Nuclearity of rapidly decreasing ultradifferentiable functions and time-frequency analysis

TL;DR

This paper demonstrates the nuclearity of certain ultradifferentiable function spaces using time-frequency analysis, establishing conditions under which these spaces are nuclear based on weight functions and Komatsu's conditions.

Contribution

It introduces new nuclearity results for ultradifferentiable function spaces using time-frequency analysis techniques, linking nuclearity to specific growth conditions.

Findings

The space $\

$ ext{S}_ ext{ω}$ is nuclear for all weight functions with $ω(t)=o(t)$.

The space $ ext{S}_{(M_p)}$ is nuclear if and only if condition $(M2)'$ holds.

Abstract

We use techniques from time-frequency analysis to show that the space $\mathcal S_\omega$ of rapidly decreasing $\omega$-ultradifferentiable functions is nuclear for every weight function $\omega(t)=o(t)$ as $t$ tends to infinity. Moreover, we prove that, for a sequence $(M_p)_p$ satisfying the classical condition $(M1)$ of Komatsu, the space of Beurling type $\mathcal S_{(M_p)}$ when defined with $L^{2}\,$norms is nuclear exactly when condition $(M2)'$ of Komatsu holds.

Equations

$$\underset{\sigma \in {\alpha }_0{\mathbb{Z}}^d\times {\beta }_0{\mathbb{Z}}^d}{\sup }{e}^{\lambda \omega (\sigma )}|V_{\phi }u(\sigma )|<+\infty ,\text{mbox}forall\lambda >0,$$

$${\tilde{\Lambda }}_{\omega }:=\{c=(c_{\sigma }{)}_{\sigma \in {\alpha }_0{\mathbb{Z}}^d\times {\beta }_0{\mathbb{Z}}^d}:\Vert c{\Vert }_k:=\underset{\sigma \in {\alpha }_0{\mathbb{Z}}^d\times {\beta }_0{\mathbb{Z}}^d}{\sup }|c_{\sigma }|e^{k\omega (\sigma )}<+\infty ,\forall k\in \mathbb{N}\}.$$

$$\displaystyle\Lambda_{\omega}:=\Big{\{}(c_{k})_{k\in\mathbb{N}_{0}}:\ \sup_{k\in\mathbb{N}_{0}}|c_{k}|e^{j\omega(k^{1/2})}<+\infty,\ \forall\,j\in\mathbb{N}_{0}\Big{\}}.$$

$$\displaystyle\qquad S_{(M_{p})}(\mathbb{R}^{d}):=\Big{\{}f\in C^{\infty}(\mathbb{R}^{d}):\ \sup_{\alpha,\beta\in\mathbb{N}_{0}^{d}}\sup_{x\in\mathbb{R}^{d}}\frac{j^{|\alpha+\beta|}}{M_{|\alpha+\beta|}}\|x^{\alpha}\partial^{\beta}f(x)\|_{\infty}<+\infty,\ \forall\,j\in\mathbb{N}\Big{\}},$$

$$s^{s/2}{M}_p\le B{C}^s{H}^{s+p}{M}_{s+p},\forall s,p\in {\mathbb{N}}_0$$

$$\exists A,H>0:M_{p+1}\le A{H}^p{M}_p,\forall p\in {\mathbb{N}}_0.$$

$$M_p^2\le M_{p-1}{M}_{p+1},\forall p\in \mathbb{N},$$

$$\omega (t_1+t_2)\le L(\omega (t_1)+\omega (t_2)+1),t_1,t_2\ge 0.$$

$${\phi }_{\omega }^{\ast }(s):=\underset{t\ge 0}{\sup }\{ts-{\phi }_{\omega }(t)\}.$$

$$\mathcal{F}(u)(\xi )=\hat{u}(\xi ):={\int }_{{\mathbb{R}}^d}u(x)e^{-i⟨x,\xi ⟩}dx,\xi \in {\mathbb{R}}^d,$$

$$\underset{x\in {\mathbb{R}}^d}{\sup }{e}^{\lambda \omega (x)}|{\partial }^{\alpha }u(x)|<+\infty \text{mbox}and\underset{\xi \in {\mathbb{R}}^d}{\sup }{e}^{\lambda \omega (\xi )}|{\partial }^{\alpha }\hat{u}(\xi )|<+\infty .$$

$$T_{x}f(y)=f(y-x),M_{\xi }f(y)=e^{i⟨y,\xi ⟩}f(y)$$

$$\Pi (z)f(y)=M_{\xi }{T}_{x}f(y)=e^{i⟨y,\xi ⟩}f(y-x)$$

$$V_{\phi }f(z):=$$

$$=$$

$$m_{\lambda }(z)=e^{\lambda \omega (z)},z\in {\mathbb{R}}^{2d},$$

$$\displaystyle L^{p,q}_{m_{\lambda}}(\mathbb{R}^{2d}):=\Big{\{}$$

$$\displaystyle L^{\infty,q}_{m_{\lambda}}(\mathbb{R}^{2d}):=\Big{\{}$$

$$\displaystyle L^{p,\infty}_{m_{\lambda}}(\mathbb{R}^{2d}):=\Big{\{}$$

$$\omega (t)=o(\lambda (t)),\text{mbox}ast\to +\infty ,$$

$$\lambda (t)=o({\omega }_0(t))=o(t),\text{mbox}ast\to +\infty .$$

$$C_{{\phi }_0}f:=⟨f,\Pi (\sigma ){\phi }_0⟩,\sigma \in \Lambda ,$$

$$D_{{\psi }_0}c=\sum _{k,n\in {\mathbb{Z}}^d}{c}_{k,n}\Pi ({\alpha }_{0}k,{\beta }_{0}n){\psi }_0.$$

$$D_{{\psi }_0}{C}_{{\phi }_0}=\mathrm{Id},\text{mbox}theidentityon{L}^2({\mathbb{R}}^d),$$

$$D_{{\psi }_0}{C}_{{\phi }_0}=\mathrm{Id},\text{mbox}on{\mathcal{S}}_{\omega }({\mathbb{R}}^d)\subset L^2({\mathbb{R}}^d).$$

$$\Vert a{\Vert }_{{\ell }_{m_{\lambda }}^{p,q}}:={\left(\sum _{n\in {\mathbb{Z}}^d}{\left(\sum _{k\in {\mathbb{Z}}^d}|a_{kn}{|}^p{m}_{\lambda }(k,n{)}^p\right)}^{q/p}\right)}^{1/q}<+\infty ,$$

$$\Vert a{\Vert }_{{\ell }_{m_{\lambda }}^{\infty ,q}}:={\left(\sum _{n\in {\mathbb{Z}}^d}{\left(\underset{k\in {\mathbb{Z}}^d}{\sup }|a_{kn}|m_{\lambda }(k,n)\right)}^q\right)}^{1/q}<+\infty ,$$

$$\Vert a{\Vert }_{{\ell }_{m_{\lambda }}^{p,\infty }}:=\underset{n\in {\mathbb{Z}}^d}{\sup }{\left(\sum _{k\in {\mathbb{Z}}^d}|a_{kn}{|}^p{m}_{\lambda }(k,n{)}^p\right)}^{1/p}<+\infty ,$$

$$a_{kn}:={\mathrm{esssup}}_{(x,\xi )\in [0,1{]}^{2d}}|F(k+x,n+\xi )|=\Vert F\cdot T_{(k,n)}{\chi }_Q{\Vert }_{L^{\infty }},$$

$$|F|\le \sum _{k,n\in {\mathbb{Z}}^d}{b}_{kn}{T}_{(k,n)}{\chi }_Q$$

$$\Vert F{\Vert }_{W(L_{m_{\lambda }}^{p,q})}=\Vert a{\Vert }_{{\ell }_{m_{\lambda }}^{p,q}}.$$

$$\mu (\lambda ):=\{\begin{array}{ll}\lambda L,& \lambda \ge 0\\ \lambda /L,& \lambda <0,\end{array}\nu (\lambda ):=\{\begin{array}{ll}\lambda L,& \lambda \ge 0\\ |\lambda |,& \lambda <0.\end{array}$$

$$\Vert F\ast G{\Vert }_{L_{m_{\lambda }}^{p,q}}\le C_{\lambda }\Vert F{\Vert }_{L_{m_{\mu (\lambda )}}^{p,q}}\Vert G{\Vert }_{L_{m_{\nu (\lambda )}}^1}$$

Full text

1. Introduction

One of the main properties of a nuclear space is that the Schwartz kernel theorem holds, which gives, for instance, a different representation of a continuous and linear pseudodifferential operator as an integral operator in terms of its kernel. This is very useful for the study of the propagation of singularities or the behaviour of wave front sets of pseudodifferential operators. See, for example, [1, 5, 10, 11, 23, 24] and the references therein.

In fact, in [5] the first three authors of the present work imposed the following condition on the weight function: there is $H>1$ such that for every $t>0$, $2\omega(t)\leq\omega(Ht)+H$ (see [8, Corollary 16 (3)]), to have that the space $\mathcal{S}_{\omega}(\mathbb{R}^{d})$ is nuclear (see [6]). Hence they could analyse the kernel of some pseudodifferential operators [5, Section 4]. In the present paper, we complete the study begun in [6] and prove that $\mathcal{S}_{\omega}(\mathbb{R}^{d})$ is nuclear for every weight function $\omega(t)=o(t)$ as $t$ tends to infinity (see Definition 2.1). Hence, now the powers of the logarithm $\omega(t)=\log^{\beta}(1+t)$, $\beta>0$, are allowed as weight functions and, in particular, we recover a known result for the weight $\omega(t)=\log(1+t),$ namely, that the Schwartz class $\mathcal{S}(\mathbb{R}^{d})$ is a nuclear space.

To see that $\mathcal{S}_{\omega}(\mathbb{R}^{d})$ is nuclear we establish an isomorphism, which is new in the literature, with some Fréchet sequence space. We use expansions in terms of Gabor frames, that are a fundamental tool in time-frequency analysis. This is motivated by the rapid decay of the Gabor coefficients of a function in $\mathcal{S}_{\omega}(\mathbb{R}^{d})$ when $\omega$ is a subadditive function, as we showed in [5]. More precisely, we proved that $u\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$ if and only if

[TABLE]

where $\alpha_{0},\beta_{0}>0$ are sufficiently small so that $\{\Pi(\sigma)\varphi\}_{\sigma\in\alpha_{0}\mathbb{Z}^{d}\times\beta_{0}\mathbb{Z}^{d}}$ is a Gabor frame in $L^{2}(\mathbb{R}^{d})$ for a fixed window function $\varphi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$, $V_{\varphi}u$ is the short-time Fourier transform of $u$ and $\Pi(\sigma)$ is the time-frequency shift defined as $\Pi(\sigma)\varphi(y)=e^{i\langle y,\beta_{0}n\rangle}\varphi(y-\alpha_{0}k)$, for $\sigma=(\alpha_{0}k,\beta_{0}n)$. The usual properties of modulation spaces in [5] hold only when the weight function $\omega$ is subadditive. However, the expansion in terms of Gabor frames is still possible when the weight is non subadditive and satisfies $\omega(t)=o(t)$ as $t$ tends to infinity. In fact, we prove here that $\mathcal{S}_{\omega}(\mathbb{R}^{d})$ is isomorphic to a topological subspace of the sequence space

[TABLE]

The isomorphism is defined in (2.6) by the restriction on its image of the analysis operator, that maps $u\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$ to its Gabor coefficients $\{V_{\varphi}u(\sigma)\}_{\sigma\in\alpha_{0}\mathbb{Z}^{d}\times\beta_{0}\mathbb{Z}^{d}}$. As a consequence, $\mathcal{S}_{\omega}(\mathbb{R}^{d})$ is nuclear by an application of Grothendieck-Pietsch criterion to the space $\tilde{\Lambda}_{\omega}$ (Proposition 3.2). This isomorphism is not the only one existing in the literature, and it should be compared with the one given by Aubry [2], only for the one-variable case, obtaining that $\mathcal{S}_{\omega}(\mathbb{R})$ is isomorphic to the different sequence space

[TABLE]

Aubry uses expansions in terms of the Hermite functions, as Langenbruch [18] did previously for spaces defined by sequences in the sense of Komatsu.

Finally, in the last section of the paper, and without using techniques from time frequency-analysis, we characterize when the Beurling space of ultradifferentiable functions $\mathcal{S}_{(M_{p})}(\mathbb{R}^{d})$ (see formula (4.1) for the definition) in the sense of Komatsu is nuclear. We can give such a characterization when the space is defined by $L^{2}$ norms. We explain and motivate a little bit this result. Pilipović, Prangoski and Vindas [22] showed that

[TABLE]

is nuclear when $(M_{p})_{p}$ satisfies the standard conditions $(M1)$ (defined below in formula (1.5)) and $(M2)$ (that we do not define here because it is not used), which is stronger than $(M2)^{\prime}$, defined below in formula (1.4). On the other hand, using the isomorphism of [18], we proved in [6] that the space $S_{(M_{p})}(\mathbb{R}^{d})$ is nuclear if $(M_{p})_{p}$ satisfies that there is $H>0$ such that for any $C>0$ there is $B>0$ with

[TABLE]

and $(M2)^{\prime}$ (stability under differential operators):

[TABLE]

The condition (1.3) is quite natural and not restrictive at all and it is used by Langenbruch [18] to show that the Hermite functions are elements of $S_{(M_{p})}(\mathbb{R}^{d})$. Moreover, Langenbruch also proves in [18, Remark 2.1] that under these two conditions (1.3) and (1.4), $S_{(M_{p})}(\mathbb{R}^{d})=\mathcal{S}_{(M_{p})}(\mathbb{R}^{d})$. If we do not assume (1.4) and consider only $\mathcal{S}_{(M_{p})}(\mathbb{R}^{d})$ (the space defined with $L^{2}$ norms), after a careful reading of the proofs of some results of [18] in the Beurling case and the use of techniques of Petzsche [20], we are able to prove here that under the additional conditions $(M1)$ (logarithmic convexity):

[TABLE]

and that $M_{p}^{1/p}\to+\infty$ as $p\to+\infty$, $\mathcal{S}_{(M_{p})}(\mathbb{R}^{d})$ is nuclear if and only if $(M_{p})_{p}$ satisfies $(M2)^{\prime}$.

The paper is organized as follows. In Section 2, we show that Gabor frames have a stable behaviour with the only assumption $\omega(t)=o(t)$ as $t$ tends to infinity on the weight function. Indeed, we see that the analysis and synthesis operators are well defined and continuous in the suitable spaces (Propositions 2.9 and 2.10), defining an isomorphism between $\mathcal{S}_{\omega}(\mathbb{R}^{d})$ and a subspace of $\tilde{\Lambda}_{\omega}$. In Section 3 we recover for this setting some known properties of Köthe echelon spaces to see that the sequence space $\tilde{\Lambda}_{\omega}$ is nuclear. And, finally, in Section 4 we characterize the nuclearity of $\mathcal{S}_{(M_{p})}(\mathbb{R}^{d})$.

2. Gabor frame operators in $\mathcal{S}_{\omega}(\mathbb{R}^{d})$

Let us condider weight functions of the form:

Definition 2.1**.**

A weight function is a continuous increasing function $\omega:\ [0,+\infty)\to[0,+\infty)$ satisfying the following properties:

• $(\alpha)$

there is $L\geq 1\mbox{ such that }\omega(2t)\leq L(\omega(t)+1),$ for each $t\geq 0;$

• $(\beta)$

$\omega(t)=o(t)\quad\mbox{as}\ t\to+\infty;$**

• $(\gamma)$

there are $a\in\mathbb{R},b>0\mbox{ such that }\omega(t)\geq a+b\log(1+t),\ t\geq 0;$

• $(\delta)$

the map $t\mapsto\varphi_{\omega}(t):=\omega(e^{t})\ \mbox{is convex}.$

For $\zeta\in\mathbb{C}^{d}$, we put $\omega(\zeta):=\omega(|\zeta|)$, where $|\zeta|$ denotes the Euclidean norm of $\zeta$.

Note that condition $(\alpha)$ implies

[TABLE]

We denote by $\varphi^{*}_{\omega}$ the Young conjugate of $\varphi_{\omega}$, defined by

[TABLE]

We recall that $\varphi^{*}_{\omega}$ is an increasing and convex function satisfying $(\varphi^{*}_{\omega})^{*}=\varphi_{\omega}$ (see [15]). Moreover $\varphi^{*}_{\omega}(s)/s$ is increasing. For a collection of further well-known properties of $\varphi^{*}_{\omega}$ we refer, for instance, to [7, Lemma 2.3].

We consider the following notation for the Fourier transform of $u\in L^{1}(\mathbb{R}^{d})$:

[TABLE]

with standard extensions to more general spaces of functions or distributions. We recover from [3] the following

Definition 2.2**.**

The space $\mathcal{S}_{\omega}(\mathbb{R}^{d})$ is the set of all $u\in L^{1}(\mathbb{R}^{d})$ such that $u,\hat{u}\in C^{\infty}(\mathbb{R}^{d})$ and for each $\lambda>0$ and each $\alpha\in\mathbb{N}_{0}^{d}$ we have

[TABLE]

The corresponding strong dual of ultradistributions will be denoted by $\mathcal{S}^{\prime}_{\omega}(\mathbb{R}^{d})$.

We denote by $T_{x}$, $M_{\xi}$ and $\Pi(z)$, respectively, the translation, the modulation and the phase-space shift operators, defined by

[TABLE]

for $x,y,\xi\in\mathbb{R}^{d}$ and $z=(x,\xi)$.

For a window function $\varphi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})\setminus\{0\}$ the short-time Fourier transform (briefly STFT) of $f\in\mathcal{S}^{\prime}_{\omega}(\mathbb{R}^{d})$ is defined, for $z=(x,\xi)\in\mathbb{R}^{2d}$, by

[TABLE]

where the brackets $\langle\cdot,\cdot\rangle$ in (2.2) and the (formal) integral in (2.3) denote the conjugate linear action of $\mathcal{S}^{\prime}_{\omega}$ on $\mathcal{S}_{\omega}$, consistent with the inner product $\langle\cdot,\cdot\rangle_{L^{2}}$.

By condition $(\gamma)$ of Definition 2.1 it is easy to deduce that $\mathcal{S}_{\omega}(\mathbb{R}^{d})\subset\mathcal{S}(\mathbb{R}^{d})$. Hence, $\mathcal{S}_{\omega}(\mathbb{R}^{d})$ can be equivalently defined as the set of all $u\in\mathcal{S}(\mathbb{R}^{d})$ that satisfy the conditions of Definition 2.2. The Fourier transform $\mathcal{F}:\mathcal{S}_{\omega}(\mathbb{R}^{d})\to\mathcal{S}_{\omega}(\mathbb{R}^{d})$ is a continuous automorphism, that can be extended in the usual way to $\mathcal{S}_{\omega}^{\prime}(\mathbb{R}^{d})$ and, moreover, the space $\mathcal{S}_{\omega}(\mathbb{R}^{d})$ is an algebra under multiplication and convolution. On the other hand, for $u,\psi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$ we have $V_{\psi}u\in\mathcal{S}_{\omega}(\mathbb{R}^{2d})$. Moreover, for $u\in\mathcal{S}^{\prime}_{\omega}(\mathbb{R}^{d})$ the short-time Fourier transform is well defined and belongs to $\mathcal{S}_{\omega}^{\prime}(\mathbb{R}^{2d})$. We refer to [3, 14, 5] for subadditive weights, and to [4, 7] for non-subadditive weights; in particular, all results of [5, Section 2] are valid in the non-subadditive case also.

We shall need the following theorem from [7]:

Theorem 2.3**.**

Given a function $u\in\mathcal{S}(\mathbb{R}^{d})$ and $1\leq p,q\leq+\infty$, we have that $u\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$ if and only if one of the following equivalent conditions is satisfied:

• $(a)$

• i)

$\forall\lambda>0,\ \alpha\in\mathbb{N}^{d}_{0}\ \exists C_{\alpha,\lambda}>0$* s.t. $\displaystyle\|e^{\lambda\omega(x)}\partial^{\alpha}u(x)\|_{L^{p}}\leq C_{\alpha,\lambda}\,$,*

• ii)

$\forall\lambda>0,\ \alpha\in\mathbb{N}^{d}_{0}\ \exists C_{\alpha,\lambda}>0$* s.t. $\displaystyle\|e^{\lambda\omega(\xi)}\partial^{\alpha}\hat{u}(\xi)\|_{L^{q}}\leq C_{\alpha,\lambda}\,$;*

• $(b)$

• i)

$\forall\lambda>0,\ \alpha\in\mathbb{N}^{d}_{0}\ \exists C_{\alpha,\lambda}>0$* s.t. $\displaystyle\|e^{\lambda\omega(x)}x^{\alpha}u(x)\|_{L^{p}}\leq C_{\alpha,\lambda}\,$,*

• ii)

$\forall\lambda>0,\ \alpha\in\mathbb{N}^{d}_{0}\ \exists C_{\alpha,\lambda}>0$* s.t. $\displaystyle\|e^{\lambda\omega(\xi)}\xi^{\alpha}\hat{u}(\xi)\|_{L^{q}}\leq C_{\alpha,\lambda}\,$;*

• $(c)$

• i)

$\forall\lambda>0\ \exists C_{\lambda}>0$* s.t. $\displaystyle\|e^{\lambda\omega(x)}u(x)\|_{L^{p}}\leq C_{\lambda}\,$,*

• ii)

$\forall\lambda>0\ \exists C_{\lambda}>0$* s.t. $\displaystyle\|e^{\lambda\omega(\xi)}\hat{u}(\xi)\|_{L^{q}}\leq C_{\lambda}\,$;*

• $(d)$

• i)

$\forall\lambda>0,\beta\in\mathbb{N}_{0}^{d}\ \exists C_{\beta,\lambda}>0$* s.t. $\displaystyle\sup_{\alpha\in\mathbb{N}_{0}^{d}}\|x^{\beta}\partial^{\alpha}u(x)\|_{L^{p}}e^{-\lambda\varphi^{*}_{\omega}\left(\frac{|\alpha|}{\lambda}\right)}\leq C_{\beta,\lambda}\,$,*

• ii)

$\forall\mu>0,\alpha\in\mathbb{N}_{0}^{d}\ \exists C_{\alpha,\mu}>0$* s.t. $\displaystyle\sup_{\beta\in\mathbb{N}_{0}^{d}}\|x^{\beta}\partial^{\alpha}u(x)\|_{L^{q}}e^{-\mu\varphi^{*}_{\omega}\left(\frac{|\beta|}{\mu}\right)}\leq C_{\alpha,\mu}\,$;*

• $(e)$

$\forall\mu,\lambda>0\ \exists C_{\mu,\lambda}>0$* s.t. $\displaystyle\sup_{\alpha,\beta\in\mathbb{N}_{0}^{d}}\|x^{\beta}\partial^{\alpha}u(x)\|_{L^{p}}e^{-\lambda\varphi^{*}_{\omega}\left(\frac{|\alpha|}{\lambda}\right)}e^{-\mu\varphi^{*}_{\omega}\left(\frac{|\beta|}{\mu}\right)}\leq C_{\mu,\lambda}\,$;*

• $(f)$

$\forall\lambda>0\ \exists C_{\lambda}>0$* s.t. $\displaystyle\sup_{\alpha,\beta\in\mathbb{N}_{0}^{d}}\|x^{\beta}\partial^{\alpha}u(x)\|_{L^{p}}e^{-\lambda\varphi^{*}_{\omega}\left(\frac{|\alpha+\beta|}{\lambda}\right)}\leq C_{\lambda}\,$;*

• $(g)$

$\forall\mu,\lambda>0\ \exists C_{\mu,\lambda}>0$* s.t. $\displaystyle\sup_{\alpha\in\mathbb{N}_{0}^{d}}\|e^{\mu\omega(x)}\partial^{\alpha}u(x)\|_{L^{p}}e^{-\lambda\varphi^{*}_{\omega}\left(\frac{|\alpha|}{\lambda}\right)}\leq C_{\mu,\lambda}\,$;*

• $(h)$

Given $\psi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})\setminus\{0\}$, $\forall\lambda>0\ \exists C_{\lambda}>0$ s.t. $\displaystyle\|V_{\psi}u(z)e^{\lambda\omega(z)}\|_{L^{p,q}}\leq C_{\lambda}$.

Let us set, for $\lambda\in\mathbb{R}\setminus\{0\}$,

[TABLE]

and consider the weighted $L^{p,q}$ spaces

[TABLE]

for $1\leq p,q<+\infty$, and

[TABLE]

for $1\leq p,q\leq+\infty$ with $p=+\infty$ or $q=+\infty$ respectively. If $p=q$ we write $L^{p}_{m_{\lambda}}(\mathbb{R}^{d})=L^{p,q}_{m_{\lambda}}(\mathbb{R}^{d})$.

Here we consider generic weight functions $\omega$ satisfying $(\alpha)$ of Definition 2.1 (weaker than subadditivity). In this case modulation spaces lack several properties. Hence we prove directly some results on Gabor frames in $\mathcal{S}_{\omega}(\mathbb{R}^{d})$ without using modulation spaces. If $\omega$ is subadditive we know, by Theorem 4.2 of [14], that for any fixed $\varphi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$ its dual window $\psi$, in the sense of the theory of Gabor frames (see Gröchenig [12]), belongs to $\mathcal{S}_{\omega}(\mathbb{R}^{d})$ (see also [13, Thm. 4.2] and [16]). In our case we will fix $\varphi_{0}(x):=e^{-|x|^{2}}$ the Gaussian function, $\alpha_{0},\beta_{0}>0$ such that $\{\Pi(\sigma)\varphi_{0}\}_{\sigma\in\alpha_{0}\mathbb{Z}^{d}\times\beta_{0}\mathbb{Z}^{d}}$ is a Gabor frame for $L^{2}(\mathbb{R}^{d})$, and then prove that the canonical dual window $\psi_{0}$ of $\varphi_{0}$ is in $\mathcal{S}_{\omega}(\mathbb{R}^{d})$. To this aim we start by the following

Lemma 2.4**.**

Let $\omega$ be a weight function. There exists then a subadditive weight function $\sigma$ such that $\omega(t)=o(\sigma(t))$ as $t\to+\infty$.

Proof.

Let us consider $\omega_{0}(t)=\max\{0,t-1\}$. This is a continuous increasing function $\omega_{0}:\ [0,+\infty)\to[0,+\infty)$ that satisfies $(\alpha)$, $(\gamma)$ and $(\delta)$ of Definition 2.1 and moreover $\omega_{0}|_{[0,1]}\equiv 0$ and $\omega_{0}$ is concave on $[1+\infty)$.

Then, by Lemma 1.7 and Remark 1.8(1) of [9], there exists a weight function $\lambda$ satisfying $(\alpha)$, $(\gamma)$ and $(\delta)$ and such that $\lambda|_{[0,1]}\equiv 0$, $\lambda$ concave on $[1,+\infty)$ and

[TABLE]

Since $\lambda(t+1)$ is concave on $[0,+\infty)$ with $\lambda(1)=0$, we have that $\sigma(t):=\lambda(t)+\lambda(2)$ is the required weight function. ∎

Proposition 2.5**.**

Let $\varphi_{0}(x)=e^{-|x|^{2}}$ be the Gaussian function and let $\psi_{0}$ be a dual window of $\varphi_{0}$. Then $\varphi_{0},\psi_{0}\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$ for every weight function $\omega$.

Proof.

Let $\omega$ be a weight function as in Definition 2.1. By Lemma 2.4 there exists a subadditive weight function $\sigma$ such that $\omega(t)=o(\sigma(t))$ as $t\to+\infty$. Then $\mathcal{S}_{\sigma}(\mathbb{R}^{d})\subseteq\mathcal{S}_{\omega}(\mathbb{R}^{d})$.

Clearly $\varphi_{0}\in\mathcal{S}_{\sigma}(\mathbb{R}^{d})\subseteq\mathcal{S}_{\omega}(\mathbb{R}^{d})$ by condition $(\beta)$. Since $\sigma$ is subadditive, by [14, Thm. 4.2], its dual window $\psi_{0}\in\mathcal{S}_{\sigma}(\mathbb{R}^{d})\subseteq\mathcal{S}_{\omega}(\mathbb{R}^{d})$ and the proof is complete. ∎

We fix, once and for all, $\varphi_{0}(x)=e^{-|x|^{2}}$, $\alpha_{0},\beta_{0}>0$ such that $\{\Pi(\sigma)\varphi_{0}\}_{\sigma\in\alpha_{0}\mathbb{Z}^{d}\times\beta_{0}\mathbb{Z}^{d}}$ is a Gabor frame for $L^{2}$ and $\psi_{0}$ the canonical dual window of $\varphi_{0}$ (see [12, Section 7.3]). For the lattice $\Lambda:=\alpha_{0}\mathbb{Z}^{d}\times\beta_{0}\mathbb{Z}^{d}$, we consider the analysis operator $C_{\varphi_{0}}$ acting on a function $f\in L^{2}(\mathbb{R}^{d})$

[TABLE]

and the synthesis operator $D_{\psi_{0}}$ acting on a sequence $c=(c_{k,n})_{k,n\in\mathbb{Z}^{d}}$

[TABLE]

It is well known (see, for instance, [12]) that

[TABLE]

since $\psi_{0}$ is the canonical dual window of $\varphi_{0}$, and then

[TABLE]

Later on we shall explain more precisely this identity on $\mathcal{S}_{\omega}(\mathbb{R}^{d})$.

We denote by $\ell^{p,q}_{m_{\lambda}}$, for $1\leq p,q\leq+\infty$ and $\lambda\in\mathbb{R}\setminus\{0\}$, the space of all sequences $a=(a_{kn})_{k,n\in\mathbb{Z}^{d}}$, with $a_{kn}\in\mathbb{C}$ for every $k,n\in\mathbb{Z}^{d}$, such that

[TABLE]

if $1\leq p,q<+\infty$,

[TABLE]

for $1\leq p,q\leq+\infty$ with $p=+\infty$ or $q=+\infty$ respectively.

Then we say that a measurable function $F$ on $\mathbb{R}^{2d}$ belongs to the amalgam space $W(L^{p,q}_{m_{\lambda}})$ for the sequence

[TABLE]

where $\chi_{Q}$ is the characteristic function of the cube $Q=[0,1]^{2d}$, when $a=(a_{kn})_{k,n\in\mathbb{Z}^{d}}\in\ell^{p,q}_{m_{\lambda}}$. Equivalently, $F\in W(L^{p,q}_{m_{\lambda}})$ if and only if

[TABLE]

for some $b=(b_{kn})_{k,n\in\mathbb{Z}^{d}}\in\ell^{p,q}_{m_{\lambda}}$ (cf. [12, pg. 222]). The amalgam space $W(L^{p,q}_{m_{\lambda}})$ is endowed with the norm

[TABLE]

In what follows we shall need the Young estimate for $L^{p,q}_{m_{\lambda}}$:

Proposition 2.6**.**

Let $\omega$ be a weight function and $L$ as in (2.1). Set, for every $\lambda\in\mathbb{R}$,

[TABLE]

Then, for $F\in L^{p,q}_{m_{\mu(\lambda)}}$ and $G\in L^{1}_{m_{\nu(\lambda)}}$, with $1\leq p,q\leq+\infty$, we have that $F*G\in L^{p,q}_{m_{\lambda}}$ and

[TABLE]

for a constant $C_{\lambda}>0$ depending on $\lambda$.

Proof.

Let us first assume $1\leq p,q<+\infty$. From the definition of convolution

[TABLE]

Now, for $\lambda\geq 0$ we have, by (2.1),

[TABLE]

so that

[TABLE]

By the standard Young’s inequality for (non weighted) $L^{p,q}$ spaces we obtain

[TABLE]

For $\lambda<0$ we have, by (2.1),

[TABLE]

and then, as before,

[TABLE]

for some $C_{\lambda}>0$. The proof for $p=+\infty$ and/or $q=+\infty$ is similar. ∎

We have the following proposition, analogous to [12, Prop. 11.1.4]. We give the proof for the convenience of the reader.

Proposition 2.7**.**

Let $\omega$ be a weight function, $L$ as in (2.1) and $\lambda>0$. If $F\in W(L^{p,q}_{m_{\lambda L}})$ is continuous, then for every $\alpha,\beta>0$ there exists a constant $C_{\alpha,\beta,\lambda}>0$ such that

[TABLE]

for $\tilde{m}_{\lambda}(k,n):=m_{\lambda}(\alpha k,\beta n)$.

Proof.

The continuity of $F$ is necessary in order that $F(\alpha k,\beta n)$ is well defined. For $(\alpha k,\beta n)\in(r,s)+[0,1]^{2d}$ with $(r,s)\in\mathbb{Z}^{d}\times\mathbb{Z}^{d}$ we have

[TABLE]

for $C_{\lambda}=e^{\lambda L}\sup_{(x,\xi)\in[0,1]^{2d}}e^{\lambda L\omega(x,\xi)}$. Then

[TABLE]

Since there are at most $\tilde{C}_{\alpha}:=\left(\left[\frac{1}{\alpha}\right]+1\right)^{d}$ points $\alpha k\in r+[0,1]^{d}$ we obtain

[TABLE]

Analogously, there are at most $\tilde{C}_{\beta}:=\left(\left[\frac{1}{\beta}\right]+1\right)^{d}$ points $\beta n\in s+[0,1]^{d}$ and therefore

[TABLE]

∎

Proposition 2.8**.**

Let $\omega$ be a weight function, $L$ as in (2.1) and $\lambda>0$. If $F\in L^{p,q}_{m_{\lambda L}}$ and $G\in L^{1}_{m_{\lambda L^{2}}}$, then $F*G\in W(L^{\infty}_{m_{\lambda}})$ and

[TABLE]

Proof.

From the definition of the norm in $W(L^{\infty}_{m_{\lambda}})$ we have

[TABLE]

By (2.1), it is easy to see that

[TABLE]

Therefore, we obtain

[TABLE]

Since $(x,\xi)\in[0,1]^{2d}$ we have that $e^{\lambda L^{2}\omega(x,\xi)}$ is bounded by a constant depending on $\lambda$ (and $L$), so we obtain

[TABLE]

for some $C_{\lambda}>0$.

By Young’s inequality we finally deduce

[TABLE]

∎

Now, our aim is to show that there is an isomorphism between $\mathcal{S}_{\omega}(\mathbb{R}^{d})$ and its image through the analysis operator $C_{\varphi_{0}}$:

[TABLE]

where $\tilde{\Lambda}_{\omega}$ is defined in (1.1).

The following proposition holds for every window function $\varphi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})\setminus\{0\}$ and in particular for our fixed window $\varphi_{0}\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$:

Proposition 2.9**.**

Let $\omega$ be a weight function and $\varphi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})\setminus\{0\}$. The analysis operator

[TABLE]

is continuous.

Proof.

It is known that if $f\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$ then for every $\lambda>0$ there exists $C_{\lambda}>0$ such that

[TABLE]

In fact, this property is proved in [14] when $\omega$ is subadditive, but it is still true in the general case (Theorem 2.3). Since $C_{\varphi}f=(V_{\varphi}f(\sigma))_{\sigma\in\Lambda}$ we have $C_{\varphi}f\in\tilde{\Lambda}_{\omega}$.

Now, we prove that the operator $C_{\varphi}$ is continuous. By [12, Lemma 11.3.3]

[TABLE]

By Propositions 2.7 and 2.8, for every fixed $\lambda>0$ we obtain

[TABLE]

for $\tilde{m}_{\lambda}(k,n)=m_{\lambda}(\alpha_{0}k,\beta_{0}n)$ and for some $C_{\lambda},C^{\prime}_{\lambda}>0$ ($\alpha_{0}$ and $\beta_{0}$ are fixed). Observe that, since $f,\varphi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$, then $V_{\varphi}f\in L^{\infty}_{m_{\lambda L^{2}}}$ and $V_{\varphi}\varphi\in L^{1}_{m_{\lambda L^{3}}}$ for every $\lambda>0$ by Theorem 2.3(h).

Therefore, for every fixed $\lambda>0$ there exists a constant $C^{\prime\prime}_{\lambda}=C^{\prime}_{\lambda}\|V_{\varphi}\varphi\|_{L^{1}_{m_{\lambda L^{3}}}}>0$ such that

[TABLE]

This gives the continuity by Theorem 2.3(h). ∎

The following proposition is valid for any $\psi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})\setminus\{0\}$.

Proposition 2.10**.**

Let $\omega$ be a weight function and $\psi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})\setminus\{0\}$. Then the synthesis operator

[TABLE]

is continuous.

Proof.

Let $c=(c_{\sigma})_{\sigma\in\Lambda}\in\tilde{\Lambda}_{\omega}$. For simplicity, we denote $c_{\sigma}$ by $c_{kn}$ for $\sigma=(\alpha_{0}k,\beta_{0}n)$. We start proving that $D_{\psi}c\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$. We shall apply Theorem 2.3(c) with $p=+\infty$. So, first, we have to see that $D_{\psi}c\in\mathcal{S}(\mathbb{R}^{d})$.

By definition

[TABLE]

Now, we see that $D_{\psi}c\in C^{\infty}(\mathbb{R}^{d})$. To that aim we show that for each $\gamma\in\mathbb{N}_{0}^{d}$, the series

[TABLE]

is uniformly convergent on $t\in\mathbb{R}^{d}$. Let us compute

[TABLE]

Since $(c_{kn})_{k,n\in\mathbb{Z}^{d}}\in\tilde{\Lambda}_{\omega}$, for every $\lambda>0$ there exists $C_{\lambda}>0$ such that

[TABLE]

Now, since $\omega$ is increasing it is obvious that $\omega(t,s)\geq\frac{1}{2}(\omega(t)+\omega(s)).$ Therefore

[TABLE]

Since

[TABLE]

we obtain

[TABLE]

Then we have, by (2.9), for $C_{\lambda,\gamma}=C^{\prime}_{\lambda}\max_{\mu\leq\gamma}|\beta_{0}|^{|\mu|-|\gamma|}$, since $\psi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$ (see Definition 2.2),

[TABLE]

for some $C^{\prime}_{\lambda,\gamma}>0$. Hence, for $\lambda>0$ sufficiently large the series

[TABLE]

is uniformly convergent on $t\in\mathbb{R}^{d}$. This implies that $D_{\psi}c\in C^{\infty}(\mathbb{R}^{d})$ for every $c\in\tilde{\Lambda}_{\omega}$.

In particular we can differentiate $D_{\psi}c$ in (2.7) term by term, so that, to prove that $D_{\psi}c\in\mathcal{S}(\mathbb{R}^{d})$, we can estimate, for every $\gamma,\mu\in\mathbb{N}_{0}^{d}$,

[TABLE]

Since

[TABLE]

we obtain

[TABLE]

for some $C_{\lambda,\gamma,\mu}>0$ because $\psi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$, by Theorem 2.3(b). Since the series in (2.13) converges for $\lambda>0$ sufficiently large, we have $D_{\psi}c\in\mathcal{S}(\mathbb{R}^{d})$.

By Theorem 2.3(c), to see that $D_{\psi}c\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$ it is now enough to prove that, for every $\tilde{\lambda}>0$, the following two conditions hold:

[TABLE]

To prove (2.14) we use the calculations in (2.11) and obtain, for every $\lambda\geq 4L\tilde{\lambda}$,

[TABLE]

for some $\tilde{C}_{\lambda}>0$, since $\psi\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$. For $\lambda$ sufficiently large the series in (2.17) converges and hence (2.14) is proved.

To prove (2.15) let us now consider

[TABLE]

Since the series

[TABLE]

converges uniformly and moreover, by (2.16) with $\tilde{\lambda}=0$ and $\lambda$ large enough,

[TABLE]

by the Dominated Convergence Theorem

[TABLE]

Then

[TABLE]

and since $\hat{\psi}\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$ satisfies the same estimates as $\psi$ the proof of (2.15) is similar to that of (2.14) and so $D_{\psi}c\in\mathcal{S}_{\omega}(\mathbb{R}^{d})$.

Now, we see that $D_{\psi}$ is continuous. To this aim we have to estimate (2.14) and (2.15), for every $\tilde{\lambda}>0$, by some seminorm of $c=(c_{kn})_{k,n\in\mathbb{Z}^{d}}$ in $\tilde{\Lambda}_{\omega}$. Writing, for every $\lambda>0$,

[TABLE]

and proceeding as to obtain (2.17), with $\sup_{k,n\in\mathbb{Z}^{d}}\left(|c_{kn}|e^{\lambda\omega(\alpha_{0}k,\beta_{0}n)}\right)$ instead of $C_{\lambda}$ in (2.10), we obtain that for every $\tilde{\lambda}>0$ there exist $\lambda>0$ and $C_{\tilde{\lambda}}>0$ such that

[TABLE]

Similarly, from (2.19),

[TABLE]

for some $C^{\prime}_{\tilde{\lambda}}>0$. Therefore $D_{\psi}$ is continuous and the proof is complete. ∎

We already know from the general theory of Gabor frames that $D_{\psi_{0}}C_{\varphi_{0}}=\mathop{\rm Id}\nolimits$ on $\mathcal{S}_{\omega}(\mathbb{R}^{d})$, as already observed in (2.4). Hence the operator in (2.6) is injective, surjective, continuous and its inverse $D_{\psi_{0}}|_{\mathop{\rm Im}\nolimits C_{\varphi_{0}}}$ is continuous. Since we consider on $\mathop{\rm Im}\nolimits C_{\varphi_{0}}$ the topology induced by $\tilde{\Lambda}_{\omega}$, to see that $\mathcal{S}_{\omega}(\mathbb{R}^{d})$ is nuclear it is enough to check that $\tilde{\Lambda}_{\omega}$ is nuclear [19, Prop. 28.6].

3. Nuclearity of $\mathcal{S}_{\omega}(\mathbb{R}^{d})$

In this section we show that $\tilde{\Lambda}_{\omega}$ is nuclear by an application of Grothendieck-Pietsch criterion. For a countable lattice $\Lambda$, we consider a matrix

[TABLE]

of Köthe type with positive entries, in the sense that $A$ satisfies

[TABLE]

We denote

[TABLE]

We put

[TABLE]

Analogously, we define $\tilde{\ell}^{\infty}$ and $\tilde{c}_{0}$. The spaces $\tilde{\ell}^{p}$, for $1\leq p\leq+\infty$, and $\tilde{c}_{0}$ are Banach spaces, while $\tilde{\lambda}^{p}(A)$, for $1\leq p\leq+\infty$, and $\tilde{c}_{0}(A)$ are Fréchet spaces. We consider the canonical basis $(e_{\eta})_{\eta\in\Lambda}$:

[TABLE]

Since $\Lambda$ is countable, it is obvious that $(e_{\eta})_{\eta\in\Lambda}$ is a Schauder basis for $\tilde{c}_{0}(A)$ and $\tilde{\lambda}^{p}(A)$, for $1\leq p<+\infty$.

The following result is analogous to [19, Prop. 28.16]. We give the proof in the case of lattices for the sake of completeness.

Theorem 3.1**.**

Let $A$ be as in (3.1) a matrix of Köthe type with positive entries. The following are equivalent:

• (a)

$\tilde{\lambda}^{p}(A)$* is nuclear for some $1\leq p\leq+\infty$;*

• (b)

$\tilde{\lambda}^{p}(A)$* is nuclear for all $1\leq p\leq+\infty$;*

• (c)

$\forall k\in\mathbb{N}\,\exists m\in\mathbb{N},m\geq k\ \mbox{s.t.}\ \sum_{\sigma\in\Lambda}a_{\sigma,k}a_{\sigma,m}^{-1}<+\infty$.

Proof.

If $1\leq p<+\infty$, then $\tilde{\lambda}^{p}(A)$ is a Fréchet space with the increasing fundamental system of seminorms $(\|\cdot\|_{m})_{m\in\mathbb{N}}$ and the Schauder basis $(e_{\eta})_{\eta\in\Lambda}$. We can then apply Grothendieck-Pietsch criterion (see [19, Thm. 28.15] or [21]) to $\tilde{\lambda}^{p}(A)$ and obtain that $\tilde{\lambda}^{p}(A)$ is nuclear if and only if

[TABLE]

Since

[TABLE]

the thesis is clear for $p<+\infty$.

Now, we treat the case $p=+\infty$. Assume that $\tilde{\lambda}^{\infty}(A)$ is nuclear. We prove that

[TABLE]

To this aim, for every $k\in\mathbb{N}$, we denote

[TABLE]

the local space of $\tilde{\lambda}^{\infty}(A)$. This is a Banach space with the norm $\|\cdot\|_{k}$ (observe that $a_{\sigma,k}>0$ for all $\sigma\in\Lambda,k\in\mathbb{N}$). The operator

[TABLE]

is an isometric isomorphism and $A_{k}(E_{k})=\tilde{\ell}^{\infty}$. For every $k\in\mathbb{N}$, the inclusion

[TABLE]

is compact by [19, Lemma 24.17]. Indeed, $\tilde{\lambda}^{\infty}(A)$ is a locally convex space, which is nuclear (by assumption) and hence Schwartz by [19, Cor. 28.5]; moreover $E_{k}$ is a Banach space and hence we can apply [19, Lemma 24.17] and obtain that there exists a neighbourhood $V$ of 0 in $\tilde{\lambda}^{\infty}(A)$, that we can take of the form $\{c\in\tilde{\lambda}^{\infty}(A):\ \|c\|_{m}<\varepsilon\}$, for some $\varepsilon>0$ and with $m\geq k$ (the family of seminorms $(\|\cdot\|_{m})_{m\in\mathbb{N}}$ is increasing), whose image through $i_{k}$ is precompact, and hence compact. Moreover, for $m\geq k$ clearly $E_{m}\subseteq E_{k}$. So, for every $k\in\mathbb{N}$ there exists $m\geq k$ such that the inclusion $i^{k}_{m}=i_{k}|_{E_{m}}$

[TABLE]

is compact (and also $i^{k}_{m^{\prime}}$ for all $m^{\prime}\geq m$).

Then, we put $D:=A_{k}\circ i^{k}_{m}\circ A_{m}^{-1}$:

[TABLE]

The operator $D$ is clearly compact. The restriction \tilde{D}:=D\big{|}_{\tilde{c}_{0}} satisfies $\tilde{D}(\tilde{c}_{0})\subseteq\tilde{c}_{0}$, for $m\geq k$, since

[TABLE]

for $c=(c_{\sigma})_{\sigma\in\Lambda}\in\tilde{c}_{0}$. The operator $\tilde{D}$ is also compact.

For every $\varepsilon>0$ we define, for $m\geq k,$

[TABLE]

and also

[TABLE]

The operator $T_{\varepsilon}:\,\tilde{c}_{0}\to\tilde{c}_{0}$ is continuous since

[TABLE]

Now we consider

[TABLE]

Hence, $P_{\varepsilon}$ is a compact projection on

[TABLE]

Since $\tilde{c}_{0}$ is a Banach space we can apply [19, Cor. 15.6] and obtain that the kernel $\ker(\mathop{\rm Id}\nolimits-P_{\varepsilon})$ is finite dimensional. But $P_{\varepsilon}$ is a projection and hence its image $\mathop{\rm Im}\nolimits(P_{\varepsilon})=\ker(\mathop{\rm Id}\nolimits-P_{\varepsilon})$ is finite dimensional and $I_{\varepsilon}$ must be finite for every $\varepsilon>0$. Then

[TABLE]

and (3.5) is proved.

This implies that $\tilde{\lambda}^{\infty}(A)=\tilde{c}_{0}(A)$. Indeed, if $c=(c_{\sigma})_{\sigma\in\Lambda}\in\tilde{\lambda}^{\infty}(A)$ then for every $k\in\mathbb{N}$ we find $m\in\mathbb{N}$, $m\geq k$ such that (3.5) holds and we get

[TABLE]

since $|c_{\sigma}|a_{\sigma,m}$ is bounded because $c\in\tilde{\lambda}^{\infty}(A)$ and $a_{\sigma,k}a_{\sigma,m}^{-1}\to 0$ by (3.5). Therefore $c\in\tilde{c}_{0}(A)$.

Now, $\tilde{c}_{0}(A)$ is a Fréchet space endowed with the increasing fundamental system of seminorms $(\|~{}\cdot~{}\|_{m})_{m\in\mathbb{N}}$ and the Schauder basis $(e_{\eta})_{\eta\in\Lambda}$. We can then apply Grothendieck-Pietsch criterion (3.4) to $\tilde{c}_{0}(A)$ for

[TABLE]

Since $\tilde{c}_{0}(A)=\tilde{\lambda}^{\infty}(A)$ is nuclear by assumption, then (3.4) implies $(c)$.

On the contrary, if $(c)$ holds then $\tilde{c}_{0}(A)$ is nuclear by the Grothendiech-Pietsch criterion (3.4). We see again that $\tilde{\lambda}^{\infty}(A)=\tilde{c}_{0}(A)$. If $c=(c_{\sigma})_{\sigma\in\Lambda}\in\tilde{\lambda}^{\infty}(A)$, we have

[TABLE]

since $|c_{\sigma}|a_{\sigma,m}$ is bounded for $c\in\tilde{\lambda}^{\infty}(A)$ and (3.5) holds by the convergence of the series in $(c)$. Therefore $c\in\tilde{c}_{0}(A)$. ∎

Observe that, for $\Lambda=\alpha_{0}\mathbb{Z}^{d}\times\beta_{0}\mathbb{Z}^{d}$ as fixed in Section 2, the matrix

[TABLE]

satisfies (3.2) and (3.3). Hence the space $\tilde{\Lambda}_{\omega}$ defined in (1.1) is, in fact,

[TABLE]

Proposition 3.2**.**

The sequence space $\tilde{\Lambda}_{\omega}$ is nuclear.

Proof.

By Theorem 3.1 we have that $\tilde{\Lambda}_{\omega}=\tilde{\lambda}^{\infty}(\tilde{A})$ is nuclear if and only if

[TABLE]

Since, by condition $(\gamma)$ of Definition 2.1,

[TABLE]

we have, for $m>k+\frac{2d}{b}$,

[TABLE]

∎

As we explained at the end of Section 2, we deduce:

Theorem 3.3**.**

The space $\mathcal{S}_{\omega}(\mathbb{R}^{d})$ is nuclear.

4. Nuclearity of $\mathcal{S}_{(M_{p})}(\mathbb{R}^{d})$ with $L^{2}$ norms

Let $(M_{p})_{p\in\mathbb{N}_{0}}$ be a sequence such that $M_{p}^{1/p}\to+\infty$ as $p\to+\infty$ and consider the locally convex space of rapidly decreasing ultradifferentiable functions

[TABLE]

where $\|\cdot\|_{2}$ denotes the $L^{2}$ norm. We write the associated function in the usual way:

[TABLE]

Langenbruch [18] uses (1.3) to show that the Hermite functions $H_{\gamma}$, for $\gamma\in\mathbb{N}_{0}^{d}$, are an absolute Schauder basis in $\mathcal{S}_{(M_{p})}(\mathbb{R}^{d})$, where

[TABLE]

and the Hermite polynomials $h_{\gamma}$ are given by

[TABLE]

Here we consider a matrix $A^{*}$ of Köthe type with positive entries as in Section 3 for $\Lambda=\mathbb{N}_{0}^{d}$, defined by

[TABLE]

where $M(t)$ is the associated function defined by (4.2). We characterize when $\mathcal{S}_{(M_{p})}(\mathbb{R}^{d})$ is nuclear with Theorem 3.5 of [20], that we state here in our setting, for the convenience of the reader. In what follows we denote $\lambda^{1}:=\tilde{\lambda}^{1}(A^{*})$ and $\lambda^{\infty}:=\tilde{\lambda}^{\infty}(A^{*})$.

Theorem 4.1**.**

Assume that the inclusion $j:\,\lambda^{1}\to\lambda^{\infty}$ has dense image. Let $E$ be a locally convex space such that we have a commutative diagram of continuous linear operators of the form

\lambda^{1}$$\lambda^{\infty}$$T$$S$$j$$E

with $S$ injective or $T$ with dense image. Then $\lambda^{1}$ is nuclear if and only if $E$ is nuclear.

We can now prove the following:

Proposition 4.2**.**

Let $(M_{p})_{p}$ be a sequence satisfying $M_{p}^{1/p}\to+\infty$ as $p\to+\infty$, condition (1.3) and $(M1)$. Then $\mathcal{S}_{(M_{p})}(\mathbb{R}^{d})$ is nuclear if and only if the associated function $M(t)$ satisfies

[TABLE]

Proof.

We shall use Theorem 4.1 with $E=\mathcal{S}_{(M_{p})}(\mathbb{R}^{d})$. We observe that $\lambda^{1}\subseteq\lambda^{\infty}$ and denote by $j$ the inclusion

[TABLE]

Let us consider the linear map

[TABLE]

where

[TABLE]

are the Hermite coefficients of $f$, and then the linear map

[TABLE]

In Theorem 3.4 of [18] it was proved that condition (1.3) implies that $S$ and $T$ are continuous. Note also that the diagram in Theorem 4.1 commutes by the uniqueness of the coefficients with respect to the Schauder basis $(H_{\gamma})_{\gamma\in\mathbb{N}_{0}^{d}}$.

Let us prove that $j$ has dense image. By conditions $M_{p}^{1/p}\to+\infty$ and (M1), and by [20, Lemma 3.2], we have

[TABLE]

Therefore, for every $k\in\mathbb{N}$ there exists $m\in\mathbb{N}$, $m>k$, such that

[TABLE]

and hence $\lambda^{\infty}=\tilde{c}_{0}(A^{*})$, by the same arguments we used to prove that (3.5) implies $\tilde{\lambda}^{\infty}(A)=\tilde{c}_{0}(A)$ in Section 3. Then $j(\lambda^{1})$ is dense in $\lambda^{\infty}=\tilde{c}_{0}(A^{*})$ because

[TABLE]

is dense in $\tilde{c}_{0}(A^{*})$ and is contained in $\lambda^{1}$.

Moreover, $S$ is injective. Hence, by Theorem 4.1, $E=\mathcal{S}_{(M_{p})}(\mathbb{R}^{d})$ is nuclear if and only if $\lambda^{1}$ is nuclear. By Theorem 3.1, the sequence space $\lambda^{1}$ is nuclear if and only if

[TABLE]

The series in (4.5) converges if and only if

[TABLE]

for some $C_{N,H}>0$ and $N>2d$ (see the proof of [6, Thm. 1]). This gives the conclusion since (4.6) is equivalent to (4.4) (see again the proof of [6, Thm. 1]). ∎

Theorem 4.3**.**

Let $(M_{p})_{p}$ be a sequence satisfying $M_{p}^{1/p}\to+\infty$ as $p\to+\infty$, condition (1.3) and $(M1)$. Then $\mathcal{S}_{(M_{p})}(\mathbb{R}^{d})$ is nuclear if and only if $(M2)^{\prime}$ holds.

Proof.

It follows from Proposition 4.2 because, under condition (M1), condition (M2)′ is equivalent to condition (4.4) (see [6, Rem. 1]). ∎

If $(M2)^{\prime}$ is satisfied then $\mathcal{S}_{(M_{p})}(\mathbb{R}^{d})$ can be equivalently defined with $L^{\infty}$ norms as in (1.2) (see [18, Remark 2.1]) and hence $S_{(M_{p})}(\mathbb{R}^{d})$ is nuclear (cf. [6, Corollary 1]), but we cannot derive a characterization in terms of $(M2)^{\prime}$ from the results of Langenbruch [18].

Acknowledgments. The first three authors were partially supported by the Project FFABR 2017 (MIUR), and by the Projects FIR 2018 and FAR 2018 (University of Ferrara). The first and third authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The research of the second author was partially supported by the project MTM2016-76647-P and the grant BEST/2019/172 from Generalitat Valenciana. The fourth author is supporded by FWF-project J 3948-N35.