpha )\vartheta _{r}(\alpha )|^2 \right) ^{\frac{1}{2}}, \end{aligned}$$

and $\mathcal {A}_{[\vartheta _{r}]}^2(\mathbf {C}^{d}) = A^2_{(\omega )}(\mathbf {C}^{d})$ with equality in norms.

Proof

Since

$$\begin{aligned} e^{-M_{1, r}(z)}= \prod _{j=1}^de^{-r_j|z_j|^{\frac{2}{2\tau +1}}}, \end{aligned}$$

Lemma 2.7 shows that we may assume that $d=1$, giving that $r=r_1$ and $\alpha _0=1$.

In view of Theorem 2.6 we have $\mathcal {A}_{[\vartheta ]}^2(\mathbf {C}^{d})= A_{(\omega _1)}^2(\mathbf {C}^{d})$ with equality in norms, when

$$\begin{aligned} \vartheta (\alpha ) = \left( \frac{1}{\alpha !} \int _0^{\infty } e^{-2r t^{\frac{1}{2\tau +1}} } t^{\alpha } \, dt \right) ^{\frac{1}{2}}. \end{aligned}$$

By $u= 2rt^{\frac{1}{2\tau +1}}$ as new variables of integration we obtain

$$\begin{aligned} \vartheta (\alpha )&= \left( (2\tau +1) (2r)^{-(2\tau +1)(\alpha +1)} \frac{1}{\alpha !}\int _0^{\infty } e^{-u} u^{\alpha (2\tau +1)+2\tau } \, du \right) ^{\frac{1}{2}}\\&= \left( (2\tau +1) (2r)^{-(2\tau +1)(\alpha +1)} \left( \frac{\Gamma \left( (2\tau +1)(\alpha +1) \right) }{\alpha !} \right) \right) ^{\frac{1}{2}} , \end{aligned}$$

which implies that $\theta =\theta _{1,r}$, and the result follows. $\square$

We also need the following version of Stirling’s formula.

Lemma 3.17

Let $\alpha \ge 0$ be an integer and let $\tau >-\frac{1}{2}$. Then

$$\begin{aligned} \frac{\displaystyle {\Gamma \left( (2\tau +1)(\alpha +1) \right) }}{\alpha !}&\asymp \left( 2\tau +1 \right) ^{(2\tau +1) \cdot \alpha } (\alpha +1)^{\tau }\cdot \alpha !^{2\tau }. \end{aligned}$$

(3.17)

Lemma 3.17 follows by repeated applications of Stirling’s formula and the standard limit

$$\begin{aligned} \lim _{t\rightarrow \infty } \left( 1+\frac{x}{t} \right) ^{t} = e^x \end{aligned}$$

for every $x\in \mathbf {R}$. In order to be self-contained we present the arguments.

Proof

The result is obviously true for $\alpha =0$. For $\alpha \ge 1$ we have $\alpha ^{\tau } \asymp (\alpha +1)^{\tau }$. A combination of the the latter relations and Stirling’s formula gives

$$\begin{aligned}&\frac{\displaystyle {\Gamma \left( (2\tau +1)(\alpha +1) \right) }}{\alpha !} \asymp \frac{\displaystyle { \left( (2\tau +1) \alpha + 2\tau \right) ^{(2\tau +1) \cdot \alpha +2\tau +\frac{1}{2}}}}{\displaystyle {e^{\alpha + 2\tau (\alpha +1)}}} \cdot \frac{e^\alpha }{\alpha ^{\alpha + \frac{1}{2}}}\nonumber \\&\quad = \frac{\displaystyle { \left( 2\tau +1 \right) ^{(2\tau +1) \cdot \alpha + 2\tau +\frac{1}{2}} \left( 1 + \frac{2\tau }{(2\tau +1)\alpha } \right) ^{(2\tau +1) \cdot \alpha + 2\tau +\frac{1}{2}} \alpha ^{2\tau (\alpha +1)} }}{e^{2\tau (\alpha + 1)}}\nonumber \\&\quad \asymp \left( 2\tau +1 \right) ^{(2\tau +1) \cdot \alpha } \frac{\displaystyle { \alpha ^{2\tau (\alpha +1)} }}{e^{2\tau \alpha }} \asymp \left( 2\tau +1 \right) ^{(2\tau +1)\cdot \alpha } (\alpha +1)^{\tau }\alpha !^{2\tau }, \end{aligned}$$

(3.18)

and the result follows. $\square$

Proposition 3.9 essentially follows from the following lemma.

Lemma 3.18

Let $\tau >-\frac{1}{2}$ and $r_0,r\in \mathbf {R}^{d}_+$ be such that $r_0<r$. Then the following is true:

• (1) If

  $$\begin{aligned} |c(F,\alpha )| \lesssim \left( \frac{2 r_0}{2\tau +1} \right) ^{\frac{2\tau +1}{2} \cdot \alpha }\alpha !^{-\tau }, \end{aligned}$$

  (3.19)

  then

  $$\begin{aligned} |F(z)|\lesssim e^{r_1|z_1|^{\frac{2}{2\tau +1}} +\cdots + r_d|z_d| ^{\frac{2}{2\tau +1}}} ; \end{aligned}$$

  (3.20)

• (2) If

  $$\begin{aligned} |F(z)|\lesssim e^{r_{0,1}|z_1|^{\frac{2}{2\tau +1}} +\cdots + r_{0,d}|z_d|^{\frac{2}{2\tau +1}}} \end{aligned}$$

  (3.21)

  then

  $$\begin{aligned} |c(F,\alpha )| \lesssim \left( \frac{2 r}{2\tau +1} \right) ^{\frac{2\tau +1}{2} \cdot \alpha }\alpha !^{-\tau } \end{aligned}$$

  (3.22)

Proof

Let $\vartheta _{r}$ be the same as in Lemma 3.16 and $r_1\in \mathbf {R}^{d}_+$ be such that $r_0<r_1<r$. First we prove (1). Suppose that (3.19) holds and let $\alpha _0=(1,\dots ,1)\in \mathbf {N}^{d}$. Also let $M_{1,r}$ be the same as in Lemma 3.15. Then Lemmas 3.15 and 3.16 give

$$\begin{aligned} \Vert F\cdot e^{-M_{1,r}}\Vert _{L^\infty (\mathbf {C}^{d})}\lesssim & {} \Vert F\cdot e^{-M_{1,r_1}}\Vert _{L^2 (\mathbf {C}^{d})}\asymp \left( \sum _{\alpha \in \mathbf {N}^{d}} \left| c(F,\alpha ) \vartheta _{r_1}(\alpha ) \right| ^2 \right) ^{\frac{1}{2}}\\= & {} \left( \sum _{\alpha \in \mathbf {N}^{d}} \left| c(F,\alpha ) \vartheta _{r_0}(\alpha ) \left( \frac{r_0}{r_1} \right) ^{(\alpha +\alpha _0)\frac{2\tau +1}{2}} \right| ^2 \right) ^{\frac{1}{2}}\\\lesssim & {} \sup _{\alpha \in \mathbf {N}^{d}}\left| c(F,\alpha ) \vartheta _{r_0}(\alpha ) \prod _{j=1}^d(\alpha _j+1)^{-\frac{\tau }{2}} \right| \\ \asymp&\sup _{\alpha \in \mathbf {N}^{d}} \left( |c(F,\alpha )| \left( \frac{2\tau +1}{2r_0} \right) ^{\frac{2\tau +1}{2}\cdot \alpha } \alpha !^{\tau } \right) . \end{aligned}$$

Here the second inequality follows from the fact that

$$\begin{aligned} \sum _{\alpha \in \mathbf {N}^{d}} \left( \left( \prod _{j=1}^d (\alpha _j+1)^{\frac{\tau }{2}} \right) \left( \frac{r_0}{r_1} \right) ^{(2\tau +1) \cdot (\alpha +\alpha _0)} \right) \end{aligned}$$

is convergent since $r_0<r_1$, and the fifth relation follows from Lemma 3.17. This implies that (3.20) holds and (1) follows.

Next assume that (3.21) holds. By Lemma 3.15 we get

$$\begin{aligned} \Vert F\cdot e^{-M_{1,r_0}}\Vert _{L^\infty (\mathbf {C}^{d})}\gtrsim & {} \Vert F\cdot e^{-M_{1,r_1}}\Vert _{L^2 (\mathbf {C}^{d})}\\ \asymp&\left( \sum _{\alpha \in \mathbf {N}^{d}} \left| c(F,\alpha ) \vartheta _{r_1}(\alpha ) \right| ^2 \right) ^{\frac{1}{2}} \ge \sup _{\alpha \in \mathbf {N}^{d}} \left( \left| c(F,\alpha ) \right| \vartheta _{r_1}(\alpha ) \right) \\\gtrsim & {} \sup _{\alpha \in \mathbf {N}^{d}} \left( |c(F,\alpha ) | \left( \frac{2\tau +1}{2r_1} \right) ^{\frac{2\tau +1}{2}\alpha } \prod _{j=1}^d(\alpha _j+1)^{\frac{\tau }{2}} \alpha !^{\tau } \right) \\\gtrsim & {} \sup _{\alpha \in \mathbf {N}^{d}} \left( |c(F,\alpha ) | \left( \frac{2\tau +1}{2r} \right) ^{\frac{2\tau +1}{2}\alpha } \alpha !^{\tau } \right) , \end{aligned}$$

where the third inequality follows from Lemma 3.17. This gives (2). $\square$

Proposition 3.11 mainly follows from the following result.

Lemma 3.19

Let $r,r_0\in \mathbf {R}^{d}_+$ be such that $r_0<r$, $s\in (0,\frac{1}{2})$ and let $F\in A(\mathbf {C}^{d})$. Then the following is true:

• (1) if (3.7) holds, then

  $$\begin{aligned} |c(F,\alpha )|\lesssim e^{-(R_{1}|\alpha _1|^{\frac{1}{2s}} +\cdots + R_{d}|\alpha _d|^{\frac{1}{2s}})},\quad R=s \left( \frac{1-2s}{r} \right) ^{\frac{1-2s}{2s}} ; \end{aligned}$$

  (3.23)

• (2) if $R_0\in \mathbf {R}^{d}_+$ is given by

  $$\begin{aligned} |c(F,\alpha )|\lesssim e^{-(R_{0,1}|\alpha _1|^{\frac{1}{2s}} +\cdots + R_{0,d}|\alpha _d|^{\frac{1}{2s}})},\quad R_0=s \left( \frac{1-2s}{r_0} \right) ^{\frac{1-2s}{2s}}, \end{aligned}$$

  (3.24)

  then (3.7) holds with r in place of $r_0$.

Proof

Let $r_1,r_2,r_3\in \mathbf {R}^{d}_+$ be such that $r_1<r<r_2<r_3$. By Lemma 2.7 we may assume that $d=1$, and by Lemma 3.15 the result follows if we prove

$$\begin{aligned} \sup _{\alpha \in \mathbf {N}} \left( |c(F,\alpha ) e^{R_{3}\cdot \alpha ^{\frac{1}{2s}}}|^2 \right)\lesssim & {} \int _{\mathbf {C}}|F(z) e^{-r(\log \langle z\rangle )^{\frac{1}{1-2s}}}|^2\, d\lambda (z)\nonumber \\\lesssim & {} \sup _{\alpha \in \mathbf {N}} \left( |c(F,\alpha ) e^{R_{1}\cdot \alpha ^{\frac{1}{2s}}}|^2 \right) , \end{aligned}$$

(3.25)

where $R_1,R_2,R_3\in \mathbf {R}_+$ satisfy

$$\begin{aligned} R_j = \left( \frac{1-2s}{r_j} \right) ^{\frac{1-2s}{2s}},\quad j=1,2,3. \end{aligned}$$

By Theorem 2.6 we have

$$\begin{aligned} \int _{\mathbf {C}}|F(z) e^{-r(\log \langle z\rangle )^{\frac{1}{1-2s}}}|^2\, d\lambda (z) \asymp \sum _{\alpha \in \mathbf {N}} |c(F,\alpha )\vartheta _{r}(\alpha )|^2, \end{aligned}$$

where

$$\begin{aligned} \vartheta _r(\alpha )= \left( \frac{\pi }{2\alpha !} \int _0^\infty e^{-r(\log \langle t\rangle )^{\frac{1}{1-2s}}}t^\alpha \, dt \right) ^{\frac{1}{2}}. \end{aligned}$$

Let

$$\begin{aligned} \theta = \frac{1}{1-2s} >1\quad \text {and}\quad g_{r,\alpha } (t) = e^{-r(\log t)^\theta }t^\alpha . \end{aligned}$$

In order to prove (3.25), we need to show that

$$\begin{aligned} e^{R_2\alpha ^{\frac{1}{2s}}} \lesssim \vartheta _{r}(\alpha ) \lesssim e^{R_1\alpha ^{\frac{1}{2s}}}, \end{aligned}$$

(3.26)

which shall be reached by modifying the proof of (15) in [4].

We have

$$\begin{aligned} \vartheta _{r}(\alpha )^2\lesssim & {} \int _e^\infty e^{(r_1-r)(\log t)^\theta }g_{r_1,\alpha } (t)\, dt\\\lesssim & {} \sup _{t\ge e}(g_{r_1,\alpha } (t)) \int _e^\infty e^{(r_1-r)(\log t)^\theta }\, dt \asymp \sup _{t\ge e}(g_{r_1,\alpha } (t)). \end{aligned}$$

By straight-forward computations it follows that $g_{r,\alpha } (t)$ attains its global maximum for

$$\begin{aligned} t_{r,\alpha } = \exp \left( \left( \frac{\alpha }{\theta r} \right) ^{\frac{1-2s}{2s}} \right) , \end{aligned}$$

(3.27)

and that

$$\begin{aligned} g_{r_1,\alpha }(t_{r_1,\alpha } ) = \exp \left( {r_1^{1-\frac{1}{2s}}}{\theta ^{-\frac{1}{2s}} (\theta -1)\alpha ^{\frac{1}{2s}}} \right) = e^{2R_1\cdot \alpha ^{\frac{1}{2s}}}, \end{aligned}$$

and the second inequality in (3.26) follows.

In order to prove the first inequality in (3.26), we claim that for some c which is independent of $\alpha$ we have

$$\begin{aligned} \left( 1-\frac{c}{t_{r_2,\alpha }} \right) ^\alpha \gtrsim e^{(r-r_2)(\log t_{r_2,\alpha })^\theta }. \end{aligned}$$

(3.28)

In fact, by (3.27) it follows that

$$\begin{aligned} \mathop {\mathrm{lim}}\limits _{\alpha \rightarrow \infty }\alpha ^{-1}t_{r_{2,\alpha }}=\infty . \end{aligned}$$

This together with the fact that $r<r_2$ give

$$\begin{aligned} \mathop {\mathrm{lim}}\limits _{\alpha \rightarrow \infty }\left( 1-\frac{c}{t_{r_{2,\alpha }}}\right) ^\alpha =\mathop {\mathrm{lim}}\limits _{\alpha \rightarrow \infty }\left( \left( 1-\frac{c}{t_{r_{2,\alpha }}}\right) ^{t_{r_{2,\alpha }}}\right) ^{\frac{\alpha }{t_{r_{2,\alpha }}}}=(e^{-c})^0=1 \end{aligned}$$

and

$$\begin{aligned} \mathop {\mathrm{lim}}\limits _{\alpha \rightarrow \infty }e^{(r-r_2)((1-2s)\alpha /r_2)^\frac{1}{2s}}=0,\end{aligned}$$

and (3.28) follows.

By (3.28) and the fact that $\frac{1}{2s}>1$ we get

$$\begin{aligned} \vartheta _r(\alpha )^2\gtrsim & {} \frac{1}{\alpha !} \int _{t_{r_2,\alpha }-c}^{t_{r_2,\alpha }}e^{-r(\log t)^\theta }t^\alpha \, dt \gtrsim \frac{1}{\alpha !}e^{-r(\log t_{r_2,\alpha })^\theta } (t_{r_2,\alpha }-c)^\alpha \\\gtrsim & {} \frac{1}{\alpha !}e^{-r_2(\log t_{r_2,\alpha })^\theta } t_{r_2,\alpha }^\alpha = \frac{1}{\alpha !}e^{2R_2\cdot \alpha ^{\frac{1}{2s}}} \gtrsim e^{2R_3\cdot \alpha ^{\frac{1}{2s}}}. \end{aligned}$$

This gives the result. $\square$

3.3 Proofs of main results

Next we prove Proposition 3.9 and thereby Theorem 3.2.

Proof of Proposition 3.9

It is clear that (2) $\Rightarrow$ (3) $\Rightarrow$ (4) $\Rightarrow$ (5). By Proposition 3.13 it follows that (5) implies (1). We need to prove that (1) implies (2).

Suppose (1) holds and let $r_4=r$. Choose $r_1,r_2,r_3\in \mathbf {R}^{d}_+$ such that $r_0<r_1< r_2\le r_3<r_4$ and $r_1r_4<r_2^2$, $\chi \in \mathcal {R}_{r_2,r_3}^\infty (\mathbf {C}^{d})$, and let $F_{\alpha ,\chi }$ be as in Lemma 3.14 with $r_2$ and $r_3$ in place of $r_1$ and $r_2$. If $\tau = 1$, then Lemma 3.18 (2) gives

$$\begin{aligned} |c(F,\alpha )| \lesssim r_1^\alpha \alpha !^{-\frac{1}{2}}. \end{aligned}$$

(3.29)

Let

$$\begin{aligned} F_0(z) = \sum _{\alpha \in \mathbf {N}^{d}}c(F,\alpha ) \varsigma _\alpha z^\alpha . \end{aligned}$$

(3.30)

We claim that the series in (3.30) is uniformly convergent with respect to z in $\overline{D_R(0)}$.

In fact, if $|z_j|\le r_{4,j}$, then (3.14) gives

$$\begin{aligned} |c(F,\alpha )\varsigma _\alpha z^\alpha |\lesssim \langle \alpha \rangle ^d r_2^{-2\alpha } r_1^\alpha r_4^\alpha = \langle \alpha \rangle ^d \rho ^\alpha , \end{aligned}$$

where $\rho \in \mathbf {R}^{d}_+$ satisfies $\rho _j= \frac{r_{1,j}r_{4,j}}{r_{2,j}^2}<1$. Since $\sum _{\alpha \in \mathbf {N}^{d}}\langle \alpha \rangle ^d\rho ^\alpha$ is convergent, Weierstrass theorem shows that (3.30) is uniformly convergent and defines an analytic function in $\overline{D_{r_{4,j}}(0)}$. Hence, $F_0\in A(\overline{D_{r_{4,j}}(0)})$. Furthermore, by the support of $\chi$ we get

$$\begin{aligned} \Pi _A(F_0\cdot \chi )= & {} \Pi _A \left( \sum _{\alpha \in \mathbf {N}^{d}} c(F,\alpha )\varsigma _\alpha z^\alpha \cdot \chi \right) \nonumber \\= & {} \sum _{\alpha \in \mathbf {N}^{d}} c(F,\alpha )\varsigma _\alpha \Pi _A (z^\alpha \cdot \chi ) = \sum _{\alpha \in \mathbf {N}^{d}}c(F,\alpha )e_\alpha = F. \end{aligned}$$

(3.31)

Hence (2) holds, and the proof is complete. $\square$

For future references we observe that if $\varsigma _\alpha$ and $\chi$ are the same as in Lemma 3.14, then (3.30) shows that the relationship between $c(F,\alpha )$ and $c(F_0,\alpha )$ is given by

$$\begin{aligned} c(F_0,\alpha ) = c(F,\alpha )\varsigma _\alpha \alpha !^{\frac{1}{2}}. \end{aligned}$$

(3.32)

Proof of Theorem 3.4

The equivalence between (1) and (2) is clear. It is also obvious that (3) implies (2) in view of Proposition 3.9. We shall prove the equivalence between (1) and (3).

Suppose (1) holds. Then (3.29) holds for every $r_1\in \mathbf {R}^{d}_+$. Let $r,R\in \mathbf {R}^{d}_+$ be chosen such that $\chi \in \mathcal {R}_{r,R}^\infty (\mathbf {C}^{d})$, $F_{\alpha ,\chi }$ be as in Lemma 3.14 and let $F_0$ be given by (3.30). By (3.29) we have

$$\begin{aligned} |c(F,\alpha ) \varsigma _\alpha | \lesssim \langle \alpha \rangle ^dr^{-2\alpha }r_1^{\alpha } \end{aligned}$$

for every $r_1\in \mathbf {R}^{d}_+$, giving that

$$\begin{aligned} |c(F,\alpha ) \varsigma _\alpha | \lesssim r_0^\alpha \end{aligned}$$

for every $r_0\in \mathbf {R}^{d}_+$. This implies that the series in (3.30) is locally uniformly convergent with respect to z and defines an entire function on $\mathbf {C}^{d}$. Hence $F_0\in A(\mathbf {C}^{d})$. Moreover, by (3.31) it follows that $\Pi _A(F_0\chi )=F$, and we have proved that (1) implies (3).

Next suppose that (3) holds. Then

$$\begin{aligned} |c(F_0,\alpha )|\lesssim r^\alpha \alpha !^{\frac{1}{2}} \end{aligned}$$

for every $r\in \mathbf {R}^{d}_+$. By (3.14) and (3.32) we get

$$\begin{aligned} |c(F,\alpha )|= |c(F_0,\alpha )| \varsigma _\alpha ^{-1}\alpha !^{-\frac{1}{2}} \lesssim r^\alpha \varsigma _\alpha ^{-1} \lesssim r^{\alpha }R^{2\alpha }\alpha !^{-\frac{1}{2}}. \end{aligned}$$

Since $r\in \mathbf {R}^{d}_+$ can be chosen arbitrarily small we get

$$\begin{aligned} |c(F,\alpha )|\lesssim r_0^\alpha \alpha !^{-\frac{1}{2}} \end{aligned}$$

for every $r_0\in \mathbf {R}^{d}_+$. This implies that $F\in \mathcal {A}_{0,\flat _1}(\mathbf {C}^{d})$. That is, we have proved that (3) implies (1), and the result follows. $\square$

Next we prove Proposition 3.10.

Proof of Proposition 3.10

Suppose (3.3) holds, and let $r_1,r_2\in \mathbf {R}^{d}_+$ be such that $r_0<r_1<r_2<r$. Then Lemma 3.18 gives

$$\begin{aligned} |c(F,\alpha )| \lesssim \left( \frac{2r_1}{2\tau +1} \right) ^{\frac{2\tau +1}{2}\cdot \alpha } \alpha !^{-\tau }. \end{aligned}$$

By Lemma 3.14 and (3.32) we get

$$\begin{aligned} |c(F_0,\alpha )|\lesssim & {} \left( \frac{2r_1}{2\tau +1} \right) ^{\frac{2\tau +1}{2}\cdot \alpha } \alpha !^{-\tau } \langle \alpha \rangle ^d\alpha ! t_1^{-2\alpha }\\\lesssim & {} \left( \frac{2r_2}{2\tau +1} \right) ^{\frac{2\tau +1}{2}\cdot \alpha } \alpha !^{-(\tau -1)} t_1^{-2\alpha } = \left( \frac{2R_2}{2\tau -1} \right) ^{\frac{2\tau -1}{2}\cdot \alpha } \alpha !^{-(\tau -1)}, \end{aligned}$$

when

$$\begin{aligned} R_2 = \frac{2\tau -1}{2} \left( \frac{2r_2}{2\tau +1} \right) ^{\frac{2\tau +1}{2\tau -1}}t_1^{-\frac{4}{2\tau -1}}. \end{aligned}$$

This proves (1).

Suppose instead that $r_0,R_0,r_1,r_2\in \mathbf {R}^{d}_+$ are such that (3.6) hold and $r_0<r_1<r_2<r$, and that $F_0\in A(\mathbf {C}^{d})$ satisfies (3.4). Also let $F=\Pi _A(F_0\cdot \chi )$. Then

$$\begin{aligned} \left( \frac{2R_0}{2\tau -1} \right) ^{\frac{2\tau -1}{2}} < \left( \frac{2r_1}{2\tau +1} \right) ^{\frac{2\tau +1}{2}}t_2^{-2}. \end{aligned}$$

By combining the latter estimate with Lemma 3.18 we get

$$\begin{aligned} |c(F_0,\alpha )| \lesssim \left( \frac{2r_1}{2\tau +1} \right) ^{\frac{2\tau +1}{2}\cdot \alpha }t_2^{-2\alpha }\alpha !^{1-\tau } \end{aligned}$$

Hence Lemma 3.14 and (3.32) give

$$\begin{aligned} |c(F,\alpha )| \lesssim \left( \frac{2r_1}{2\tau +1} \right) ^{\frac{2\tau +1}{2}\cdot \alpha }\alpha !^{-\tau } \end{aligned}$$

and Lemma 3.18 again implies that (3.3) holds with $r_2$ in place of $r_0$. This gives the result. $\square$

Proof of Theorems 3.5–3.7

First suppose $\sigma \in (\frac{1}{2},1)$, and let $\sigma _0 =\frac{\sigma }{2\sigma -1}$ and $\tau =\frac{1}{2\sigma }$. Then

$$\begin{aligned} \frac{2\sigma }{\sigma +1} = \frac{2}{2\tau +1} \quad \text {and}\quad \frac{2\sigma _0}{\sigma _0 -1} = \frac{2}{2\tau -1}. \end{aligned}$$

Theorem 3.5 now follows from these observations and Proposition 3.10 in the case $\tau \in (\frac{1}{2},1)$.

In the same way, Theorem 3.6 follows by choosing $\tau =1$ in Proposition 3.10.

Finally, suppose $\sigma \in (0,\frac{1}{2})$, and let $\sigma _0 =\frac{\sigma }{1-2\sigma }$ and $\tau =\frac{1}{2\sigma }$. Then

$$\begin{aligned} \frac{2\sigma }{\sigma +1} = \frac{2}{2\tau +1} \quad \text {and}\quad \frac{2\sigma _0}{\sigma _0 +1} = \frac{2}{2\tau -1}, \end{aligned}$$

and Theorem 3.7 follows from these observations and Proposition 3.10 in the case $\tau >1$. $\square$

Next we prove Propositions 3.11 and 3.12 and thereby Theorem 3.8.

Proof of Propositions 3.11 and 3.12

Let $r,r_j,R_j\in \mathbf {R}^{d}_+$, $j=0,1,2,3$, be such that

$$\begin{aligned} r_0<r_1<r_2<r_3<r\quad \text {and}\quad R_j=s \left( \frac{1-2s}{r_j} \right) ^{\frac{1-2s}{2s}}. \end{aligned}$$

First suppose that $F\in A(\mathbf {C}^{d})$ satisfies (3.7) and let $F_0$ be the formal power series expansion with coefficients given by (3.32). Then $F=\Pi _A(F_0\cdot \chi )$.

By Lemma 3.19 we get

$$\begin{aligned} |c(F,\alpha )|\lesssim e^{-(R_{1,1}|\alpha _1|^{\frac{1}{2s}} +\cdots + R_{1,d}|\alpha _d|^{\frac{1}{2s}})}. \end{aligned}$$

Hence Lemma 3.14 and (3.32) give

$$\begin{aligned} |c(F_0,\alpha )|\lesssim & {} \alpha ! t_1^{-2\alpha } e^{-(R_{1,1}|\alpha _1|^{\frac{1}{2s}} +\cdots + R_{1,d}|\alpha _d|^{\frac{1}{2s}})}\\\lesssim & {} e^{-(R_{2,1}|\alpha _1|^{\frac{1}{2s}} +\cdots + R_{2,d}|\alpha _d|^{\frac{1}{2s}})}. \end{aligned}$$

In the last inequality we have used the fact that $s<\frac{1}{2}$, which implies that $R_2<R_1$ and

$$\begin{aligned} t_1^{-2\alpha }\alpha ! \lesssim e^{(R_{1,1}-R_{2,1})|\alpha _1|^{\frac{1}{2s}} +\cdots + (R_{1,d}-R_{2,d})|\alpha _d|^{\frac{1}{2s}}}. \end{aligned}$$

By applying Lemma 3.19 again it follows that $F_0$ satisfies (3.8) with $r_3$ in place of $r_0$. This gives (1).

Suppose instead that $F_0\in A(\mathbf {C}^{d})$ satisfies (3.8) and let $F=\Pi _A(F_0\cdot \chi )$. Then Lemma 3.19 gives

$$\begin{aligned} |c(F_0,\alpha )| \lesssim e^{-(R_{1,1}|\alpha _1|^{\frac{1}{2s}} +\cdots + R_{1,d}|\alpha _d|^{\frac{1}{2s}})}, \end{aligned}$$

and it follows from Lemma 3.14 and (3.32) that

$$\begin{aligned} |c(F,\alpha )|\lesssim & {} \alpha !^{-1}t_2^{2\alpha } e^{-(R_{1,1}|\alpha _1|^{\frac{1}{2s}} +\cdots + R_{1,d}|\alpha _d|^{\frac{1}{2s}})}\\\lesssim & {} e^{-(R_{1,1}|\alpha _1|^{\frac{1}{2s}} +\cdots + R_{1,d}|\alpha _d|^{\frac{1}{2s}})}. \end{aligned}$$

By applying Lemma 3.19 again we deduce (3.7) with $r_2$ in place of $r_0$, and Proposition 3.11 follows.

Proposition 3.12 is a straight-forward consequence of (3.32). The details are left for the reader. $\square$

4 Characterizations of Pilipović spaces

In this section we combine Lemma 2.10 with Theorems 3.2–3.8 to get characterizations of Pilipović spaces.

We begin with the following characterization of $\mathcal {H}_{\flat _1}$. The result is a straight-forward combination of Lemma 2.10 and Theorem 3.2. The details are left for the reader.

Proposition 4.1

Let $\phi$ be as in (2.7), $r\in \mathbf {R}^{d}_+$, $\chi _r$ be the characteristic function for $\overline{D_r(0)}$ and let $s>0$. Then the following conditions are equivalent:

• (1) $f\in \mathcal {H}_{\flat _1}(\mathbf {R}^{d})$;

• (2) $f=V_\phi ^*F$ for some $F\in \mathcal {E}_s'(\mathbf {R}^{2d})$;

• (3) $f=V_\phi ^*F$ for some $F\in \mathscr {E}'(\mathbf {R}^{2d}) \cap L^\infty (\mathbf {R}^{2d})$;

• (4) $f=V_\phi ^*F$ for some $F\in \mathscr {E}'(\mathbf {R}^{2d}) \cap L^\infty (\mathbf {R}^{2d})$ which satisfies

  $$\begin{aligned} F(x,\xi )= F_0(x-i\xi )e^{-i\langle x,\xi \rangle }\chi (x,\xi ) \end{aligned}$$

  (4.1)

  for some $r\in \mathbf {R}^{d}_+$, $\chi =\chi _r$ and $F_0\in A(\overline{D_r(0)})$.

Remark 4.2

It is clear that $\chi$ in Proposition 4.1 can be chosen as any $\chi \in \mathcal {R}^\infty (\mathbf {C}^{d})$ with suitable support properties.

Remark 4.3

In (2.32) there is a factor $e^{\frac{1}{2}(|x|^2+|\xi |^2)}$ which is absent in (4.1). We notice that this factor is not needed in (4.1) because $\mathcal {R}^\infty _{t_1,t_2}(\mathbf {C}^{d})$ is invariant under multiplications of such functions.

The next results follow from Lemma 2.10 and Theorems 3.4–3.8. The details are left for the reader.

Proposition 4.4

Let $\phi$ be as in (2.7) and $\chi \in \mathcal {R}^\infty (\mathbf {C}^{d})$. Then the following conditions are equivalent:

• (1) $f\in \mathcal {H}_{0,\flat _1}(\mathbf {R}^{d})$;

• (2) $f=V_\phi ^*F$ for some $F\in \mathscr {E}'(\mathbf {R}^{2d}) \cap L^\infty (\mathbf {R}^{2d})$ which satisfies (4.1) for some $F_0\in A(\mathbf {C}^{d})$.

Proposition 4.5

Let $\phi$ be as in (2.7), $\chi \in \mathcal {R}^\infty (\mathbf {C}^{d})$, $\sigma \in (\frac{1}{2},1)$ and let

$$\begin{aligned} \sigma _0 = \frac{\sigma }{2\sigma -1}. \end{aligned}$$

Then the following conditions are equivalent:

• (1) $f\in \mathcal {H}_{\flat _\sigma }(\mathbf {R}^{d})\ ( f\in \mathcal {H}_{0,\flat _\sigma }(\mathbf {R}^{d}))$;

• (2) $f=V_\phi ^*F$ for some $F\in \mathscr {E}'(\mathbf {R}^{2d}) \cap L^\infty (\mathbf {R}^{2d})$ which satisfies (4.1) for some $F_0\in \mathcal {A}_{0,\flat _{\sigma _0}}'(\mathbf {C}^{d})\ (F_0\in \mathcal {A}_{\flat _{\sigma _0}}'(\mathbf {C}^{d}))$.

Proposition 4.6

Let $\phi$ be as in (2.7), $\chi \in \mathcal {R}^\infty (\mathbf {C}^{d})$ and let $\sigma =\frac{1}{2}$. Then the following conditions are equivalent:

• (1) $f\in \mathcal {H}_{\flat _{\sigma }}(\mathbf {R}^{d})\ (f\in \mathcal {H}_{0,\flat _\sigma }(\mathbf {R}^{d}))$;

• (2) $f=V_\phi ^*F$ for some $F\in \mathscr {E}'(\mathbf {R}^{2d}) \cap L^\infty (\mathbf {R}^{2d})$ which satisfies (4.1) for some $F_0\in \mathcal {A}_{0,1/2}'(\mathbf {C}^{d})\ (F_0\in \mathcal {A}_{0,1/2}(\mathbf {C}^{d}))$.

Proposition 4.7

Let $\phi$ be as in (2.7), $\chi \in \mathcal {R}^\infty (\mathbf {C}^{d})$, $\sigma \in (0,\frac{1}{2})$ and let

$$\begin{aligned} \sigma _0 = \frac{\sigma }{1-2\sigma }. \end{aligned}$$

Then the following conditions are equivalent:

• (1) $f\in \mathcal {H}_{\flat _\sigma }(\mathbf {R}^{d})\ (f\in \mathcal {H}_{0,\flat _\sigma }(\mathbf {R}^{d}))$;

• (2) $f=V_\phi ^*F$ for some $F\in \mathscr {E}'(\mathbf {R}^{2d}) \cap L^\infty (\mathbf {R}^{2d})$ which satisfies (4.1) for some $F_0\in \mathcal {A}_{\flat _{\sigma _0}}(\mathbf {C}^{d})\ (F_0\in \mathcal {A}_{0,\flat _{\sigma _0}}(\mathbf {C}^{d}))$.

Proposition 4.8

Let $\phi$ be as in (2.7), $\chi \in \mathcal {R}^\infty (\mathbf {C}^{d})$ and let $s \in (0,\frac{1}{2})$. Then the following conditions are equivalent:

• (1) $f\in \mathcal {H}_{s}(\mathbf {R}^{d})\ (f\in \mathcal {H}_{s}(\mathbf {R}^{d}))$;

• (2) $f=V_\phi ^*F$ for some $F\in \mathscr {E}'(\mathbf {R}^{2d}) \cap L^\infty (\mathbf {R}^{2d})$ which satisfies (4.1) for some $F_0\in \mathcal {A}_{s}(\mathbf {C}^{d})\ (F_0\in \mathcal {A}_{0,s}(\mathbf {C}^{d}))$.