Factorizations and Singular Value Estimates of Operators with Gelfand–Shilov and Pilipović kernels

Abstract

We prove that any linear operator with kernel in a Pilipović or Gelfand–Shilov space can be factorized by two operators in the same class. We also give links on numerical approximations for such compositions. We apply these composition rules to deduce estimates of singular values and establish Schatten–von Neumann properties for such operators.

1 Introduction

The singular values and their decays are strongly related to possibilities of obtaining suitable finite rank approximations of operators. For a linear and compact operator which acts between Hilbert spaces, the singular values are the eigenvalues in decreasing order of the modulus of the operator. If more generally, the linear operator T is continuous from the quasi-Banach space $\mathscr {B}_1$ to (another) quasi-Banach space $\mathscr {B}_2$, then the singular value of order $k\ge 1$ is given by

$$\begin{aligned} \varvec{\sigma }_k(T)= \varvec{\sigma }_k(T,\mathscr {B}_1,\mathscr {B}_2) \equiv \inf \Vert T-T_0\Vert _{\mathscr {B}_1\rightarrow \mathscr {B}_2} \end{aligned}$$

(1.1)

where the infimum is taken over all linear operators $T_0$ from $\mathscr {B}_1$ to $\mathscr {B}_2$ of rank at most $k-1$. (See Sect. 2 for notations). It follows that T is compact when $\varvec{\sigma }_k(T)$ decreases to zero as k tends to infinity, or equivalently, T can be approximated by finite rank operators with arbitrarily small errors.

In this paper we deduce estimates of $\varvec{\sigma }_k(T)$ when $\mathscr {B}_1$ and $\mathscr {B}_2$ stay between small test function spaces, denoted by $\mathcal H_s(\mathbf R^{d})$ and $\mathcal H_{0,s}(\mathbf R^{d})$, and their (large) duals. The spaces $\mathcal H_s(\mathbf R^{d})$ and $\mathcal H_{0,s}(\mathbf R^{d})$ are invariant under the Fourier transform, depend on the parameter $s\ge 0$ and are obtained by imposing certain exponential type estimates on the Hermite coefficients of the involved functions. More precisely, the set $\mathcal H_s(\mathbf R^{d})$ ($\mathcal H_{0,s}(\mathbf R^{d})$) consists of all

$$\begin{aligned} f = \sum _{\alpha }c_\alpha h_\alpha \end{aligned}$$

such that $|c_\alpha |\lesssim e^{-c|\alpha |^{\frac{1}{2s}}}$ for some (for every) $c>0$. It follows that $\mathcal H_s(\mathbf R^{d})$ and $\mathcal H_{0,s}(\mathbf R^{d})$ increase with s, and are continuously embedded and dense in $\mathscr {S}(\mathbf R^{d})$.

In [26] the spaces $\mathcal H_s(\mathbf R^{d})$ and $\mathcal H_{0,s}(\mathbf R^{d})$ and their duals were characterized in different ways. For example, the images under the Bargmann transform were given, and it was proved that $f\in \mathcal H_s(\mathbf R^{d})$ ($f\in \mathcal H_{0,s}(\mathbf R^{d})$), if and only if f satisfies

$$\begin{aligned} |H^Nf(x)| \lesssim h^NN!^{2s} \end{aligned}$$

(1.2)

for some $h>0$ (for every $h>0$), where H is the harmonic oscillator $|x|^2-\Delta$ on $\mathbf R^{d}$. In this context we recall that Pilipović introduced in [19] function spaces whose elements obey estimates of the form (1.2) for certain choices of s. For this reason, we call $\mathcal H_s(\mathbf R^{d})$ and $\mathcal H_{0,s}(\mathbf R^{d})$ the Pilipović spaces of Roumieu and Beurling type, respectively, of degree $s\ge 0$ (cf. [26]).

In [19], it is also proved that $\mathcal H_{s_1}(\mathbf R^{d})$ and $\mathcal H_{0,s_2}(\mathbf R^{d})$ agree with the Gelfand–Shilov spaces $\mathcal S_{s_1}(\mathbf R^{d})$ and $\Sigma _{s_2}(\mathbf R^{d})$, respectively, when $s_1\ge \frac{1}{2}$ and $s_2>\frac{1}{2}$, while $\mathcal H_{0,\frac{1}{2}}(\mathbf R^{d})$ is different from the trivial space $\Sigma _{\frac{1}{2}}(\mathbf R^{d})=\{ 0\}$. The family of Pilipović spaces therefore contains all Gelfand–Shilov spaces which are invariant under Fourier transformations.

In Sect. 5 we consider linear operators whose kernels belong to $\mathcal H_s(\mathbf R^{2d})$. Some parts of the approach here is related to the analysis in [6, 7], where Schatten–von Neumann properties for certain types of integral operators on compact manifolds are deduced. We show that the singular values of such operator satisfies the estimate

$$\begin{aligned} \varvec{\sigma }_k(T,\mathscr {B}_1,\mathscr {B}_2) \lesssim e^{-rk^{\frac{1}{2ds}}} \end{aligned}$$

(1.3)

for some $r>0$, when $\mathscr {B}_j$ stays between $\mathcal H_s(\mathbf R^{d})$ and its dual. If the $\mathcal H_s$-spaces and their duals are replaced by $\mathcal H_{0,s}$-spaces and their duals, then we also prove that (1.3) is true for every $r>0$. Furthermore, if $\mathcal H_s$-spaces and their duals are replaced by Schwartz spaces and their duals, then we prove

$$\begin{aligned} \varvec{\sigma }_k(T,\mathscr {B}_1,\mathscr {B}_2) \lesssim k^{-N} \end{aligned}$$

(1.4)

for every $N\ge 0$, which should be available in the literature.

These singular-value estimates are based on the fact that the operator classes here above possess convenient factorization properties, which are deduced in Sect. 4. More precisely, an operator class $\mathcal M$ is called a factorization algebra, if for every $T\in \mathcal M$, there exist $T_1,T_2\in \mathcal M$ such that $T=T_1\circ T_2$. (In [28] the term decomposition algebra is used instead of factorization algebra). Evidently, $\mathscr {L}(\mathscr {B})$, the set of continuous linear operators on the quasi-Banach space $\mathscr {B}$ is a factorization algebra, since we may choose $T_1$ as the identity operator and $T_2=T$. A more challenging situation appears when $\mathcal M$ does not contain the identity operator, and in this situation it is easy to find operator classes which are not factorization algebras. For example, any Schatten–von Neumann class of finite order is not a factorization algebra.

If $\mathscr {B}$ above is a Hilbert space and $\mathcal M$ is the set of compact operators on $\mathscr {B}$, then it follows by an application of the spectral theorem that $\mathcal M$ is a factorization algebra. It is also well-known that the set of linear operators with kernels in the Schwartz space is a factorization algebra (see e. g. [2, 17, 21, 28, 30]). Furthemore, similar facts hold true for the set of operators with kernels in a fixed Gelfand–Shilov space (cf. [28]).

In Sect. 4 we extend the latter property such that all Pilipović spaces are included. That is, we prove that the set of operators with kernels in a fixed (but arbitrarily chosen) Pilipović space is a factorization algebra.

Since the singular values of the operators under considerations either satisfy conditions of the form (1.3) or (1.4) for every $N\ge 0$, it follows that the sequence $\{ \varvec{\sigma }_k(T)\} _{k=1}^\infty$ belongs to $\ell ^p$ for every $p>0$. This implies that any such operator is a Schatten–von Neumann operator of degree p for every $p>0$.

Here we remark that the latter conclusions in the Gelfand–Shilov situation, were deduced in [28] in slightly different ways, which enables to replace the quasi-Banach spaces $\mathscr {B}_1$ and $\mathscr {B}_2$ by convenient Hilbert spaces. The main property behind the latter reduction concerns [25, Proposition 3.8], where it is proved that if $s\ge \frac{1}{2}$ and

$$\begin{aligned} \mathcal H_s(\mathbf R^{d})\subseteq \mathscr {B}\subseteq \mathcal H_s^{\prime }(\mathbf R^{d}), \end{aligned}$$

then there are Hilbert spaces $\mathscr {H}_1$ and $\mathscr {H}_2$ such that

$$\begin{aligned} \mathcal H_s(\mathbf R^{d})\subseteq \mathscr {H}_1 \subseteq \mathscr {B}\subseteq \mathscr {H}_2\subseteq \mathcal H_s^{\prime }(\mathbf R^{d}). \end{aligned}$$

(1.5)

The Schatten–von Neumann properties are then obtained in straight-forward ways by the factorization properties in combination with the exact formulas, for Hilbert–Schmidt norms of operators acting between Hilbert spaces. We remark that extensions of (1.5) which include Pilipović spaces can be found in [5].

Our investigations also include analysis of operators with kernels in $\mathcal H_{\flat _\sigma }$, $\mathcal H_{0,\flat _\sigma }$, $\sigma >0$, or their duals. These spaces were carefully investigated in [10, 26] and the Hermite coefficients of the involved functions should be bounded by expressions of the form $h^{|\alpha |}(\alpha !)^{-\frac{1}{2\sigma }}$. In [26], these spaces are characterized in different ways. For example, it is here proved that the Bargmann transform is bijective from $\mathcal H_{\flat _\sigma }(\mathbf R^{d})$ to the set of all entire functions F on $\mathbf C^{d}$ such that

$$\begin{aligned} |F(z)|\lesssim e^{c|z|^{\frac{2\sigma }{\sigma +1}}} \end{aligned}$$

for some constant $C>0$.

In Sect. 3 we deduce kernel theorems for operators with kernels in these spaces, or related distribution spaces. In Sect. 4 we show certain factorization properties of operators with kernels in $\mathcal H_{\flat _\sigma }$ or in $\mathcal H_{0,\flat _\sigma }$. These factorization results are slightly weaker compared to what is deduced for operators with kernels in $\mathcal H_s$ and $\mathcal H_{0,s}$ when $s\ge 0$ is real.

In Sect. 5 we apply these factorization properties to obtain singular value estimates for operators with kernels in $\mathcal H_{\flat _\sigma }$ or in $\mathcal H_{0,\flat _\sigma }$. In particular we show that if T is an operator on $L^2(\mathbf R^{d})$ with kernel in $\mathcal H_{\flat _\sigma }(\mathbf R^{2d})$, then the singular values of T satisfy

$$\begin{aligned} \varvec{\sigma }_k(T)\lesssim h^k (k!)^{-\frac{1}{2\sigma d}} \end{aligned}$$

for some constant $h>0$.

Finally, in Sect. 6 we apply the results from the first sections to obtain certain characterizations of operators with kernels in $\mathcal H_s$ and $\mathcal H_{0,s}$. Some arguments here involve estimates with modulation space norms, and a short introduction to modulation spaces are therefore included in Sect. 2.

2 Preliminaries

In this section we recall some basic facts. We start by discussing Pilipović spaces and their properties. Thereafter we consider suitable spaces of formal Hermite series expansions, and discuss their links with Pilipović spaces.

2.1 The Pilipović Spaces

We start by considering spaces which are obtained by suitable estimates of Gelfand–Shilov or Gevrey type when using powers of the harmonic oscillator $H\equiv |x|^2-\Delta$, $x\in \mathbf R^{d}$.

Let $h>0$, $s\ge 0$ and let $\varvec{\mathcal S}_{\! h,s}(\mathbf R^{d})$ be the set of all $f\in C^\infty (\mathbf R^{d})$ such that

$$\begin{aligned} \Vert f\Vert _{\varvec{\mathcal S}_{\! h,s}}\equiv \sup _{N\ge 0} \frac{\Vert H^Nf\Vert _{L^\infty }}{h^N(N!)^{2s}}<\infty . \end{aligned}$$

(2.1)

Then $\varvec{\mathcal S}_{\! h,s}(\mathbf R^{d})$ is a Banach space. If $h_\alpha$ is the Hermite function

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

on $\mathbf R^{d}$ of order $\alpha$, then $Hh_\alpha =(2|\alpha |+d)h_\alpha$. This implies that $\varvec{\mathcal S}_{\! h,s}(\mathbf R^{d})$ contains all Hermite functions when $s>0$, and if $s=0$, and $\alpha \in {\mathbf N}^{d}$ satisfies $2|\alpha |+d\le h$, then $h_\alpha \in \varvec{\mathcal S}_{\! h,s}(\mathbf R^{d})$.

We let

$$\begin{aligned} \varvec{\Sigma }_s(\mathbf R^{d}) \equiv \bigcap _{h>0}\varvec{\mathcal S}_{\! h,s}(\mathbf R^{d}) \quad \text {and}\quad \varvec{\mathcal S}_{\! s}(\mathbf R^{d}) \equiv \bigcup _{h>0}\varvec{\mathcal S}_{\! h,s}(\mathbf R^{d}) \end{aligned}$$

and equip these spaces by projective and inductive limit topologies, respectively, of $\varvec{\mathcal S}_{\! h,s}(\mathbf R^{d})$, $h>0$. (Cf. [13, 18, 19, 26].)

In [18, 19], Pilipović proved that if $s_1\ge \frac{1}{2}$ and $s_2>\frac{1}{2}$, then $\varvec{\mathcal S}_{\! s_1}(\mathbf R^{d})$ and $\varvec{\Sigma }_{s_2}(\mathbf R^{d})$ agree with the Gelfand–Shilov spaces $\mathcal S_{s_1}(\mathbf R^{d})$ and $\Sigma _{s_2}(\mathbf R^{d})$ ¹, respectively, and that

$$\begin{aligned} \varvec{\Sigma }_{1/2} (\mathbf R^{d})\ne \Sigma _{1/2} (\mathbf R^{d}) = \{ 0 \} . \end{aligned}$$

(See e. g. [26] for notations).

By the definitions it follows that the latter relations extend into

$$\begin{aligned} \varvec{\mathcal S}_{\! s_1}(\mathbf R^{d})&= \mathcal S_{s_1}(\mathbf R^{d}),&\quad \varvec{\Sigma }_{s_2}(\mathbf R^{d})&= \Sigma _{s_2}(\mathbf R^{d}),&\quad s_1&\ge \frac{1}{2},\ s_2 > \frac{1}{2} \end{aligned}$$

and

$$\begin{aligned} \varvec{\mathcal S}_{\! s_1}(\mathbf R^{d})&\ne \mathcal S_{s_1}(\mathbf R^{d})= \{ 0\},&\quad \varvec{\Sigma }_{s_2}(\mathbf R^{d})&\ne \Sigma _{s_2}(\mathbf R^{d})=\{ 0\} ,&\quad s_1&<\frac{1}{2},\ 0<s_2 \le \frac{1}{2} . \end{aligned}$$

The space $\varvec{\Sigma }_s(\mathbf R^{d})$ is called the Pilipović space (of Beurling type) of order $s\ge 0$ on $\mathbf R^{d}$. Similarly, $\varvec{\mathcal S}_{\! s}(\mathbf R^{d})$ is called the Pilipović space (of Roumieu type) of order $s\ge 0$ on $\mathbf R^{d}$.

The dual spaces of $\varvec{\mathcal S}_{\! h,s}(\mathbf R^{d})$, $\varvec{\Sigma }_s(\mathbf R^{d})$ and $\varvec{\mathcal S}_{\! s}(\mathbf R^{d})$ are denoted by $\varvec{\mathcal S}_{\! h,s}^{\prime }(\mathbf R^{d})$, $\varvec{\Sigma }_s^{\prime }(\mathbf R^{d})$ and $\varvec{\mathcal S}_{\! s}^{\prime }(\mathbf R^{d})$, respectively. We have

$$\begin{aligned} \varvec{\Sigma }_s^{\prime }(\mathbf R^{d}) = \bigcup _{h>0} \varvec{\mathcal S}_{\! h,s}^{\prime }(\mathbf R^{d}) \end{aligned}$$

when $s>0$ and

$$\begin{aligned} \varvec{\mathcal S}_{\! s}^{\prime }(\mathbf R^{d}) = \bigcap _{h>0} \varvec{\mathcal S}_{\! h,s}^{\prime }(\mathbf R^{d}) \end{aligned}$$

when $s\ge 0$, with inductive respectively projective limit topologies of $\varvec{\mathcal S}_{\! h,s}^{\prime }(\mathbf R^{d})$, $h>0$ (cf. [26]).

2.2 Quasi-Banach Spaces, Singular Values and Schatten–von Neumann Operators

Let $\mathscr {B}$ be a vector space. A quasi-norm $\Vert \, \cdot \, \Vert _{\mathscr {B}}$ on $\mathscr {B}$ is a non-negative and real-valued function on $\mathscr {B}$ which is non-degenerate in the sense that

$$\begin{aligned} \Vert f\Vert _{\mathscr {B}}=0\qquad \Longleftrightarrow \qquad f=0,\quad f\in \mathscr {B}, \end{aligned}$$

and fulfills

$$\begin{aligned} \Vert \alpha f\Vert _{\mathscr {B}}&= |\alpha |\cdot \Vert f\Vert _{\mathscr {B}},&\qquad f&\in \mathscr {B},\ \alpha \in \mathbf C\nonumber , \end{aligned}$$

and

$$\begin{aligned} \Vert f+g\Vert _{\mathscr {B}}&\le D(\Vert f\Vert _{\mathscr {B}} +\Vert g\Vert _{\mathscr {B}}),&\qquad f,g&\in \mathscr {B}, \end{aligned}$$

(2.2)

for some constant $D\ge 1$ which is independent of $f,g\in \mathscr {B}$. Then $\mathscr {B}$ is a topological vector space when the topology for $\mathscr {B}$ is defined by $\Vert \, \cdot \, \Vert _{\mathscr {B}}$, and $\mathscr {B}$ is called a quasi-Banach space if $\mathscr {B}$ is complete under this topology.

Let $\mathscr {B}_1$ and $\mathscr {B}_2$ be (quasi-)Banach spaces, and let T be a linear map between $\mathscr {B}_1$ and $\mathscr {B}_2$. Then the singular values of order $k\ge 1$ of T is given by (1.1), where the infimum is taken over all linear operators $T_0$ from $\mathscr {B}_1$ to $\mathscr {B}_2$ with rank at most $k-1$. Therefore, $\varvec{\sigma }_1(T)$ equals $\Vert T\Vert _{\mathscr {B}_1\rightarrow \mathscr {B}_2}$, and $\varvec{\sigma }_k(T)$ are non-negative and non-increasing with respect to k.

For any $p\in (0,\infty ]$, $\mathscr {I}_p(\mathscr {B}_1,\mathscr {B}_2)$, the set of Schatten–von Neumann operators of order p is the quasi-Banach space which consists of all linear and continuous operators T from $\mathscr {B}_1$ to $\mathscr {B}_2$ such that

$$\begin{aligned} \Vert T\Vert _{\mathscr {I}_p} = \Vert T\Vert _{\mathscr {I}_p(\mathscr {B}_1,\mathscr {B}_2)} \equiv \Vert ( \varvec{\sigma }_k(T,\mathscr {B}_1,\mathscr {B}_2) ) _{k=1}^\infty \Vert _{l^p} \end{aligned}$$

is finite.

2.3 Spaces of Hermite Series Expansions

Next we recall the definitions of topological vector spaces of Hermite series expansions, given in [26]. As in [26], it is convenient to use the sets $\mathbf R_\flat$ and $\overline{\mathbf R_\flat }$ when indexing our spaces.

Definition 2.1

The sets $\mathbf R_\flat$ and $\overline{\mathbf R_\flat }$ are given by

$$\begin{aligned} {\textstyle {\mathbf R_\flat = \mathbf R_+ \underset{\sigma >0}{\textstyle {\bigcup }} \{ \flat _\sigma \} }} \quad \text {and}\quad {\textstyle {\overline{\mathbf R_\flat } = \mathbf R_\flat \bigcup \{ 0 \} }}. \end{aligned}$$

Moreover, beside the usual ordering in $\mathbf R$, the elements $\flat _\sigma$ in $\mathbf R_\flat$ and $\overline{\mathbf R_\flat }$ are ordered by the relations $x_1<\flat _{\sigma _1}<\flat _{\sigma _2}<x_2$, when $\sigma _1<\sigma _2$, $x_1<\frac{1}{2}$ and $x_2\ge \frac{1}{2}$ are real.

A function $\vartheta$ on a discrete set $\Lambda$ is called a weight (on $\Lambda$) if it is real-valued and positive.

Definition 2.2

Let $p\in (0,\infty ]$, $s\in {\mathbf R_\flat }$, $r\in \mathbf R$, $\vartheta$ be a weight on ${\mathbf N}^{d}$, and let

$$\begin{aligned} \vartheta _{r,s}(\alpha )\equiv {\left\{ \begin{array}{ll} e^{r|\alpha |^{\frac{1}{2s}}}, &{}\quad \text {when}\quad s\in \mathbf R_+, \\ r^{|\alpha |}(\alpha !)^{\frac{1}{2\sigma }}, &{}\quad \text {when}\quad s = \flat _\sigma , \quad \qquad \alpha \in {\mathbf N}^{d}. \end{array}\right. } \end{aligned}$$

Then,

• (1) $\ell _0^{\prime } ({\mathbf N}^{d})$ is the set of all sequences $\{c_\alpha \} _{\alpha \in {\mathbf N}^{d}} \subseteq \mathbf C$ on ${\mathbf N}^{d}$;

• (2) $\ell _{0,0}({\mathbf N}^{d})\equiv \{ 0\}$, and $\ell _0({\mathbf N}^{d})$ is the set of all sequences $\{c_\alpha \} _{\alpha \in {\mathbf N}^{d}}\subseteq \mathbf C$ such that $c_\alpha \ne 0$ for at most finite numbers of $\alpha$;

• (3) $\ell ^p_{[\vartheta ]}({\mathbf N}^{d})$ is the quasi-Banach space which consists of all sequences $\{ c_\alpha \} _{\alpha \in {\mathbf N}^{d}} \subseteq \mathbf C$ such that

  $$\begin{aligned} \Vert \{ c_\alpha \} _{\alpha \in {\mathbf N}^{d}} \Vert _{\ell ^p_{[\vartheta ]}}\equiv \Vert \{ c_\alpha \vartheta (\alpha )\} _{\alpha \in {\mathbf N}^{d}} \Vert _{\ell ^p} \end{aligned}$$

  is finite;

• (4) $\ell _{0,s}({\mathbf N}^{d})\equiv \underset{r>0}{\bigcap }\ell ^p_{[\vartheta _{r,s}]}({\mathbf N}^{d})$ and $\ell _s({\mathbf N}^{d})\equiv \underset{r>0}{\bigcup }\ell ^p_{[\vartheta _{r,s}]}({\mathbf N}^{d})$, with projective respectively inductive limit topologies of $\ell ^p_{[\vartheta _{r,s}]}({\mathbf N}^{d})$ with respect to $r>0$;

• (5) $\ell _{0,s}^{\prime }({\mathbf N}^{d})\equiv \underset{r>0}{\bigcup }\ell ^p_{[1/\vartheta _{r,s}]}({\mathbf N}^{d})$ and $\ell _s^{\prime }({\mathbf N}^{d})\equiv \underset{r>0}{\bigcap }\ell ^p_{[1/\vartheta _{r,s}]}({\mathbf N}^{d})$, with inductive respectively projective limit topologies of $\ell ^p_{[1/\vartheta _{r,s}]}({\mathbf N}^{d})$ with respect to $r>0$.

Let $p\in (0,\infty ]$, and let $\Omega _N$ be the set of all $\alpha \in {\mathbf N}^{d}$ such that $|\alpha |\le N$. Then the topology of $\ell _0({\mathbf N}^{d})$ is defined by the inductive limit topology of the sets

$$\begin{aligned} \left\{ \, \{ c_\alpha \} _{\alpha \in {\mathbf N}^{d}} \in \ell _0^{\prime }({\mathbf N}^{d})\, ;\, c_\alpha =0\ \text {when}\ \alpha \notin \Omega _N\, \right\} \end{aligned}$$

with respect to $N\ge 0$, and whose topology is given through the quasi-semi-norms

$$\begin{aligned} \{ c_\alpha \} _{\alpha \in {\mathbf N}^{d}}\mapsto \Vert \{ c_\alpha \} _{|\alpha |\le N} \Vert _{\ell ^p(\Omega _N)}. \end{aligned}$$

(2.3)

Furthermore, the topology of $\ell _0^{\prime } ({\mathbf N}^{d})$ is defined by the quasi-semi-norms (2.3). It follows that $\ell _0^{\prime }({\mathbf N}^{d})$ is a Fréchet space, and that the topologies of $\ell _0({\mathbf N}^{d})$ and $\ell _0^{\prime }({\mathbf N}^{d})$ are independent of p.

Next we consider spaces of formal Hermite series expansions

$$\begin{aligned} f=\sum _{\alpha \in {\mathbf N}^{d}}c_\alpha h_\alpha ,\quad \{c_\alpha \} _{\alpha \in {\mathbf N}^{d}}\in \ell _0^{\prime } ({\mathbf N}^{d}), \end{aligned}$$

(2.4)

which correspond to

$$\begin{aligned} \ell _{0,s}({\mathbf N}^{d}),\quad \ell _s({\mathbf N}^{d}),\quad \ell ^p_{[\vartheta ]}({\mathbf N}^{d}), \quad \ell _s^{\prime }({\mathbf N}^{d})\quad \text {and}\quad \ell _{0,s}^{\prime }({\mathbf N}^{d}). \end{aligned}$$

(2.5)

For that reason we consider the mapping

$$\begin{aligned} T : \, \{ c_\alpha \} _{\alpha \in {\mathbf N}^{d}} \mapsto \sum _{\alpha \in {\mathbf N}^{d}} c_\alpha h_\alpha , \end{aligned}$$

(2.6)

between sequences and formal Hermite series expansions.

Definition 2.3

Let $p\in (0,\infty ]$, $\vartheta$ be a weight on ${\mathbf N}^{d}$, and let $s\in \overline{\mathbf R_\flat }$.

• The spaces

  $$\begin{aligned} \mathcal H_{0,s}(\mathbf R^{d}),\quad \mathcal H_s(\mathbf R^{d}),\quad \mathcal H^p_{[\vartheta ]}(\mathbf R^{d}), \quad \mathcal H_s^{\prime }(\mathbf R^{d})\quad \text {and}\quad \mathcal H_{0,s}^{\prime }(\mathbf R^{d}) \end{aligned}$$

  (2.7)

  are the images of T in (2.6) under the corresponding spaces in (2.5). Furthermore, the topologies of the spaces in (2.7) are inherited from corresponding spaces in (2.5);

• The quasi-norm $\Vert f\Vert _{\mathcal H^p_{[\vartheta ]}}$ of $f\in \mathcal H_0^{\prime }(\mathbf R^{d})$, is given by $\Vert \{ c_\alpha \} _{\alpha \in {\mathbf N}^{d}}\Vert _{\ell ^p_{[\vartheta ]}}$, when f is given by (2.4).

By the definitions it follows that the inclusions

$$\begin{aligned} \mathcal H_0(\mathbf R^{d})&\subseteq \mathcal H_{0,s}(\mathbf R^{d})\subseteq \mathcal H_{s}(\mathbf R^{d}) \subseteq \mathcal H_{0,t}(\mathbf R^{d}) \nonumber \\&\subseteq \mathscr {S}(\mathbf R^{d}) \subseteq \mathscr {S}^{\prime }(\mathbf R^{d}) \subseteq \mathcal H_{0,t}^{\prime }(\mathbf R^{d}) \subseteq \mathcal H_{s}^{\prime }(\mathbf R^{d}) \nonumber \\&\subseteq \mathcal H_{0,s}^{\prime }(\mathbf R^{d})\subseteq \mathcal H_0^{\prime }(\mathbf R^{d}), \quad \text {when}\ s,t\in \mathbf R_\flat ,\ s<t, \end{aligned}$$

(2.8)

hold true, and are in fact continuous embeddings.

Remark 2.4

By the definition it follows that T in (2.6) is a homeomorphism between any of the spaces in (2.5) and corresponding space in (2.7).

The next result shows that the spaces in Definition 2.3 essentially agrees with the Pilipović spaces. We refer to [26] for the proof.

Proposition 2.5

Let $0\le s\in \mathbf R$. Then $\mathcal H_{0,s}(\mathbf R^{d}) = \varvec{\Sigma }_s(\mathbf R^{d})$ and $\mathcal H_s(\mathbf R^{d}) = \varvec{\mathcal S}_s(\mathbf R^{d})$.

Remark 2.6

Let T be given by (2.6), $p\in [1,2]$ and let $\vartheta$ be a weight on ${\mathbf N}^{d}$ such that $1/\vartheta \in \ell ^\infty ({\mathbf N}^{d})$. Then

$$\begin{aligned} \begin{array}{llll} \big ( \ell _s({\mathbf N}^{d}), \ell ^2({\mathbf N}^{d}),\ell _s^{\prime }({\mathbf N}^{d}) \big ) \ &{} \overset{\overset{\scriptstyle {T}}{}}{\rightarrow } &{}\ \big ( \mathcal H_s(\mathbf R^{d}), L^2(\mathbf R^{d}),\mathcal H_s^{\prime }(\mathbf R^{d}) \big ),&{} s \ge 0, \\ \big ( \ell _{0,s}({\mathbf N}^{d}), \ell ^2({\mathbf N}^{d}),\ell _{0,s}^{\prime }({\mathbf N}^{d}) \big ) \ &{} \overset{\overset{\scriptstyle {T}}{}}{\rightarrow } \ &{} \big ( \mathcal H_{0,s}(\mathbf R^{d}), L^2(\mathbf R^{d}),\mathcal H_{0,s}^{\prime }(\mathbf R^{d}) \big ),&{} s>0, \\ \big ( \ell _{[\vartheta ]}^p({\mathbf N}^{d}), \ell ^2({\mathbf N}^{d}), \ell _{[1/\vartheta ]}^{p^{\prime }}({\mathbf N}^{d}) \big ) \ &{} \overset{\overset{\scriptstyle {T}}{}}{\rightarrow } \ &{} \big ( \mathcal H_{[\vartheta ]}^p(\mathbf R^{d}), L^2(\mathbf R^{d}), \mathcal H_{[1/\vartheta ]}^{p^{\prime }}(\mathbf R^{d}) \big )&{} \end{array} \end{aligned}$$

are isometric bijections between Gelfand triples. (Cf. e. g. Sect. 4 in [26].)

2.4 Pseudo-Differential Operators

We let $\mathscr {F}$ be the Fourier transform on $\mathscr {S}^{\prime } (\mathbf R^{d})$, given by

$$\begin{aligned} (\mathscr {F}f)(\xi ) = (2\pi )^{-d/2}\int _{\mathbf R^{d}}f(x)e^{-i\langle x,\xi \rangle }\, d\xi \end{aligned}$$

when $f\in L^1(\mathbf R^{d})$. We also let $\mathscr {F}_2F$ be the partial Fourier transform of F(x, y) with respect to the y-variable. The Fourier transform restricts to homeomorphisms on $\mathscr {S}(\mathbf R^{d})$, $\mathcal H_s(\mathbf R^{d})$ and $\mathcal H_{s,0}(\mathbf R^{d})$, and extends uniquely to homeomorphisms on corresponding duals. The same holds true for $\mathscr {F}_2$ when acting on functions and distributions on $\mathbf R^{2d}$ (cf. [26]).

For every $s\ge \frac{1}{2}$, real $d\times d$-matrix A and $a\in \mathcal S_s^{\prime }(\mathbf R^{2d})$ (the symbol), the pseudo-differential operator ${\text {Op}}_A(a)$ is the linear and continuous operator from $\mathcal S_s(\mathbf R^{d})$ to $\mathcal S_s^{\prime }(\mathbf R^{d})$ with distribution kernel

$$\begin{aligned} K_{A,a}(x,y) = (2\pi )^{-{d/2}}(\mathscr {F}_2^{-1}a)(x-A(x-y),x-y). \end{aligned}$$

If $a\in L^1(\mathbf R^{2d})$ and $f\in \mathcal S_s(\mathbf R^{d})$, then ${\text {Op}}_A(a)f$ is given by

$$\begin{aligned} {\text {Op}}_A(a)f(x) = (2\pi )^{-d}\iint _{\mathbf R^{2d}}a(x-A(x-y),\xi )f(y)e^{i\langle x-y,\xi \rangle } \, dyd\xi . \end{aligned}$$

The product for compositions of pseudo-differential operators on the symbol level is denoted by ${\#}_A$. This implies that if $a,b\in \mathscr {S}(\mathbf R^{2d})$, then $a{\#}_Ab$ is defined by

$$\begin{aligned} {\text {Op}}_A(a{\#}_Ab) ={\text {Op}}_A(a)\circ {\text {Op}}_A(b). \end{aligned}$$

The product $a{\#}_Ab$ is well-defined and is uniquely extendable in different ways (see e. g. [1, 4, 16]).

2.5 Modulation Spaces

Next we discuss basic properties for modulation spaces, and start by recalling the conditions for the involved weight functions. A function $\omega$ on $\mathbf R^{d}$ is called a weight (on $\mathbf R^{d}$), if $\omega >0$ and $\omega ,1/\omega \in L^\infty _{loc}(\mathbf R^{d})$.

Let $\omega$ be a weight on $\mathbf R^{d}$, and set $\langle x\rangle \equiv 1+|x|$ when $x\in \mathbf R^{d}$. Then $\omega$ is called a weight of polynomial type, if

$$\begin{aligned} \omega (x+y) \lesssim \omega (x)\langle y\rangle ^N,\quad x,y\in \mathbf R^{d}, \end{aligned}$$

(2.9)

for some $N\ge 0$. Here and in what follows we write $A\lesssim B$ when $A,B\ge 0$ and $A\le cB$ for a suitable constant $c>0$. We also let $A\asymp B$ when $A\lesssim B$ and $B\lesssim A$. We let $\mathscr {P}(\mathbf R^{d})$ be the set of all weights on $\mathbf R^{d}$ of polynomial type.

Let $\phi \in \mathscr {S}(\mathbf R^{d}){\setminus } 0$ be fixed. Then the short-time Fourier transform $V_\phi f$ of $f\in \mathscr {S}^{\prime } (\mathbf R^{d})$ with respect to the window function $\phi$ is defined by

$$\begin{aligned} V_\phi f(x,\xi ) \equiv (\mathscr {F}_2 (U(f\otimes \phi )))(x,\xi ) = \mathscr {F}(f \, \overline{\phi (\, \cdot \, -x)})(\xi ), \end{aligned}$$

where $(UF)(x,y)=F(y,y-x)$. If $f\in \mathscr {S}(\mathbf R^{d})$, then it follows that

$$\begin{aligned} V_\phi f(x,\xi ) = (2\pi )^{-d/2}\int _{\mathbf R^{d}}f(y)\overline{\phi (y-x)}e^{-i\langle y,\xi \rangle }\, dy . \end{aligned}$$

Let $\omega \in \mathscr {P}(\mathbf R^{2d})$, $p,q\in (0,\infty ]$ and $\phi \in \mathscr {S}(\mathbf R^{d}){\setminus } 0$ be fixed. Then the mixed Lebesgue space $L^{p,q}_{(\omega )}(\mathbf R^{2d})$ consists of all measurable functions F on $\mathbf R^{2d}$ such that $\Vert F\Vert _{L^{p,q}_{(\omega )}}<\infty$. Here

$$\begin{aligned} \Vert F\Vert _{L^{p,q}_{(\omega )}} \equiv \Vert F_{p,\omega }\Vert _{L^q}, \quad \text {where} \quad F_{p,\omega }(\xi ) \equiv \Vert F(\, \cdot \, ,\xi )\omega (\, \cdot \, ,\xi )\Vert _{L^p}. \end{aligned}$$

(2.10)

We note that these quasi-norms might attain $+\infty$.

The modulation space $M^{p,q}_{(\omega )}(\mathbf R^{d})$ is the quasi-Banach space which consist of all $f\in \mathscr {S}^{\prime } (\mathbf R^{d})$ such that $\Vert f\Vert _{M^{p,q}_{(\omega )}}<\infty$, where

$$\begin{aligned} \Vert f\Vert _{M^{p,q}_{(\omega )}}\equiv \Vert V_\phi f\Vert _{L^{p,q}_{(\omega )}}. \end{aligned}$$

(2.11)

For conveniency we set $M^{p,q}=M^{p,q}_{(\omega )}$ when $\omega \equiv 1$. We remark that the definition of $M^{p,q}_{(\omega )}(\mathbf R^{d})$ is independent of the choice of $\phi \in \mathscr {S}(\mathbf R^{d}){\setminus } 0$ and different $\phi$ gives rise to equivalent quasi-norms. (See e. g. [8, 9, 11, 14] for general properties of modulation spaces).

3 Kernel Theorems

In this section we deduce suitable kernel theorems for operators between Pilipović spaces and their duals. Since the spaces under considerations can in convenient ways be formulated in terms of Hermite series expansions, we may easily reduce ourselves to kernel results for matrix operators, in similar ways as in e. g. [20].

We begin with the following result concerning kernel properties of matrix operators. Here we identify linear operators on discrete sets by their matrices.

Proposition 3.1

Let $\vartheta _k$ be weight functions on ${\mathbf N}^{d_k}$, $k=1,2$, $\vartheta (\alpha ,\beta )= \vartheta _1(\beta )^{-1}\vartheta _2(\alpha )$, and let T be a linear and continuous map from $\ell ^1_{[\vartheta _1]}({\mathbf N}^{d_1})$ to $\ell ^\infty _{[\vartheta _2]}({\mathbf N}^{d_2})$. Then the following is true:

• (1) If $A\in \mathbb \ell ^\infty _{[\vartheta ]}({\mathbf N}^{d_2+d_1})$, then the map $f\mapsto A\cdot f$ from $\ell _0({\mathbf N}^{d_1})$ to $\ell _0^{\prime }({\mathbf N}^{d_2})$ extends uniquely to a linear and continuous map from $\ell ^1_{[\vartheta _1]}({\mathbf N}^{d_1})$ to $\ell ^\infty _{[\vartheta _2]}({\mathbf N}^{d_2})$;

• (2) there is a unique element $A\in \ell ^\infty _{[\vartheta ]}({\mathbf N}^{d_2+d_1})$ such that $Tf =A\cdot f$ for every $f\in \ell ^1_{[\vartheta _1]}({\mathbf N}^{d_1})$. Furthermore,

  $$\begin{aligned} \Vert T\Vert _{\ell ^1_{[\vartheta _1]}({\mathbf N}^{d_1})\rightarrow \ell ^\infty _{[\vartheta _2]}({\mathbf N}^{d_2})} = \Vert A\Vert _{\ell ^\infty _{[\vartheta ]}}. \end{aligned}$$

  (3.1)

Proof

The assertion (1) follows by straight-forward estimates and is left for the reader.

(2) Evidently, for some unique (matrix)

$$\begin{aligned} A= (a_{\alpha ,\beta })_{(\alpha , \beta )\in {\mathbf N}^{d_2+d_1}}\in \ell _0^{\prime }({\mathbf N}^{d_2+d_1}), \end{aligned}$$

$Tf =A\cdot f$ holds for every $f\in \ell _0({\mathbf N}^{d_1})$. Moreover, let

$$\begin{aligned} \Omega _1 {=} \{ \, f_1{\in } \ell _0({\mathbf N}^{d_1})\, ;\, \Vert f_1\Vert _{\ell ^1_{[\vartheta _1]}}\le 1\, \} \quad \text {and}\quad \Omega _2 = \{ \, f_2\in \ell _0({\mathbf N}^{d_2})\, ;\, \Vert f_2\Vert _{\ell ^1_{[1/\vartheta _2]}}\le 1\, \} . \end{aligned}$$

Since $\ell _0({\mathbf N}^{d_1})$ and $\ell _0({\mathbf N}^{d_2})$ are dense in $\ell ^1_{[\vartheta _1]}({\mathbf N}^{d_1})$ and $\ell ^1_{[1/\vartheta _2]}({\mathbf N}^{d_2})$, respectively, we obtain

$$\begin{aligned}&\Vert T\Vert _{\ell ^1_{[\vartheta _1]}({\mathbf N}^{d_1})\rightarrow \ell ^\infty _{[\vartheta _2]}({\mathbf N}^{d_2})} = \sup _{f_1\in \Omega _1}\sup _{f_2\in \Omega _2} |(A\cdot f_1,f_2)_{\ell ^2}| \\&\quad = \sup _{f_1\in \Omega _1}\Vert A\cdot f_1\Vert _{\ell ^\infty _{[\vartheta _2]}} = \sup _{\beta \in {\mathbf N}^{d_1}}\sup _{\alpha \in {\mathbf N}^{d_2}}|a_{\alpha ,\beta }\vartheta _1(\beta )^{-1}\vartheta _2(\alpha )| = \Vert A\Vert _{\ell ^\infty _{[\vartheta ]}}, \end{aligned}$$

which gives (2). $\square$

By the links between $\mathcal H^p_{[\vartheta _k]}(\mathbf R^{d_k})$ and $\mathcal H^p_{[\vartheta ]}(\mathbf R^{d_2}\times \mathbf R^{d_1})$, and $\ell ^p_{[\vartheta _k]}({\mathbf N}^{d_k})$ and $\ell ^p_{[\vartheta ]}({\mathbf N}^{d_2}\times {\mathbf N}^{d_1})$, respectively, the previous proposition immediately gives the following. (Cf. Remark 2.6.)

Proposition 3.2

Let $\vartheta _k$ be weight functions on ${\mathbf N}^{d_k}$, $k=1,2$, $\vartheta (\alpha _2,\beta )= \vartheta _1(\beta )^{-1}\vartheta _2(\alpha _2)$, and let T be a linear and continuous map from $\mathcal H^1_{[\vartheta _1]}(\mathbf R^{d_1})$ to $\mathcal H^\infty _{[\vartheta _2]}(\mathbf R^{d_2})$. Then the following is true:

• (1) If $K\in \mathcal H^\infty _{[\vartheta ]}(\mathbf R^{d_2}\times \mathbf R^{d_1})$, then the map

  $$\begin{aligned} f\mapsto \big ( x_2\mapsto \langle K(x_2,\, \cdot \, ),f\rangle \big ) \end{aligned}$$

  (3.2)

  from $\mathcal H_0(\mathbf R^{d_1})$ to $\mathcal H_0^{\prime }(\mathbf R^{d_2})$ extends uniquely to a linear and continuous map from $\mathcal H^1_{[\vartheta _1]}(\mathbf R^{d_1})$ to $\mathcal H^\infty _{[\vartheta _2]}(\mathbf R^{d_2})$;

• (2) there is a unique element $K\in \mathcal H^\infty _{[\vartheta ]}(\mathbf R^{d_2}\times \mathbf R^{d_1})$ such that

  $$\begin{aligned} Tf =\big ( x_2\mapsto \langle K(x_2,\, \cdot \, ),f\rangle \big ) \end{aligned}$$

  (3.3)

  for every $f\in \mathcal H^1_{[\vartheta _1]}(\mathbf R^{d_1})$. Furthermore,

  $$\begin{aligned} \Vert T\Vert _{\mathcal H^1_{[\vartheta _1]}(\mathbf R^{d_1})\rightarrow \mathcal H^\infty _{[\vartheta _2]}(\mathbf R^{d_2})} = \Vert K\Vert _{\mathcal H^\infty _{[\vartheta ]}}. \end{aligned}$$

  (3.4)

We now have the following kernel results.

Theorem 3.3

Let $s\in \overline{\mathbf R_\flat }$, and let T be the linear and continuous map from $\mathcal H_0(\mathbf R^{d_1})$ to $\mathcal H_0^{\prime }(\mathbf R^{d_2})$. Then the following is true:

• (1) if T is a linear and continuous map from $\mathcal H_s^{\prime }(\mathbf R^{d_1})$ to $\mathcal H_s(\mathbf R^{d_2})$, then there is $K\in \mathcal H_s(\mathbf R^{d_2}\times \mathbf R^{d_1})$ such that (3.3) holds true;

• (2) if T is a linear and continuous map from $\mathcal H_s(\mathbf R^{d_1})$ to $\mathcal H_s^{\prime }(\mathbf R^{d_2})$, then there is $K\in \mathcal H_s^{\prime }(\mathbf R^{d_2}\times \mathbf R^{d_1})$ such that (3.3) holds true.

The same holds true if the $\mathcal H_s$ and $\mathcal H_s^{\prime }$ spaces are replaced by $\mathcal H_{0,s}$ and $\mathcal H_{0,s}^{\prime }$ spaces, respectively, or by $\mathscr {S}$ and $\mathscr {S}^{\prime }$ spaces, respectively.

Theorem 3.4

Let $K\in \mathcal H_0^{\prime }(\mathbf R^{d_2}\times \mathbf R^{d_1})$, $s\in \overline{\mathbf R_\flat }$ and let T be the linear and continuous map from $\mathcal H_0(\mathbf R^{d_1})$ to $\mathcal H_0^{\prime }(\mathbf R^{d_2})$, given by (3.2). Then the following is true:

• (1) if $K\in \mathcal H_s(\mathbf R^{d_2}\times \mathbf R^{d_1})$, then T extends uniquely to a linear and continuous map from $\mathcal H_s^{\prime }(\mathbf R^{d_1})$ to $\mathcal H_s(\mathbf R^{d_2})$;

• (2) if $K\in \mathcal H_s^{\prime }(\mathbf R^{d_2}\times \mathbf R^{d_1})$, then T extends uniquely to a linear and continuous map from $\mathcal H_s(\mathbf R^{d_1})$ to $\mathcal H_s^{\prime }(\mathbf R^{d_2})$.

The same holds true if the $\mathcal H_s$ and $\mathcal H_s^{\prime }$ spaces are replaced by $\mathcal H_{0,s}$ and $\mathcal H_{0,s}^{\prime }$ spaces, respectively, or by $\mathscr {S}$ and $\mathscr {S}^{\prime }$ spaces, respectively.

Proof of Theorems 3.3 and 3.4

Let $p\in [1,\infty ]$,

$$\begin{aligned} \vartheta _r (\alpha )= {\left\{ \begin{array}{ll} e^{r|\alpha |^{\frac{1}{2s}}},\quad &{}\quad s\in \mathbf R_+{\textstyle {\bigcup }} \{ 0\} , \\ r^{|\alpha |}(\alpha !)^{\frac{1}{2\sigma }},\quad &{}\quad s=\flat _\sigma , \end{array}\right. } \end{aligned}$$

and $\sigma _r(\alpha )= \langle \alpha \rangle ^r$. The results follow from Proposition 3.2, and the facts that

$$\begin{aligned}&\mathcal H_s = \underset{r>0}{\textstyle {\bigcup }} \mathcal H^p_{[\vartheta _r]}, \quad \mathcal H_{0,s}^{\prime } = \underset{r>0}{\textstyle {\bigcup }} \mathcal H^p_{[1/\vartheta _r]}, \quad \mathscr {S}^{\prime } = \underset{r>0}{\textstyle {\bigcup }} \mathcal H^p_{[1/\sigma _r]}, \end{aligned}$$

with suitable inductive limit topologies, and

$$\begin{aligned}&\mathcal H_s^{\prime } = \underset{r>0}{\textstyle {\bigcap }} \mathcal H^p_{[1/\vartheta _r]}, \quad \mathcal H_{0,s} = \underset{r>0}{\textstyle {\bigcap }} \mathcal H^p_{[\vartheta _r]}, \quad \mathscr {S}= \underset{r>0}{\textstyle {\bigcap }} \mathcal H^p_{[\sigma _r]}, \end{aligned}$$

with suitable projective limit topologies. $\square$

Evidently, the assertions on $\mathscr {S}$ and $\mathscr {S}^{\prime }$ in Theorems 3.3 and 3.4 are well-known. For the other cases, the results are straight-forward consequences of the nuclearity of $\mathcal H^1_{[\vartheta ]}(\mathbf R^{d_2}\times \mathbf R^{d_1})$ (cf. e. g. [12] or [29]).

For completeness we also write down some of the corresponding results in the matrix case. The proofs follow by similar arguments as for the proofs of Theorems 3.4 and 3.3, and are left for the reader. Here we recall that $\ell _{\mathscr {S}}({\mathbf N}^{d_2}\times {\mathbf N}^{d_1})$ is the set of all matrices $A=(a_{\alpha ,\beta }) _{(\alpha ,\beta )\in {\mathbf N}^{d_2+d_1}}$ such that

$$\begin{aligned} |a_{\alpha ,\beta }| \lesssim \langle (\alpha ,\beta )\rangle ^{-N} \quad \text {for every}\quad N\ge 0 \end{aligned}$$

and  $\ell _{\mathscr {S}}^{\prime }({\mathbf N}^{d_2}\times {\mathbf N}^{d_1}$) is the set of all such matrices such that

$$\begin{aligned}&|a_{\alpha ,\beta }| \lesssim \langle (\alpha ,\beta )\rangle ^{N} \quad \text {for some}\quad N\ge 0. \end{aligned}$$

Theorem 3.5

Let $s\in \overline{\mathbf R_\flat }$ be real and let T be the linear and continuous map from $\ell _0({\mathbf N}^{d_1})$ to $\ell _0^{\prime }({\mathbf N}^{d_2})$ with matrix $A\in \ell _0^{\prime }({\mathbf N}^{d_2}\times {\mathbf N}^{d_1})$. Then the following is true:

• (1) if $A\in \ell _s({\mathbf N}^{d_2}\times {\mathbf N}^{d_1})$, then T extends uniquely to linear and continuous mappings from $\ell _s^{\prime }({\mathbf N}^{d_1})$ to $\ell _s({\mathbf N}^{d_2})$;

• (2) if $A\in \ell _s^{\prime }({\mathbf N}^{d_2}\times {\mathbf N}^{d_1})$, then T extends uniquely to linear and continuous mappings from $\ell _s({\mathbf N}^{d_1})$ to $\ell _s^{\prime }({\mathbf N}^{d_2})$.

• (3) if T is a linear and continuous map from $\ell _s^{\prime }({\mathbf N}^{d_1})$ to $\ell _s({\mathbf N}^{d_2})$, then $A\in \ell _s({\mathbf N}^{d_2}\times {\mathbf N}^{d_1})$;

• (4) if T is a linear and continuous map from $\ell _s({\mathbf N}^{d_1})$ to $\ell _s^{\prime }({\mathbf N}^{d_2})$, then $A\in \ell _s^{\prime }({\mathbf N}^{d_2}\times {\mathbf N}^{d_1})$.

The same holds true if $\ell _s$ and their duals are replaced by $\ell _{0,s}$ and their duals, respectively, or by $\ell _{\mathscr {S}}$ and their duals, respectively.

4 Factorizations of Pilipović and Gelfand–Shilov Kernels, and Pseudo-Differential Operators

In this section we deduce convenient factorization properties for operators with kernels in Pilipović spaces.

In what follows we use the convention that if $T_0$ is a linear and continuous operator from $\mathcal H_0(\mathbf R^{d_1})$ to $\mathcal H_0^{\prime }(\mathbf R^{d_2})$, and $g \in \mathcal H_0^{\prime }(\mathbf R^{d_0})$, then $T_0\otimes g$ is the linear and continuous operator from $\mathcal H_0(\mathbf R^{d_1})$ to $\mathcal H_0^{\prime }(\mathbf R^{d_2+d_0})$, given by

$$\begin{aligned} (T_0\otimes g)\, :\, f \mapsto (T_0f)\otimes g . \end{aligned}$$

In the following definition we recall that an operator T from $\mathcal H_0(\mathbf R^{d})$ to $\mathcal H_0^{\prime }(\mathbf R^{d})$ is called positive semi-definite, if $(Tf,f)_{L^2}\ge 0$, for every $f\in \mathcal H_0(\mathbf R^{d})$, and then we write $T\ge 0$.

Definition 4.1

Let $d_2\ge d_1$ and let T be a linear operator from $\mathcal H_0(\mathbf R^{d_1})$ to $\mathcal H_0^{\prime }(\mathbf R^{d_2})$. Then T is said to be a Hermite diagonal operator if $T=T_0\otimes g$, where the Hermite functions are eigenfunctions to $T_0$, and either $d_2=d_1$ and $g=1$, or $d_2>d_1$ and g is a Hermite function.

Moreover, if $T=T_0\otimes g$ is a Hermite diagonal operator and $T_0$ is positive semi-definite, then T is said to be a positive semi-definite Hermite diagonal operator.

The first part of the following result can be found in [2, 30] (see also [17, 21] and the references therein for an elementary proof).

Theorem 4.2

Let $s\in \mathbf R$, T be a linear and continuous operator from $\mathcal H_0(\mathbf R^{d_1})$ to $\mathcal H_0^{\prime }(\mathbf R^{d_2})$ with the kernel K, and let $d_0\ge \min (d_1,d_2)$. Then the following is true:

• (1) If $s\ge 0$ and $K\in \mathcal H_{s}(\mathbf R^{d_2+d_1})$, then there are operators $T_1$ and $T_2$ with kernels $K_1\in \mathcal H_{s}(\mathbf R^{d_0+d_1})$ and $K_2\in \mathcal H_{s}(\mathbf R^{d_2+d_0})$ respectively such that $T=T_2\circ T_1$. Furthermore, if $j \in \{ 1, 2\}$ is fixed and $d _0 \ge d _j$, then $T _j$ can be chosen as a positive semi-definte Hermite diagonal operator.

• (2) If $s>0$ and $K\in \mathcal H_{0,s}(\mathbf R^{d_2+d_1})$, then there are operators $T_1$ and $T_2$ with kernels $K_1\in \mathcal H_{0,s}(\mathbf R^{d_0+d_1})$ and $K_2\in \mathcal H_{0,s}(\mathbf R^{d_2+d_0})$ respectively such that $T=T_2\circ T_1$. Furthermore, if $j \in \{ 1, 2\}$ is fixed and $d _0 \ge d _j$, then $T _j$ can be chosen as a positive semi-definte Hermite diagonal operator.

The corresponding result for $s=\flat _\sigma$ reads:

Theorem 4.3

Let $\sigma >0$, T be a linear and continuous operator from $\mathcal H_0(\mathbf R^{d _1})$ to $\mathcal H^{\prime }_0(\mathbf R^{d _2})$ with the kernel K. Then the following is true.

• (1) If $K \in \mathcal H_{\flat _\sigma }(\mathbf R^{d _2 + d _1})$, then there are operators $T _0$, $T _1$ and $T _2$ with kernels $K _0 \in \mathcal H_{1/2} (\mathbf R^{d _2 + d _1})$, $K_1 \in \mathcal H_{\flat _{2\sigma }}(\mathbf R^{d _1 + d _1})$ and $K_2 \in \mathcal H_{\flat _{2\sigma }}(\mathbf R^{d _2 + d _2})$, respectively, and $T = T _2 \circ T _0 \circ T _1$. Furthermore, $T _1$ and $T _2$ can be chosen as positive semi-definite Hermite diagonal operators;

• (2) If $K \in \mathcal H_{0, \flat _\sigma }(\mathbf R^{d _2 + d _1})$, then there are operators $T _0$, $T _1$ and $T _2$ with kernels $K _0 \in \mathcal H_{0, 1/2} (\mathbf R^{d _2 + d _1})$, $K_1 \in \mathcal H_{0, \flat _{2\sigma }}(\mathbf R^{d _1 + d _1})$ and $K_2 \in \mathcal H_{0, \flat _{2\sigma }}(\mathbf R^{d _2 + d _2})$, respectively, and $T = T _2 \circ T _0 \circ T _1$. Furthermore, $T _1$ and $T _2$ can be chosen as positive semi-definite Hermite diagonal operators.

Remark 4.4

An operator with kernel in $\mathcal H_{s}(\mathbf R^{2d})$ is sometimes called a regularizing operator with respect to $\mathcal H_{s}(\mathbf R^{d})$, because it extends uniquely to a continuous map from (the large space) $\mathcal H_{s}^{\prime }(\mathbf R^{d})$ into (the small space) $\mathcal H_{s} (\mathbf R^{d})$. Analogously, an operator with kernel in $\mathcal H_{0,s}(\mathbf R^{2d})$ ($\mathscr {S} (\mathbf R^{2d})$) is sometimes called a regularizing operator with respect to $\mathcal H_{0,s}(\mathbf R^{d})$ ($\mathscr {S}(\mathbf R^{d})$).

Proof of Theorem 4.2

First we assume that $d_0= d_1$, and start to prove (1). Let $h_{d,\alpha }(x)$ be the Hermite function on $\mathbf R^{d}$ of order $\alpha \in \mathbf N^d$. Then K posseses the expansion

$$\begin{aligned} K(x,y) = \sum _{\alpha \in \mathbf N^{d_2}}\sum _{\beta \in \mathbf N^{d_1}} a_{\alpha ,\beta }h_{d_2,\alpha }(x)h_{d_1,\beta }(y), \end{aligned}$$

(4.1)

where the coefficients $a_{\alpha ,\beta }$ satisfies

$$\begin{aligned} \sup _{\alpha ,\beta } |a_{\alpha ,\beta }e^{r(|\alpha |^{\frac{1}{2s}}+|\beta |^{\frac{1}{2s}})}| <\infty , \end{aligned}$$

(4.2)

for some $r>0$.

Now let $z\in \mathbf R^{d_1}$, and

$$\begin{aligned} \begin{array}{ll} K_{0,2}(x,z) &{}= \sum \limits _{\alpha \in \mathbf N^{d_2}}\sum \limits _{\beta \in \mathbf N^{d_1}} b_{\alpha ,\beta }h_{d_2,\alpha }(x)h_{d_1,\beta }(z), \\ K_{0,1}(z,y) &{}= \sum \limits _{\alpha ,\beta \in \mathbf N^{d_1}} c_{\alpha ,\beta }h_{d_1,\alpha }(z)h_{d_1,\beta }(y), \end{array} \end{aligned}$$

(4.3)

where

$$\begin{aligned} b_{\alpha ,\beta } = a_{\alpha ,\beta } e^{\frac{r}{2}|\beta |^{\frac{1}{2s}}}\quad \text {and}\quad c_{\alpha ,\beta } = \delta _{\alpha ,\beta }e^{-\frac{r}{2}|\alpha |^{\frac{1}{2s}}}. \end{aligned}$$

Here $\delta _{\alpha ,\beta }$ is the Kronecker delta. Then it follows that

$$\begin{aligned} \int K_{0,2}(x,z)K_{0,1}(z,y)\, dz = \sum _{\alpha \in \mathbf N^{d_2}}\sum _{\beta \in \mathbf N^{d_1}} a_{\alpha ,\beta }h_{d_2,\alpha }(x)h_{d_1,\beta }(y) = K(x,y). \end{aligned}$$

Hence, if $T_j$ is the operator with kernel $K_{0,j}$, $j=1,2$, then $T=T_2\circ T_1$. Furthermore,

$$\begin{aligned} \sup _{\alpha ,\beta } |b_{\alpha ,\beta }e^{\frac{r}{2}(|\alpha |^{\frac{1}{2s}}+|\beta |^{\frac{1}{2s}})}| \le \sup _{\alpha ,\beta } |a_{\alpha ,\beta }e^{r(|\alpha |^{\frac{1}{2s}}+|\beta |^{\frac{1}{2s}})}| <\infty \end{aligned}$$

and

$$\begin{aligned} \sup _{\alpha ,\beta } |c_{\alpha ,\beta }e^{\frac{r}{4}(|\alpha |^{\frac{1}{2s}}+|\beta |^{\frac{1}{2s}})}| = \sup _{\alpha } |e^{-\frac{r}{4} |\alpha |^{\frac{1}{2s}}}e^{\frac{r}{4} |\alpha |^{\frac{1}{2s}}}| <\infty . \end{aligned}$$

This implies that $K_{0,1}\in \mathcal H_{s}(\mathbf R^{d_1+d_1})$ and $K_{0,2}\in \mathcal H_{s}(\mathbf R^{d_2+d_1})$, and (1) follows with $K_1=K_{0,1}$ and $K_2=K_{0,2}$, in the case $d_0=d_1$.

In order to prove (2), we assume that $K\in \mathcal H_{0,s}(\mathbf R^{d_2+d_1})$, and we let $a_{\alpha ,\beta }$ be the same as the above. Then (4.2) holds for any $r>0$, which implies that if $n\ge 0$ is an integer, then

$$\begin{aligned} \Theta _n \equiv \sup \{ \, |\beta |\, ;\, |a_{\alpha ,\beta }|\ge e^{-2(n+1)(|\alpha |^{\frac{1}{2s}}+|\beta |^{\frac{1}{2s}})}\ \text {for some}\ \alpha \in \mathbf N^{d_2}\, \} \end{aligned}$$

(4.4)

is finite.

We let

$$\begin{aligned} I_1 = \{ \, \beta \in \mathbf N^{d_1}\, ;\, |\beta |\le \Theta _1+1\, \} , \end{aligned}$$

and define inductively

$$\begin{aligned} I_j = \{ \, \beta \in \mathbf N^{d_1}{\setminus } I_{j-1}\, ;\, |\beta |\le \Theta _j+j\, \} , \quad j\ge 2. \end{aligned}$$

Then

$$\begin{aligned} I_j\cap I_k=\emptyset \quad \text {when}\quad j\ne k,\quad \text {and}\quad \bigcup _{j\ge 0} I_j=\mathbf N^{d_1}, \end{aligned}$$

and by the definitions it follows that $I_j$ is a finite set for every j.

We also let $K_{0,1}$ and $K_{0,2}$ be given by (4.3), where, if $\beta \in I_j$,

$$\begin{aligned} b_{\alpha _2,\beta } = a_{\alpha _2,\beta }e^{j|\beta |^{\frac{1}{2s}}}\quad \text {and}\quad c_{\alpha _1,\beta } = \delta _{\alpha _1,\beta }e^{-j|\beta |^{\frac{1}{2s}}}, \end{aligned}$$

when $\alpha _1\in \mathbf N^{d_1}$, $\alpha _2\in \mathbf N^{d_2}$. If $T_\ell$ is the operator with kernel $K_{0,\ell }$ for $\ell =1,2$, then it follows that $T_2\circ T_1 =T$. Furthermore, if $r>0$, then we have

$$\begin{aligned} \sup _{\alpha ,\beta } |b_{\alpha ,\beta } e^{r (|\alpha |^{\frac{1}{2s}}+|\beta |^{\frac{1}{2s}})}| \le J_1+J_2, \end{aligned}$$

where

$$\begin{aligned} J_1 = \sup _{j\le r+1}\sup _\alpha \sup _{\beta \in I_j}|b_{\alpha ,\beta } e^{r (|\alpha |^{\frac{1}{2s}}+|\beta |^{\frac{1}{2s}})}|, \end{aligned}$$

(4.5)

and

$$\begin{aligned} J_2 = \sup _{j> r+1}\sup _\alpha \sup _{\beta \in I_j}|b_{\alpha ,\beta } e^{r (|\alpha |^{\frac{1}{2s}}+|\beta |^{\frac{1}{2s}})}|. \end{aligned}$$

(4.6)

Since only finite numbers of $\beta$ is involved in the suprema in (4.5), it follows from (4.2) and the definition of $b_{\alpha ,\beta }$ that $J_1$ is finite.

For $J_2$ we have

$$\begin{aligned} J_2= & {} \sup _{j> r+1}\sup _\alpha \sup _{\beta \in I_j}|a_{\alpha ,\beta } e^{r|\alpha |^{\frac{1}{2s}}+(r+j)|\beta |^{\frac{1}{2s}}}| \\\le & {} \sup _{j> r+1}\sup _\alpha \sup _{\beta \in I_j}|e^{-2j(|\alpha |^{\frac{1}{2s}}+|\beta |^{\frac{1}{2s}})} e^{r|\alpha |^{\frac{1}{2s}}+(r+j)|\beta |^{\frac{1}{2s}}}| <\infty , \end{aligned}$$

where the first inequality follows from (4.4). Hence

$$\begin{aligned} \sup _{\alpha ,\beta } |b_{\alpha ,\beta } e^{r (|\alpha |^{\frac{1}{2s}}+|\beta |^{\frac{1}{2s}})}|< \infty \end{aligned}$$

for every $r>0$, which implies that $K_{0,2}\in \mathcal H_{0,s}(\mathbf R^{d_2+d_1})$.

If we now replace $b_{\alpha ,\beta }$ with $c_{\alpha ,\beta }$ in the definition of $J_1$ and $J_2$, it follows by similar arguments that both $J_1$ and $J_2$ are finite, also in this case. This gives

$$\begin{aligned} \sup _{\alpha ,\beta } |c_{\alpha ,\beta } e^{r (|\alpha |^{\frac{1}{2s}}+|\beta |^{\frac{1}{2s}})}|< \infty \end{aligned}$$

for every $r>0$. Hence $K_1\in \mathcal H_{0,s}(\mathbf R^{d_1+d_1})$, and (2) follows in the case $d_0=d_1$.

Next assume that $d_0>d_1$, and let $d=d_0-d_1\ge 1$. Then we set

$$\begin{aligned} K_1(z_0,y) = K_{0,1}(z_1,y)h_{d,0}(z)\quad \text {and}\quad K_2(x,z_0) = K_{0,2}(x,z_1)h_{d,0}(z), \end{aligned}$$

where $K_{0,j}$ are the same as in the first part of the proofs, $z_1\in \mathbf R^{d_1}$ and $z\in \mathbf R^{d}$, giving that $z_0=(z_1,z)\in \mathbf R^{d_0}$. We get

$$\begin{aligned} \int _{\mathbf R^{d_0}}K_2(x,z_0)K_1(z_0,y)\, dz_0 = \int _{\mathbf R^{d_1}}K_{0,2}(x,z_1)K_{0,1}(z_1,y)\, dz_1 = K(x,y). \end{aligned}$$

The assertion (1) now follows in the case $d_0>d_1$ from the implications

$$\begin{aligned} K_{0,1} \in \mathcal H_{s}(\mathbf R^{d_1+d_1})\quad \Longrightarrow \quad K_1 \in \mathcal H_{s}(\mathbf R^{d_0+d_1}) \end{aligned}$$

and

$$\begin{aligned} K_{0,2} \in \mathcal H_{s}(\mathbf R^{d_2+d_1})\quad \Longrightarrow \quad K_2 \in \mathcal H_{s}(\mathbf R^{d_2+d_0}) \end{aligned}$$

Since the same implications hold after the $\mathcal H_{s}$ spaces have been replaced by $\mathcal H_{0,s}$ spaces, the assertion (2) also follows in the case $d_0>d_1$, and the theorem follows in the case $d_0\ge d_1$.

It remains to prove the result in the case $d_0\ge d_2$. By taking the adjoint, the roles of $j=1$ and $j=2$ are interchanged, and the result follows when $d_0 \ge d_2$ as well. The proof is complete. $\square$

Proof of Theorem 4.3

(1) We have

$$\begin{aligned} K(x, y) = \sum _{\alpha \in \mathbf{N} ^{d_2}} \sum _{\beta \in {\mathbf N}^{d_1}} a _{\alpha , \beta } h_{d _2, \alpha }(x) h _{d _1, \beta }(y), \end{aligned}$$

(4.7)

where

$$\begin{aligned} \sup _{\alpha , \beta }|a _{\alpha , \beta }(\alpha ! \beta !) ^{\frac{1}{2\sigma }} R ^{-(|\alpha | + |\beta |)}| < \infty \end{aligned}$$

for some $R > 1$.

Let $z _j\in \mathbf R^{d _j}$, and

$$\begin{aligned} K _0(z_2, z_1) = \sum _{\alpha \in \mathbf{N} ^{d_2}} \sum _{\beta \in {\mathbf N}^{d_1}} a _{0, \alpha , \beta } h_{d _2, \alpha }(z_2) h _{d _1, \beta }(z_1), \end{aligned}$$

(4.8)

$$\begin{aligned} K _1(z_1, y) = \sum _{\alpha \in \mathbf{N} ^{d_1}} \sum _{\beta \in {\mathbf N}^{d _1}} a _{1, \alpha , \beta } h_{d _1, \alpha }(z_1) h _{d_1, \beta }(y) \end{aligned}$$

(4.9)

and

$$\begin{aligned} K _2(x,z_2) = \sum _{\alpha \in \mathbf{N} ^{d _2}} \sum _{\beta \in {\mathbf N}^{d _2}} a _{2, \alpha , \beta } h_{d _2, \alpha }(x) h _{d_2, \beta }(z_2), \end{aligned}$$

(4.10)

where

$$\begin{aligned} a _{j, \alpha , \beta }&= (\alpha !) ^{-\frac{1}{2\sigma }} \delta _{\alpha , \beta } R^{2|\alpha |},&\quad \alpha ,\beta&\in {\mathbf N}^{d_j}, \quad j=1, 2, \end{aligned}$$

and

$$\begin{aligned} a _{0, \alpha , \beta }&= a _{\alpha , \beta } (\alpha ! \beta !) ^{\frac{1}{2\sigma }} R^ {-2(|\alpha | + |\beta |)},&\quad \alpha&\in {\mathbf N}^{d_2},\ \beta \in {\mathbf N}^{d_1}. \end{aligned}$$

Then it follows that

$$\begin{aligned} \iint _{\mathbf R^{d_2+d _1}}K _2(x, z_2) K _0(z_2, z_1) K _1(z_1, y) \, d z_2 d z_1 = K(x, y). \end{aligned}$$

Hence, if $T _j$ is the operator with kernel $K _j$, $j =0, 1, 2$, then $T = T _2 \circ T _0 \circ T _1$. Furthermore, the kernels lie in the claimed spaces since

$$\begin{aligned} \sup _{\alpha , \beta }|a _{j, \alpha , \beta }(\alpha ! \beta !) ^{\frac{1}{4\sigma }} R ^{-(|\alpha | + |\beta |)}| < \infty , \quad \alpha ,\beta \in {\mathbf N}^{d_j} \quad j = 1, 2, \end{aligned}$$

and if $0<c<\log R$, then

$$\begin{aligned} \sup _{\alpha , \beta }|a _{0, \alpha , \beta } e^{c(|\alpha | + |\beta |)}| \le \sup _{\alpha , \beta }|R ^{-(|\alpha | + |\beta |)} e^{c(|\alpha | + |\beta |)}| < {\infty }. \end{aligned}$$

Next we prove (2). Let $a_{\alpha ,\beta }$ be as in (4.7). Then

$$\begin{aligned} \sup _{\alpha , \beta }|a _{\alpha , \beta }(\alpha ! \beta !) ^{\frac{1}{2\sigma }} R^{|\alpha |+ |\beta |}| < \infty \end{aligned}$$

for every $R > 1$, which implies that

$$\begin{aligned} \Theta _{1,n}&\equiv \sup \{ \, |\beta |\, ;\, |a_{\alpha ,\beta }|\ge (n+1) ^{-6(|\alpha |+ |\beta |)} (\alpha ! \beta !) ^{-\frac{1}{2\sigma }}\ \text {for some}\ \alpha \in {\mathbf N}^{d _2}\, \} \end{aligned}$$

and

$$\begin{aligned} \Theta _{2,n}&\equiv \sup \{ \, |\alpha |\, ;\, |a_{\alpha ,\beta }|\ge (n+1) ^{-6(|\alpha |+ |\beta |)} (\alpha ! \beta !) ^{-\frac{1}{2\sigma }}\ \text {for some}\ \beta \in {\mathbf N}^{d _1}\, \} \end{aligned}$$

are finite for every $n\ge 1$.

We let

$$\begin{aligned} I_{j,1}&= \{ \, \gamma \in {\mathbf N}^{d _j}\, ;\, |\gamma |\le \Theta _{j,1}+1\, \} ,\\ \text {and define inductively}\\ I_{j,m}&= \{ \, \gamma \in {\mathbf N}^{d _j}{\setminus } I_{j,m-1}\, ;\, |\gamma |\le \Theta _{j,m}+m\, \} ,\quad m\ge 2,\ j=1,2. \end{aligned}$$

Then

$$\begin{aligned} I_{j,m}\cap I_{j,n}=\emptyset \quad \text {when}\quad m\ne n, \quad \text {and}\quad \underset{m\ge 1}{\textstyle {\bigcup }} I_{j,m}={\mathbf N}^{d _j}. \end{aligned}$$

and by the definitions it follows that $I_{j,m}$ is a finite set for every m.

We also let $K_j$, $j = 0,1, 2$ be given by (4.8)–(4.10), where

$$\begin{aligned} a _{j, \alpha , \beta }&= (\alpha !) ^{-\frac{1}{2\sigma }} \delta _{\alpha , \beta } m ^{-|\alpha +\beta |},&\quad \alpha&\in I_{j,m}, \quad j=1, 2, \end{aligned}$$

and

$$\begin{aligned} a _{0, \alpha , \beta }&= a _{\alpha , \beta } (\alpha ! \beta !) ^{\frac{1}{2\sigma }} m_2^{2|\alpha |}m_1 ^{2|\beta |},&\quad \alpha&\in I_{2,m_2},\ \beta \in I_{1,m_1}. \end{aligned}$$

If $T _j$ is the operator with kernel $K _j$ for $j = 0, 1, 2$, then it follows that $T _2 \circ T _0 \circ T _1 = T$ and $T_1, T_2$ are positive semi-definite Hermite diagonal operators. The result therefore follows if we prove

$$\begin{aligned} \begin{array}{lll} |a_{j,\alpha ,\beta }| &{}\lesssim r^{|\alpha +\beta |}(\alpha !\beta !) ^{-\frac{1}{4\sigma }},\ &{} \forall r>0,\quad j=1,2,\quad \text {and}\\ |a_{0,\alpha ,\beta }| &{}\lesssim e^{-r(|\alpha |+|\beta |)},\ &{}\forall r>0, \end{array} \end{aligned}$$

(4.11)

and since

$$\begin{aligned} \underset{m\le R+1}{\textstyle {\bigcup }}I_{j,m} \quad \text {and}\quad \underset{m_1+m_2\le 2R}{\textstyle {\bigcup }}I_{2,m_2}\times I_{1,m_1} \end{aligned}$$

are finite sets and $R>1$ is arbitrary, it suffices to prove

$$\begin{aligned} \sup _{m> R+1}\sup _{\alpha \in I _{j,m}} | (\alpha ! \beta !) ^{\frac{1}{4\sigma }}R ^{|\alpha + \beta |} a_{j, \alpha ,\beta }|&<\infty , \quad j=1,2, \end{aligned}$$

(4.12)

and

$$\begin{aligned} \sup _{(m_1,m_2)\in M_k}\sup _{\alpha \in I _{2,m_2}} \sup _{\beta \in I _{1,m_1}} | R ^{|\alpha | + |\beta |} a_{0, \alpha ,\beta }|&<\infty ,\quad k=1,2,3, \end{aligned}$$

(4.13)

where

$$\begin{aligned} M_1&= \{ \, (m_1,m_2)\in \mathbf Z^2_+\, ;\, m_1\ge 2R-1,\ m_2=1\, \} , \\ M_2&= \{ \, (m_1,m_2)\in \mathbf Z^2_+\, ;\, m_2\ge 2R-1,\ m_1=1\, \} \end{aligned}$$

and

$$\begin{aligned} \qquad \quad \; M_3&= \{ \, (m_1,m_2)\in \mathbf Z^2_+\, ;\, m_1+m_2\ge 2R,\ m_1,m_2\ge 2\, \} , \end{aligned}$$

We have

$$\begin{aligned}&\sup _{m> R+1}\sup _{\alpha \in I _{j,m}} | (\alpha ! \beta !) ^{\frac{1}{4\sigma }}R ^{|\alpha + \beta |} a_{j, \alpha ,\beta }| \\&\qquad \qquad \qquad \qquad \qquad = \sup _{m> R+1}\sup _{\alpha , \beta \in I _{j,m}} |\delta _{\alpha , \beta }R ^{|\alpha + \beta |} m ^{-|\alpha + \beta |} | < \infty , \quad j = 1, 2, \end{aligned}$$

and (4.12) follows.

Next we prove (4.13), and start with the case $k=1$. Then $\beta \in I_{1,m_1}$ gives

$$\begin{aligned} |a_{\alpha ,\beta }|\le m_1^{-6(|\alpha |+|\beta |)}(\alpha !\beta !)^{-\frac{1}{2\sigma }}, \end{aligned}$$

which, by the fact that $m_2=1$, implies

$$\begin{aligned} |a_{0,\alpha ,\beta }|=|a_{\alpha ,\beta }|(\alpha !\beta !)^{\frac{1}{2\sigma }}m_1^{2|\beta |} \le m_1^{-4(|\alpha |+|\beta |)}. \end{aligned}$$

Hence,

$$\begin{aligned} \sup _{(m_1,m_2)\in M_1}\sup _{\alpha \in I _{2,1}} \sup _{\beta \in I _{1,m_1}} | R ^{|\alpha | + |\beta |} a_{0, \alpha ,\beta }| \le \sup _{m_1>R>1}R ^{|\alpha | + |\beta |} m_1^{-4(|\alpha | + |\beta |)} <\infty , \end{aligned}$$

and (4.13) follows in the case $k=1$.

In the same way, (4.13) follows in the case $k=2$.

Next we prove (4.13) in the case $k=3$. By the definitions it follows that if $\alpha \in I_{2,m_2}$, then

$$\begin{aligned} |a_{\alpha ,\beta }| \le m_2^{-6(|\alpha |+|\beta |)}(\alpha !\beta !)^{-\frac{1}{2\sigma }}, \quad \forall \ \beta \in {\mathbf N}^{d_1}, \end{aligned}$$

and if $\beta \in I_{1,m_1}$, then

$$\begin{aligned} |a_{\alpha ,\beta }| \le m_1^{-6(|\alpha |+|\beta |)}(\alpha !\beta !)^{-\frac{1}{2\sigma }}, \quad \forall \ \alpha \in {\mathbf N}^{d_2}. \end{aligned}$$

Hence, if $\alpha \in I_{2,m_2}$ and $\beta \in I_{1,m_1}$, the geometric mean of the right-hand sides of the inequalities becomes

$$\begin{aligned} |a_{\alpha ,\beta }| \le (m_1m_2)^{-3(|\alpha |+|\beta |)}(\alpha !\beta !)^{-\frac{1}{2\sigma }}, \end{aligned}$$

giving that

$$\begin{aligned} |a_{0,\alpha ,\beta }| \le (m_1m_2)^{-(|\alpha |+|\beta |)}. \end{aligned}$$

This gives

$$\begin{aligned}&\sup _{(m_1,m_2)\in M_3}\sup _{\alpha \in I _{2,m_2}}\sup _{\beta \in I _{1,m_1}} | R ^{|\alpha | + |\beta |} a_{0, \alpha ,\beta }| \\&\qquad \qquad \qquad \qquad \qquad \le \sup _{(m_1,m_2)\in M_3}\sup _{\alpha \in I _{2,m_2}}\sup _{\beta \in I _{1,m_1}} R ^{|\alpha | + |\beta |} (m_1m_2)^{-(|\alpha |+|\beta |)} < \infty , \end{aligned}$$

and (4.13), and thereby (4.11) follows. $\square$

Remark 4.5

Let $\sigma >0$ and $T\ge 0$ be a Hermite diagonal operator on $L^2(\mathbf R^{d})$ with kernel K in $\mathcal H_{\flat _\sigma }$. By the proof of Theorem 4.3, there are Hermite diagonal operators $T_1\ge 0$ and $T_2\ge 0$ on $L^2(\mathbf R^{d})$ with kernels $K_1$ and $K_2$ such that

$$\begin{aligned} K_1\in \mathcal H_{\flat _\sigma }(\mathbf R^{2d}),\quad K_2\in \mathcal H_{1/2}(\mathbf R^{2d}) \quad \text {and}\quad T=T_1\circ T_2=T_2\circ T_1. \end{aligned}$$

In fact, if K is given by (4.7) with $d_1=d_2=d$, it suffices to let $K_1$ and $K_2$ be given by (4.9) and (4.10), where

$$\begin{aligned} a _{1, \alpha , \beta } = R^{|\alpha +\beta |}(a _{\alpha , \beta })^{1/2} (\alpha !\beta !)^{-\frac{1}{4\sigma }} \quad \text {and}\quad a _{2, \alpha , \beta } = R^{-|\alpha +\beta |}(a _{\alpha , \beta })^{1/2}(\alpha !\beta !)^{\frac{1}{4\sigma }} \end{aligned}$$

with $R\ge 1$ sufficiently large.

Remark 4.6

From the construction of $K_1$ and $K_2$ in the proofs of Theorems 4.2 and 4.3, it follows that it is not so difficult to use numerical methods for approximations of candidates to $K_1$ and $K_2$. In fact, $K_1$ and $K_2$ are formed explicitly by the elements of the matrix for T, when the Hermite functions are used as basis for $\mathscr {S}$, $\mathcal H_s$ and $\mathcal H_{0,s}$.

We finish the section by presenting some consequences in the calculus of pseudo-differential operators. The following result is an immediate consequence of Theorem 4.2 and the fact that the map $a\mapsto K_{A,a}$ is continuous and bijective on $\mathcal S_{s_1}(\mathbf R^{2d})$, and on $\Sigma _{s_2}(\mathbf R^{2d})$, for every $s_1\ge \frac{1}{2}$ and $s_2>\frac{1}{2}$.

Theorem 4.7

Let A be a real $d\times d$-matrix, $s_1\ge \frac{1}{2}$ and let $s_2>\frac{1}{2}$. Then the following is true:

• (1) if $a\in \mathcal H_{s_1}(\mathbf R^{2d})$, then there are $a_1,a_2\in \mathcal H_{s_1}(\mathbf R^{2d})$ such that $a=a_1{\#}_Aa_2$;

• (2) if $a\in \mathcal H_{0,s_2}(\mathbf R^{2d})$, then there are $a_1,a_2\in \mathcal H_{0,s_2}(\mathbf R^{2d})$ such that $a=a_1{\#}_Aa_2$.

Remark 4.8

Extensions of Theorem 4.7 to the case where $s_1$ and $s_2$ are allowed to be smaller than $\frac{1}{2}$ is not so smooth, because those Pilipović spaces which are not Gelfand–Shilov spaces, are not invariant under dilations (cf. [26, Proposition 7.4]). However, if A is a real $d\times d$ matrix and $a\in \mathscr {S}(\mathbf R^{2d})$ is such that the kernel $K_{A,a}$ belongs to $\mathcal H_s(\mathbf R^{2d})$, then we may apply Theorem 4.2 in this situation as well.

Therefore, let $\mathcal G_{A,s}(\mathbf R^{2d})$ ($\mathcal G_{0,A,s}(\mathbf R^{2d})$) be the set of all $a\in \mathscr {S}(\mathbf R^{2d})$ such that $K_{A,a}\in \mathcal H_s(\mathbf R^{2d})$ ($K_{A,a}\in \mathcal H_{0,s}(\mathbf R^{2d})$). If $a\in \mathcal G_{A,s}(\mathbf R^{2d})$ ($a\in \mathcal G_{0,A,s}(\mathbf R^{2d})$), then there are elements $a_1,a_2\in \mathcal G_{A,s}(\mathbf R^{2d})$ ($a_1,a_2\in \mathcal G_{0,A,s}(\mathbf R^{2d})$) such that $a=a_1{\#}_Aa_2$.

5 Singular Value Estimates and Schatten–von Neumann Properties for Operators with Pilipović Kernels

In this section we use Theorem 4.2 to obtain estimates of the form (1.3) for operators T with kernels in Pilipović spaces of order s, provided $\mathscr {B}_1$ and $\mathscr {B}_2$ stay between the given Pilipović space and its dual. In particular it follows that any such operator belongs to any Schatten–von Neumann class.

In the following result we show that the singular values for operators $T_K$ with kernels K in Pilipović spaces or Schwartz spaces, obey estimates of the form

$$\begin{aligned} \varvec{\sigma }_k(T_K,\mathscr {B}_1,\mathscr {B}_2)&\lesssim e^{-rk^{\frac{1}{2ds}}}, \end{aligned}$$

(5.1)

$$\begin{aligned} \varvec{\sigma }_k(T_K,\mathscr {B}_1,\mathscr {B}_2)&\lesssim R^{k} (k!)^{-\frac{1}{2\sigma d}} \end{aligned}$$

(5.2)

or

$$\begin{aligned} \varvec{\sigma }_k(T_K,\mathscr {B}_1,\mathscr {B}_2)&\lesssim k^{-N}. \end{aligned}$$

(5.3)

Here $V_1\hookrightarrow V_2$ means that the topological space $V_1$ is continuously embedded in the topological space $V_2$.

Theorem 5.1

Let $p\in (0,\infty ]$, $s\ge 0$ be real, $\sigma >0$ and let $d=\min (d_1,d_2)$. Then the following is true:

• (1) if $K\in \mathcal H_s(\mathbf R^{d_2+d_1})$, and $\mathscr {B}_1$ and $\mathscr {B}_2$ are quasi-Banach spaces such that $\mathscr {B}_1\hookrightarrow \mathcal H_s^{\prime }(\mathbf R^{d_1})$ and $\mathcal H_s(\mathbf R^{d_2})\hookrightarrow \mathscr {B}_2$, then (5.1) holds for some $r>0$. In particular, $T_K\in \mathscr {I}_p(\mathscr {B}_1,\mathscr {B}_2)$;

• (2) if $K\in \mathcal H_{0,s}(\mathbf R^{d_2+d_1})$, and $\mathscr {B}_1$ and $\mathscr {B}_2$ are quasi-Banach spaces such that $\mathscr {B}_1\hookrightarrow \mathcal H_{0,s}^{\prime }(\mathbf R^{d_1})$ and $\mathcal H_{0,s}(\mathbf R^{d_2})\hookrightarrow \mathscr {B}_2$, then (5.1) holds for every $r>0$. In particular, $T_K\in \mathscr {I}_p(\mathscr {B}_1,\mathscr {B}_2)$;

• (3) if $K\in \mathcal H_{\flat _\sigma }(\mathbf R^{d_2+d_1})$, and $\mathscr {B}_1$ and $\mathscr {B}_2$ are quasi-Banach spaces such that $\mathscr {B}_1\hookrightarrow \mathcal H_{1/2}^{\prime }(\mathbf R^{d_1})$ and $\mathcal H_{1/2}(\mathbf R^{d_2})\hookrightarrow \mathscr {B}_2$, then (5.2) holds for some $R>0$. In particular, $T_K\in \mathscr {I}_p(\mathscr {B}_1,\mathscr {B}_2)$;

• (4) if $K\in \mathcal H_{0,\flat _\sigma }(\mathbf R^{d_2+d_1})$, and $\mathscr {B}_1$ and $\mathscr {B}_2$ are quasi-Banach spaces such that $\mathscr {B}_1\hookrightarrow \mathcal H_{0,1/2}^{\prime }(\mathbf R^{d_1})$ and $\mathcal H_{0,1/2}(\mathbf R^{d_2})\hookrightarrow \mathscr {B}_2$, then (5.2) holds for every $R>0$. In particular, $T_K\in \mathscr {I}_p(\mathscr {B}_1,\mathscr {B}_2)$;

• (5) if $K\in \mathscr {S}(\mathbf R^{d_2+d_1})$, and $\mathscr {B}_1$ and $\mathscr {B}_2$ are quasi-Banach spaces such that $\mathscr {B}_1\hookrightarrow \mathscr {S}^{\prime } (\mathbf R^{d_1})$ and $\mathscr {S}(\mathbf R^{d_2})\hookrightarrow \mathscr {B}_2$, then (5.3) holds for every $N>0$. In particular, $T_K\in \mathscr {I}_p(\mathscr {B}_1,\mathscr {B}_2)$.

We observe that Theorem 5.1 (5) should be available in the literature.

We need some preparations for the proof. First we recall that if $\mathscr {B}_j$, $j=0,1,2$, are quasi-Banach spaces and $T_j$ are linear and continuous mappings from $\mathscr {B}_{j-1}$ to $\mathscr {B}_j$, $j=1,2$, then there is a constant C such that

$$\begin{aligned} \varvec{\sigma }_k(T_2\circ T_1,\mathscr {B}_0,\mathscr {B}_2)&\le C \Vert T_1\Vert _{\mathscr {B}_0\rightarrow \mathscr {B}_1} \varvec{\sigma }_k(T_2,\mathscr {B}_1,\mathscr {B}_2) \end{aligned}$$

(5.4)

and

$$\begin{aligned} \varvec{\sigma }_k(T_2\circ T_1,\mathscr {B}_0,\mathscr {B}_2)&\le C \Vert T_2\Vert _{\mathscr {B}_1\rightarrow \mathscr {B}_2} \varvec{\sigma }_k(T_1,\mathscr {B}_0,\mathscr {B}_1). \end{aligned}$$

(5.5)

In fact, if $\Omega _{j,l}(k)$ is the set of all linear operators from $\mathscr {B}_j$ to $\mathscr {B}_l$ with rank at most $k-1$, then

$$\begin{aligned} \varvec{\sigma }_k(T_2\circ T _1,\mathscr {B}_0,\mathscr {B}_2)&= \inf _{S\in \Omega _{0,2}(k)} \Vert T_2\circ T_1-S\Vert _{\mathscr {B}_0\rightarrow \mathscr {B}_2} \\&\le \inf _{T_0\in \Omega _{0,1}(k)} \Vert T_2\circ T_1-T_2\circ T_0\Vert _{\mathscr {B}_0\rightarrow \mathscr {B}_2} \\&\lesssim \Vert T_2\Vert _{\mathscr {B}_1\rightarrow \mathscr {B}_2}\left( \inf _{T_0\in \Omega _{0,1}(k)} \Vert T_1-T_0\Vert _{\mathscr {B}_0\rightarrow \mathscr {B}_1} \right) \\&=\Vert T_2\Vert _{\mathscr {B}_1\rightarrow \mathscr {B}_2} \varvec{\sigma }_k(T _1,\mathscr {B}_0,\mathscr {B}_1), \end{aligned}$$

which gives (5.5). In the same way (5.4) is obtained. (See also [22]).

Proof of Theorem 5.1

We only prove (1), (3) and (5). The assertions (2) and (4) follow by similar arguments and are left for the reader.

(1) By Theorem 4.2 we get

$$\begin{aligned} T_K = T_{K_3}\circ T_{K_2}\circ T_{K_1}, \end{aligned}$$

(5.6)

where the kernels $K_1$, $K_2$ and $K_3$ of the operators $T_{K_1}$, $T_{K_2}$ and $T_{K_3}$ belong to $\mathcal H_s(\mathbf R^{d_1+d_1})$, $\mathcal H_s(\mathbf R^{{d_2}+{d_1}})$ and $\mathcal H_s(\mathbf R^{d_2+d_2})$, respectively, and $T_{K_2}$ is a positive semi-definite Hermite diagonal operator.

It follows that $T_{K_1}$ is continuous from $\mathscr {B}_1$ to $L^2(\mathbf R^{d_1})$, and $T_{K_3}$ is continuous from $L^2(\mathbf R^{d_2})$ to $\mathscr {B}_2$. Hence, by (5.4) and (5.5) it suffices to prove that, if $T=T_{K_2}$, then

$$\begin{aligned} \varvec{\sigma }_k = \varvec{\sigma }_k(T,L^2(\mathbf R^{d_1}),L^2(\mathbf R^{d_2})) \le Ce^{-ck^{\frac{1}{2ds}}}, \end{aligned}$$

(5.7)

for some positive constants c and C.

By the constructions we have, setting $d_0=|d_1-d_2|$,

$$\begin{aligned} K_2(x,y)&= K_{0,2}(x,y_1)h(y_2),&\quad y&=(y_1,y_2),&\ x,y_1&\in \mathbf R^{d},&\ y_2&\in \mathbf R^{d_0}, \end{aligned}$$

when $d_1\ge d_2,$ and

$$\begin{aligned} K_2(x,y)&= h(x_1)K_{0,2}(x_2,y),&\quad x&=(x_1,x_2),&\ x_2,y&\in \mathbf R^{d},&\ x_1&\in \mathbf R^{d_0} \end{aligned}$$

when $d_2\ge d_1$, where

$$\begin{aligned} K_{0,2}(x,y)&= \sum _{\alpha \in {\mathbf N}^{d}}c_\alpha h_\alpha (x)h_\alpha (y), \quad x,y\in \mathbf R^{d}, \end{aligned}$$

(5.8)

with

$$\begin{aligned} 0\le c_\alpha&\lesssim e^{-r|\alpha |^{\frac{1}{2s}}} \end{aligned}$$

(5.9)

for some constant $r>0$. Here h is a fixed Hermite function on $\mathbf R^{d_0}$ when $d_0>0$, and $h\equiv 1$ otherwise. We observe that (5.8) describes the spectral decomposition of $T_{K_{0,2}}$, with $\{ h_\alpha \} _{\alpha \in {\mathbf N}^{d_0}}$ as the orthonormal basis of eigenfunctions, and with eigenvalues $\{ c_\alpha \} _{\alpha \in {\mathbf N}^{d_0}}$. Furthermore, it is evident that

$$\begin{aligned} \varvec{\sigma }_k(T_{K_{0,2}},L^2(\mathbf R^{d}),L^2(\mathbf R^{d})) = \varvec{\sigma }_k(T_{K_2},L^2(\mathbf R^{d_1}),L^2(\mathbf R^{d_2})),\quad k\ge 1. \end{aligned}$$

Hence it suffices to prove (5.7) in the case $d_1=d_2=d$.

Let $M_{N,d}$ be the number of all multi-indices $\alpha \in {\mathbf N}^{d}$ such that $|\alpha |\le N$. Then $M_{N,d}\asymp \langle N\rangle ^d$. Since the singular values are the eigenvalues of $T_{K_2}$ in non-increasing order, (5.9) gives

$$\begin{aligned} \varvec{\sigma }_k(T_{K_2}) \lesssim e^{-rN ^{\frac{1}{2s}}} \end{aligned}$$

for some $r>0$ when $M_{N-1,d}<k\le M_{N,d}$. For such k we also have $k\asymp N^d$, since $(N-1)^d\asymp N^d$. By combining these estimates we get

$$\begin{aligned} \varvec{\sigma }_k (T_K)\lesssim e^{-rN ^{\frac{1}{2s}}} \lesssim e^{-r_0k ^{\frac{1}{2ds}}} \end{aligned}$$

for some constant $r_0$. This gives (5.1).

(3) By Theorem 4.3 and Remark 4.5, we get

$$\begin{aligned} T _K = T _{K _{0,2}} \circ T _{K _2} \circ T _{K _0} \circ T _{K _1} \circ T _{K _{0,1}}, \end{aligned}$$

(5.10)

where the corresponding kernels satisfy

$$\begin{aligned} K_{0,j} \in \mathcal H_{1/2} (\mathbf R^{d_j+d _j}), \quad K_j \in \mathcal H_{\flat _{2\sigma }}(\mathbf R^{d_j+d_j}) ,\quad \text {and}\quad K_0\in \mathcal H_{1/2} (\mathbf R^{d_2+d_1}), \end{aligned}$$

$j=1,2$. Furthermore, all kernels except $K_0$ to the operators in (5.10) are positive semi-definite Hermite diagonal operators.

It follows that

$$\begin{aligned}&T _{K _{0,1}}\, :\, \mathscr {B}_1 \rightarrow L ^2(\mathbf R^{d_1}), \quad T_{K_0}\, :\, L^2(\mathbf R^{d_1}) \rightarrow L^2(\mathbf R^{d_2}) \\&\qquad \qquad \qquad \text {and}\quad T _{K _{0,2}}\, :\, L ^2(\mathbf R^{d_2}) \rightarrow \mathscr {B}_2, \end{aligned}$$

are continuous. By similar arguments as in the proof of (1), we get

$$\begin{aligned} \varvec{\sigma }_k(T _{K _{j}}, L ^2({\mathbf R^{d_j}}), L ^2({\mathbf R^{d_j}})) \lesssim R^{k} (k!)^{-\frac{1}{2\sigma d_j}}, \quad j = 1, 2. \end{aligned}$$

Hence,

$$\begin{aligned} \varvec{\sigma }_k(T _K,\mathscr {B}_1, \mathscr {B}_2)\lesssim R^{k} (k!)^{-\frac{1}{2\sigma d_j}}, \quad j=1,2, \end{aligned}$$

in view of (5.4)–(5.5). This gives (3).

(5) By [2, 17, 21, 28, 30] we get

$$\begin{aligned} T_K = T_{K_3}\circ T_{K_2}\circ T_{K_1}, \end{aligned}$$

(5.11)

where the kernels $K_1$, $K_2$ and $K_3$ of the operators $T_{K_1}$, $T_{K_2}$ and $T_{K_3}$ belong to $\mathscr {S}(\mathbf R^{d_1+d_1})$, $\mathscr {S}(\mathbf R^{d_2+d_1})$ and $\mathscr {S}(\mathbf R^{d_2+d_2})$, respectively. Furthermore, we may assume that $T_{K_2}$ is a positive semi-definite Hermite diagonal operator (cf. e. g. [28]).

It follows that $T_{K_1}$ is continuous from $\mathscr {B}_1$ to $L^2(\mathbf R^{d_1})$, and $T_{K_3}$ is continuous from $L^2(\mathbf R^{d_2})$ to $\mathscr {B}_2$. Hence, by (5.4) and (5.5) it suffices to prove that for every $N>0$ there is a constant $C>0$ such that

$$\begin{aligned} \varvec{\sigma }_k = \varvec{\sigma }_k(T_{K_{0,2}},L^2(\mathbf R^{d}),L^2(\mathbf R^{d})) \le Ck^{-N}. \end{aligned}$$

(5.12)

where $K_{0,2}$ is the same as in the proof of (1). By the construction, $c_\alpha$ in (5.8) fulfills

$$\begin{aligned} 0\le c_\alpha \lesssim \langle \alpha \rangle ^{-N} \end{aligned}$$

for every $N>0$, and by similar arguments as in the proof of (1) we get

$$\begin{aligned} \varvec{\sigma }_k \lesssim k^{-\frac{N}{d}} \end{aligned}$$

for every N, and (5) follows.

Finally, by (5.1)–(5.3) it also follows that $\{ \varvec{\sigma }_k(T,\mathscr {B}_1,\mathscr {B}_2)\}$ belongs to $\ell ^p$ for every $p>0$. This gives the second parts of (1)–(5). $\square$

6 Discrete Characterizations of Kernels to Smoothing Operators

In this section we show that any operators with kernels in Gelfand–Shilov, Pilipović or Schwartz spaces can be characterized by convenient expansions of the form

$$\begin{aligned} K \ = \ \sum _{j=1}^\infty \lambda _j f_{1,j}\otimes f_{2,j}, \qquad \{\lambda _j \} _{j=1}^\infty \subseteq \mathbf R_+ \end{aligned}$$

(6.1)

for some

$$\begin{aligned} \{f_{k,j} \} _{j=1}^\infty \subseteq L^2(\mathbf R^{d_k}),\quad k=1,2. \end{aligned}$$

(6.2)

In fact, the following result is an extension of Lemma 3.2 in [24].

Theorem 6.1

Let $p\in [1,\infty ]$ and T be a linear and continuous operator from $\mathcal H_0(\mathbf R^{d_1})$ to $\mathcal H_0^{\prime }(\mathbf R^{d_2})$ with kernel K. Then the following is true:

• (1) if $K\in \mathscr {S}(\mathbf R^{d_2}\times \mathbf R^{d_1})$, then (6.1) holds for some orthogonal sequences in (6.2) such that

  $$\begin{aligned} \sup _{j\ge 1} \big (j^N\lambda _j \big )<\infty \quad \text {and}\quad \sup _{j\ge 1}\left( j^N\Vert x^\alpha D^\beta f_{k,j}\Vert _{L^p(\mathbf R^{d_k})}\right) <\infty \end{aligned}$$

  (6.3)

  for $k=1,2$, $\alpha ,\beta \in {\mathbf N}^{d}$ and every $N\ge 0$.

• (2) if $K\in C^\infty (\mathbf R^{d_2}\times \mathbf R^{d_1})$ satisfies (6.1) and (6.3) for $k=1,2$ and every $N\ge 1$, then $K\in \mathscr {S}(\mathbf R^{d_2}\times \mathbf R^{d_1})$.

The corresponding characterizations of operators with Pilipović kernels are given in the following theorem. Here recall that the harmonic oscillator is given by $H= |x|^2-\Delta$, $x\in \mathbf R^{d}$.

Theorem 6.2

Let $p\in [1,\infty ]$, $s>0$, $d=\min (d_1,d_2)$ and T be a linear and continuous operator from $\mathcal H_0(\mathbf R^{d_1})$ to $\mathcal H_0^{\prime }(\mathbf R^{d_2})$ with kernel K. Then the following is true:

• (1) if $K\in \mathcal H_s (\mathbf R^{d_2}\times \mathbf R^{d_1})$ ($K\in \mathcal H_{0,s} (\mathbf R^{d_2}\times \mathbf R^{d_1})$), then (6.1) holds for some orthogonal sequences in (6.2) such that

  $$\begin{aligned} \sup _{j\ge 1} \big (e^{r\cdot j^{\frac{1}{2ds}}} \lambda _j \big )<\infty \quad \text {and}\quad \sup \left( \frac{e^{r\cdot j^{\frac{1}{2ds}}} \Vert H^N f_{k,j}\Vert _{L^p(\mathbf R^{d_k})}}{h^N(N!)^{2s}}\right) <\infty \end{aligned}$$

  (6.4)

  for $k=1,2$ and some $h> 0$ and $r>0$ (every $h> 0$ and $r>0$), where the latter supremum is taken over all $j\ge 0$ and $N\ge 0$;

• (2) if $K\in C^\infty (\mathbf R^{d_2}\times \mathbf R^{d_1})$ satisfies (6.1) and (6.4) for $k=1,2$ and some $r>0$ (every $r>0$), then $K\in \mathcal H_s (\mathbf R^{d_2}\times \mathbf R^{d_1})$ ($K\in \mathcal H_{0,s} (\mathbf R^{d_2}\times \mathbf R^{d_1})$).

We need some preparations for the proof. First we observe that $\mathcal H^p_{[\vartheta ]}$ possesses the expected interpolation properties. (Cf. [3].)

Lemma 6.3

Let $\theta \in [0,1]$, $\vartheta$, $\vartheta _1$ and $\vartheta _2$ be weights on ${\mathbf N}^{d}$, and let $p,p_1,p_2\in [1,\infty ]$ be such that

$$\begin{aligned} \frac{1}{p}=\frac{1-\theta }{p_1}+\frac{\theta }{p_2} \quad \text {and}\quad \vartheta = \vartheta _1^{1-\theta }\vartheta _2^\theta . \end{aligned}$$

Then

$$\begin{aligned} (\mathcal H^{p_1}_{[\vartheta _1]}(\mathbf R^{d}),\mathcal H^{p_2}_{[\vartheta _2]}(\mathbf R^{d})) _{[\theta ]} =\mathcal H^p_{[\vartheta ]}(\mathbf R^{d}). \end{aligned}$$

Proof

The result follows from the fact that the map

$$\begin{aligned} \{ c_\alpha \} _{\alpha \in {\mathbf N}^{d}} \mapsto \sum _{\alpha \in {\mathbf N}^{d}} c_\alpha h_\alpha \end{aligned}$$

is bijective and isometric from $\ell ^p_{[\vartheta ]}({\mathbf N}^{d})$ to $\mathcal H^p_{[\vartheta ]}(\mathbf R^{d})$, and that

$$\begin{aligned} \left( \ell ^{p_1}_{[\vartheta _1]}({\mathbf N}^{d}),\ell ^{p_2}_{[\vartheta _2]}({\mathbf N}^{d})\right) _{[\theta ]} =\ell ^p_{[\vartheta ]}({\mathbf N}^{d}). \end{aligned}$$

$\square$

We also need the following extension of [27, Proposition 5.5] on powers of non-negative self-adjoint operators on $L^2(\mathbf R^{d})$.

Proposition 6.4

Let $s\ge 0$, $t>0$ and let T be a self-adjoint and non-negative operator on $L^2(\mathbf R^{d})$ with kernel K in $\mathcal H_s(\mathbf R^{d}\times \mathbf R^{d})$. Then the following is true:

• (1) the kernel of $T^t$ belongs to $\mathcal H_s(\mathbf R^{d}\times \mathbf R^{d})$;

• (2) $T^t$ is continuous from $\mathcal H_s^{\prime }(\mathbf R^{d})$ to $\mathcal H_s(\mathbf R^{d})$.

The same holds true if the $\mathcal H_s$ and $\mathcal H_s^{\prime }$ spaces are replaced by $\mathcal H_{0,s}$ and $\mathcal H_{0,s}^{\prime }$ spaces, respectively, or by $\mathscr {S}$ and $\mathscr {S}^{\prime }$ spaces, respectively.

Proof

We only prove the result when $K\in \mathcal H_s(\mathbf R^{d}\times \mathbf R^{d})$. The other cases follow by similar arguments and are left for the reader.

Let

$$\begin{aligned} \Omega = \{ \, z\in \mathbf C\, ;\, 0<{\text {Re}}(z)<1\, \} \end{aligned}$$

and $T_0(z)=T^z$ when $z\in \overline{\Omega }$. Then the map $z\mapsto T(z)$ with values in $\mathscr {L}(L^2(\mathbf R^{d}))$ is continuous on $\overline{\Omega }$ and analytic on $\Omega$.

Furthermore, by writing $T^z=T^x\circ T^{iy}$ when $z=x+iy$, and using that $T^{iy}$ is bounded on $L^2(\mathbf R^{d})$ when $y\in \mathbf R$, it follows from the assumptions that

$$\begin{aligned} \sup _{y\in \mathbf R}\Vert T_0(iy)\Vert _{L^2(\mathbf R^{d})\rightarrow L^2(\mathbf R^{d})}&\le 1, \\ \sup _{y\in \mathbf R}\Vert T_0(1+iy)\Vert _{L^2(\mathbf R^{d})\rightarrow \mathcal H^2_{[\vartheta _r]}(\mathbf R^{d})}&<\infty \\ \end{aligned}$$

and

$$\begin{aligned} \sup _{z\in \overline{\Omega }}\Vert T_0(z)\Vert _{L^2(\mathbf R^{d})\rightarrow L^2(\mathbf R^{d})}&\le \sup _{0\le x\le 1} \Vert T^x\Vert _{L^2(\mathbf R^{d})\rightarrow L^2(\mathbf R^{d})} \end{aligned}$$

for some $r>0$, where $\vartheta _r(\alpha )=e^{r|\alpha |^{\frac{1}{2s}}}$.

It now follows from Lemma 6.3 and Calderon-Lion’s interpolation theorem (cf. Theorem IX.20 in [20]) that $T^t$ is continuous from $L^2(\mathbf R^{d})$ to $\mathcal H^2_{[\vartheta _{rt}]}(\mathbf R^{d})$. Duality gives that

$$\begin{aligned} T^t\, :\, L^2(\mathbf R^{d}) \rightarrow \mathcal H^2_{[\vartheta _{rt}]}(\mathbf R^{d}) \end{aligned}$$

and

$$\begin{aligned} T^t\, :\, \mathcal H^2_{[1/\vartheta _{rt}]}(\mathbf R^{d}) \rightarrow L^2(\mathbf R^{d}) \end{aligned}$$

are continuous. Hence, by interpolation we obtain that

$$\begin{aligned}&T^t\, :\, \mathcal H^2_{[1/\vartheta _{rt/2}]}(\mathbf R^{d}) \rightarrow \mathcal H^2_{[\vartheta _{rt/2}]}(\mathbf R^{d}) \end{aligned}$$

is continuous (cf. Remark 2.6), and the result follows from

$$\begin{aligned} \mathcal H_s(\mathbf R^{d}) = \bigcup _{r>0} \mathcal H^2_{[\vartheta _{r}]}(\mathbf R^{d}) \quad \text {and}\quad \mathcal H_s^{\prime }(\mathbf R^{d}) = \bigcap _{r>0} \mathcal H^2_{[1/\vartheta _{r}]}(\mathbf R^{d}). \end{aligned}$$

$\square$

We also need the following characterization of Pilipović kernels. Here recall that $\mathscr {P}(\mathbf R^{d})$ is the set of polynomially moderated weights on $\mathbf R^{d}$ (cf. Sect. 2).

Lemma 6.5

Let $p, q \in (0, \infty ]$, $\omega \in \mathscr {P}(\mathbf R^{ 2d _2} \times \mathbf R^{2d_1})$, $s>0$, $K\in \mathcal H^{\prime }_0 (\mathbf R^{d _2} \times \mathbf R^{d_1})$,

$$\begin{aligned} H_1=|x_1|^2-\Delta _{x_1} \quad and\quad H_2=|x_2|^2-\Delta _{x_2},\qquad x = (x_2,x_1)\in \mathbf R^{d_2}\times \mathbf R^{d_1}. \end{aligned}$$

Also let $H=H_2+H_1$ be the Harmonic oscillator on $\mathbf R^{d_2}\times \mathbf R^{d_1}$. Then the following conditions are equivalent:

• (1) $K\in \mathcal H_s(\mathbf R^{d_2}\times \mathbf R^{d_1})$    ($K\in \mathcal H_{0,s}(\mathbf R^{d_2}\times \mathbf R^{d_1})$);

• (2) for some $h>0$    (for every $h>0$) it holds

  $$\begin{aligned} \Vert H ^N K\Vert _{L ^2} \lesssim h ^N (N !) ^{2s}, \qquad N \ge 0\textit{;} \end{aligned}$$

  (6.5)

• (3) for some $h>0$    (for every $h>0$) it holds

  $$\begin{aligned} \Vert H_1^{N_1}H_2^{N_2}K \Vert _{L ^2}\lesssim h^{(N_1+N_2)} (N_1!N_2!)^{2s},\qquad N _1, N _2\ge 0\textit{;} \end{aligned}$$

  (6.6)

• (4) for some $h>0$    (for every $h>0$) it holds

  $$\begin{aligned} \Vert H_1^{N_1}H_2^{N_2}K \Vert _{M ^{p, q} _{(\omega )}}\lesssim h^{N_1+N_2} (N_1!N_2!)^{2s},\qquad N _1 \ge N _{0, 1},\ N _2 \ge N _{0,2}. \end{aligned}$$

  (6.7)

Proof

The assertion (1) and (2) are equivalent in view of [26, Proposition 5.1]. Next we prove that (2) and (3) are equivalent. Assume that (6.5) holds. Since $K \in \mathcal H^{\prime }_0 (\mathbf R^{d _2} \times \mathbf R^{d_1})$, K has a formal Hermite series expansions

$$\begin{aligned} K = \sum _{\alpha _1\in {\mathbf N}^{d_1}}\sum _{\alpha _2\in {\mathbf N}^{d_2}} c_\alpha (K) h _{\alpha _2} \otimes h _{\alpha _1}, \end{aligned}$$

where the Hermite coefficients satisfy

$$\begin{aligned} |c_\alpha (K)| \lesssim e^{-\frac{1}{h}|\alpha |^{\frac{1}{2s}}}, \qquad \alpha =(\alpha _2,\alpha _1)\in {\mathbf N}^{d_2}\times {\mathbf N}^{d_1}, \end{aligned}$$

for some (every) $h>0$. By Parseval’s inequality we obtain

$$\begin{aligned}&\Vert H_1^{N_1}H_2^{N_2}K\Vert _{L^2 } \\&\qquad \le \sum _{\alpha _1\in {\mathbf N}^{d_1}}\sum _{\alpha _2\in {\mathbf N}^{d_2}} (2|\alpha _1|+d_1)^{N_1}(2|\alpha _2|+d_2)^{N_2} e^{-\frac{1}{h}|\alpha |^{\frac{1}{2s}}} \le I_1\cdot I_2, \end{aligned}$$

where

$$\begin{aligned} I_k = \sum _{\alpha _j \in {\mathbf N}^{d_k}} (2|\alpha _j |+d_k)^{N_k} e^{-\frac{1}{h_0}|\alpha _j |^{\frac{1}{2s}}} \end{aligned}$$

with $h_0=ch$, for some constant $c>0$ which only depends on s.

By Lemma 5.7 in [26] and its proof we get

$$\begin{aligned} I_k \lesssim (3(4sh_0)^{2s})^{N_k}(N_k!)^{2s}\asymp h^{N_k}(N_k!)^{2s} \end{aligned}$$

for some $h>0$ and a combination of these estimates shows that (2) implies (3).

Assume instead that (6.6) holds. Then

$$\begin{aligned}&\Vert H^N K\Vert _{L^ 2} = \Vert (H_1+H_2)^N K\Vert _{L^ 2} \le \sum _{k=0}^N{N \atopwithdelims ()k}\Vert H_1^{N-k}H_2^k K\Vert _{L^ 2} \\&\quad \lesssim h^N\sum _{k=0}^N{N \atopwithdelims ()k} ((N-k)!k!)^{2s} \le h^N(N!)^{2s}\sum _{k=0}^N{N \atopwithdelims ()k} = (2h)^N(N!)^{2s}, \end{aligned}$$

and it follows that (3) implies (2).

It remains to prove the equivalence between (4) and (1)–(3). First we show that

$$\begin{aligned} \Vert H ^N _1 K\Vert _{M ^{p, q} _{(\omega )}}\lesssim h ^N (N !) ^{2s}, \quad N \ge N _0 \end{aligned}$$

(6.8)

is independent on $N _0$ and $\omega$ when $p, q \ge 1$. If (6.8) is true for $N_0=0$, then it is also true for $N_0>0$. If $0\le N\le N_0$, $N_1=N_0-N\ge 0$ and (6.8) holds for some $N _0 \ge 0$, then by the fact that

$$\begin{aligned}&H^N _1 \, :\, M^{p,q}_{(v_N\omega )}(\mathbf R^{d _2} \times \mathbf R^{d _1})\rightarrow M^{p,q}_{(\omega )}(\mathbf R^{d _2} \times \mathbf R^{d _1}), \end{aligned}$$

(6.9)

with

$$\begin{aligned} v_N(x _1,\xi _1, x _2,\xi _2 ) = C(1+|x _1|^2+|\xi _1 |^2)^N, \end{aligned}$$

(6.10)

is a homeomorphism (cf. e. g. [23, Theorem 3.10]), it follows that

$$\begin{aligned} \Vert H^N _1 K \Vert _{M^{p,q}_{(\omega )}} \lesssim \Vert H^{N_0} _1 K \Vert _{M^{p,q}_{(\omega /v_{N_1})}} \lesssim \Vert H^{N_0} _1 K\Vert _{M^{p,q}_{(\omega )}}<\infty , \end{aligned}$$

and (6.8) holds for $N _0 = 0$ as well. This shows that (6.8) is independent of $N _0 \ge 0$ when $p, q \ge 1$.

Since $\omega \in \mathscr {P}(\mathbf R^{2 d _2} \times \mathbf R^{2d _1} )$, there exists an integer $N _0 \ge 0$ such that

$$\begin{aligned} 1/v_{N_0}\lesssim \omega \lesssim v_{N_0} , \end{aligned}$$

and then

$$\begin{aligned} \Vert K\Vert _{M^{p,q}_{( 1 /v_{N_0})}}\lesssim \Vert K\Vert _{M^{p,q}_{(\omega )}} \lesssim \Vert K\Vert _{M^{p,q}_{(v_{N_0})}}. \end{aligned}$$

(6.11)

Hence the stated invariance follows if we prove that (6.8) holds for $\omega =v_{N_0}$, if it is true for $\omega =1/v_{N_0}$.

Therefore, assume that (6.8) holds for $\omega =1/v_{N_0}$. If $N\ge 2N_0$, then the bijectivity of (6.9) gives

$$\begin{aligned}&\frac{\Vert H_1^N K\Vert _{M^{p,q}_{(v_{N_0})}}}{h^N(N!)^{2s}} \lesssim \frac{\Vert H_1^{N+2N_0}K\Vert _{M^{p,q}_{(1/v_{N_0})}}}{h^N(N!)^{2s}} \\&\quad = h^{2N_0}{{N+2N_0}\atopwithdelims (){2N_0}}^{2s}((2N_0)!)^{2s} \frac{\Vert H_1^{N+2N_0}K\Vert _{M^{p,q}_{(1/v_{N_0})}}}{h^{N+2N_0}((N+2N_0)!)^{2s}} \\&\quad \asymp {{N+2N_0}\atopwithdelims (){2N_0}}^{2s} \frac{\Vert H_1^{N+2N_0}K\Vert _{M^{p,q}_{(1/v_{N_0})}}}{h^{N+2N_0}((N+2N_0)!)^{2s}} \lesssim \frac{\Vert H_1^{N+2N_0}K\Vert _{M^{p,q}_{(1/v_{N_0})}}}{h_1^{N+2N_0}((N+2N_0)!)^{2s}}, \end{aligned}$$

where $h_1=\frac{h}{4^s}$. This shows that (6.8) is independent of $\omega$ in the case $p, q \ge 1$.

By repeating these arguments, it follows that (6.7) is independent of $N _{0,1}$, $N _{0, 2}$, $\omega$ and $p,q\in [1,\infty ]$. For general $p,q\in (0,\infty ]$, the invariance of (6.7) with respect to $N _{0,1}$, $N _{0, 2}$, $\omega$, p and q, is now a consequence of the embeddings

$$\begin{aligned} M^\infty _{(v_N\omega )}(\mathbf R^{d _2} \times \mathbf R^{d _1}) \subseteq M^{p,q} _{(\omega )}(\mathbf R^{d _2} \times \mathbf R^{d _1}) \subseteq M^\infty _{(\omega )}(\mathbf R^{d _2} \times \mathbf R^{d _1}) \end{aligned}$$

when

$$\begin{aligned} \qquad N >\frac{d_1+d_2}{\min (p,q)} \end{aligned}$$

(see e. g. [11]).

The equivalence bewteen (3) and (4) now follows from these invariance properties and the fact that $L^{2} = M^{2,2}$, and the result follows.

In the next proof we let ${\text {ON}}_d$ be the set of all orthonormal sequences in $L^2(\mathbf R^{d})$.

Proof of Theorems 6.1 and 6.2

We only prove Theorem 6.2 in the Roumieu case. The other cases (Theorem 6.2 in the Beurling case, and Theorem 6.1) follow by similar arguments and are left for the reader.

In the sequel we employ the same notation used in the proof of Lemma 6.5

(1) Assume that $K\in \mathcal H_s (\mathbf R^{d_2}\times \mathbf R^{d_1})$. By polar decomposition we have

$$\begin{aligned} K(x_2,x_1)&= \sum _{j=1}^\infty \lambda _{0,j}g_j(x_2) \overline{f_j(x_1)},&\quad x_1&\in \mathbf R^{d_1},\ x_2\in \mathbf R^{d_2}, \end{aligned}$$

where $\lambda _{0,j}\ge 0$ are the singular values of T, $\{ f_j\} _{j=1}^\infty \in {\text {ON}}_{d_1}$ and $\{ g_j\} _{j=1}^\infty \in {\text {ON}}_{d_2}$. Now let $K_1$ and $K_2$ be the kernels of $T_1\equiv (T^*\circ T)^{\frac{1}{4}}$ and $T_2\equiv (T\circ T^*)^{\frac{1}{4}}$, respectively. Then

$$\begin{aligned} K_1(x_2,x_1) = \sum _{j=1}^\infty \sqrt{\lambda _{0,j}}\, f_j(x_2) \overline{f_j(x_1)}, \quad x_1,x_2 \in \mathbf R^{d_1} \end{aligned}$$

and

$$\begin{aligned}&K_2(x_2,x_1) = \sum _{j=1}^\infty \sqrt{\lambda _{0,j}}\, g_j(x_2) \overline{g_j(x_1)}, \quad x_1,x_2 \in \mathbf R^{d_2}. \end{aligned}$$

By Theorem 5.1 we get

$$\begin{aligned} \lambda _{0,j} \lesssim e^{-rj^{\frac{1}{2ds}}} \end{aligned}$$

(6.12)

for some constant $r>0$.

Since $K_1\in \mathcal H_s (\mathbf R^{d_1}\times \mathbf R^{d_1})$, Lemma 6.5 gives

$$\begin{aligned} \sum _{j=1}^\infty \sqrt{\lambda _{0,j}} \Vert H^Nf_j\Vert _{L^2}^2 = \Vert H_1^NH_2^NK_1\Vert _{\text {Tr}} \le \Vert H_1^NH_2^NK_1\Vert _{M^{1,1}} \lesssim h^N(N!)^{4s}, \end{aligned}$$

where $\Vert \, \cdot \, \Vert _{{\text {Tr}}}$ is the trace-class norm. Here we have identified operators with their kernels, and used the fact that operators with kernels in $M^{1,1}(\mathbf R^{2d})$ are of trace-class (cf. [15, 26]). Hence,

$$\begin{aligned} \lambda _{0,j}^{\frac{1}{4}}\Vert H^Nf_j\Vert _{L^2}\lesssim h_0^N(N!)^{2s}, \end{aligned}$$

where $h_0=\sqrt{h}$. Hence, if $f_{1,j} = \lambda _{0,j}^{\frac{1}{3}}f_j$ we obtain

$$\begin{aligned} \Vert H^Nf_{1,j}\Vert _{L^2}\lesssim \lambda _{0,j}^{\frac{1}{12}}h_0^N(N!)^{2s} \lesssim e^{-r\cdot j^{\frac{1}{2ds}}}h_0^N(N!)^{2s} \end{aligned}$$

for some $r>0$. By considering $K_2$ instead of $K_1$ and letting $f_{2,j}= \lambda _{0,j}^{\frac{1}{3}}g_j$, the same computations give

$$\begin{aligned} \Vert H^Nf_{2,j}\Vert _{L^2}\lesssim e^{-r\cdot j^{\frac{1}{2ds}}}h_0^N(N!)^{2s} \end{aligned}$$

for some $r>0$ and $h_0>0$.

The assertion now follows if we let $\lambda _j = \lambda _{0,j}^{\frac{1}{3}}$.

(2) By the assumptions and Cauchy-Schwartz inequality, we obtain

$$\begin{aligned} \Vert H ^{N _1} _1 H ^{N _2} _2 K\Vert _{L ^2}= & {} \left\| H ^{N _1} _1 H ^{N _2} _2 \left( \sum \lambda _j f _{1,j} \otimes f _{2, j}\right) \right\| _{L ^2}\\= & {} \left( \iint _{\mathbf R^{d _2+d _1}} \left| \sum \lambda _j H ^{N _1} _1f _{1,j} \otimes H ^{N _2} _2 f _{2, j}\right| ^2\, dx_1 dx _2 \right) ^{1/2}\\\le & {} \left( \iint _{\mathbf R^{d _2+d _1}} \Big (\sum \lambda _j ^2\Big ) \Big (\sum |H ^{N _1} _1f _{1,j} \otimes H ^{N _2} _2 f _{2, j} | ^2\Big ) \, dx_1 dx _2 \right) ^{1/2}\\\le & {} \sum \lambda _j \left( \sum \Vert H ^{N _1} _1 f_{1, j}\Vert _{L ^2}^2 \Vert H ^{N _2} _2 f_{2, j}\Vert _{L ^2}^2 \right) ^{1/2}\\\lesssim & {} h ^{N _1 + N _2}(N _1! N _2 !) ^{2s} \sum e ^{- r \cdot j^{1/{2ds}}} \lesssim h ^{N _1 + N _2}(N _1! N _2 !) ^{2s}. \end{aligned}$$

Hence, $K\in \mathcal H_s (\mathbf R^{d_2}\times \mathbf R^{d_1})$ in view of Lemma 6.5. $\square$