Solution to the Stieltjes moment problem in Gelfand-Shilov spaces

TL;DR

This paper characterizes the conditions under which the Stieltjes moment problem is solvable in Gelfand-Shilov spaces, providing new insights into ultraholomorphic function spaces and their moment mappings.

Contribution

It offers a complete characterization of the surjectivity and right inverse existence of the Stieltjes moment mapping in Gelfand-Shilov spaces based on their weight sequences.

Findings

Characterization of surjectivity conditions for the moment mapping.

Existence criteria for continuous linear right inverses.

New results on the Borel-Ritt problem in ultraholomorphic spaces.

Abstract

We characterize the surjectivity and the existence of a continuous linear right inverse of the Stieltjes moment mapping on Gelfand-Shilov spaces, both of Beurling and Roumieu type, in terms of their defining weight sequence. As a corollary, we obtain some new results about the Borel-Ritt problem in spaces of ultraholomorphic functions on the upper half-plane.

Equations

$${\int }_0^{\infty }{x}^p\mathrm{d}F(x)=a_p,p\in \mathbb{N}.$$

$${\int }_0^{\infty }{x}^p\phi (x)\mathrm{d}x=a_p,p\in \mathbb{N},$$

$$\underset{p\in \mathbb{N}}{\sup }\underset{x\ge 0}{\sup }\frac{h^p{x}^p|{\phi }^{(n)}(x)|}{M_p}<\infty$$

$$\underset{p\in \mathbb{N}}{\sup }\frac{h^p|a_p|}{M_p}<\infty$$

$$\mathcal{M}:{\mathcal{S}}_{\ast }(0,\infty )\to {\Lambda }_{\ast }:\phi \to {\left({\int }_0^{\infty }{x}^p\phi (x)\mathrm{d}x\right)}_{p\in \mathbb{N}}$$

$$\mathcal{F}(\phi )(\xi )=\hat{\phi }(\xi ):={\int }_{-\infty }^{\infty }\phi (x)e^{ix\xi }\mathrm{d}x,\phi \in \mathcal{S}(\mathbb{R}).$$

$${\mu }_p(\phi ):={\int }_{-\infty }^{\infty }{x}^p\phi (x)\mathrm{d}x.$$

$$\mathcal{B}:C^{\infty }(\mathbb{R})\to {\mathbb{C}}^{\mathbb{N}}:\phi \to ({\phi }^{(p)}(0){)}_{p\in \mathbb{N}}$$

$$\mathcal{M}:\mathcal{S}(\mathbb{R})\to {\mathbb{C}}^{\mathbb{N}}:\phi \to ({\mu }_p(\phi ){)}_{p\in \mathbb{N}}.$$

$$M(t):=\underset{p\in \mathbb{N}}{\sup }\log \frac{t^p}{M_p},t>0.$$

$$N_{p}:=\left\{\begin{array}[]{ll}M_{q},&\mbox{$p=2q$, $q\in\mathbb{N}$},\\ \\ (M_{q}M_{q+1})^{1/2},&\mbox{$p=2q+1$, $q\in\mathbb{N}$}.\end{array}\right.$$

$$\Vert \phi {\Vert }_{{\mathcal{D}}_{[-1,1]}^{N_p,h}}:=\underset{p\in \mathbb{N}}{\sup }\underset{x\in [-1,1]}{\max }\frac{h^p|{\phi }^{(p)}(x)|}{M_p}<\infty .$$

$${\mathcal{D}}_{[-1,1]}^{(N_p)}:=\underset{h\to \infty }{\underleftarrow{\lim }}{\mathcal{D}}_{[-1,1]}^{N_p,h},{\mathcal{D}}_{[-1,1]}^{\{N_p\}}:=\underset{h\to 0^{+}}{\underrightarrow{\lim }}{\mathcal{D}}_{[-1,1]}^{N_p,h}.$$

$$\Vert a{\Vert }_{{\Lambda }_{N_p,h}}:=\underset{p\in \mathbb{N}}{\sup }\frac{h^p|a_p|}{N_p}<\infty .$$

$${\Lambda }_{(N_p)}:=\underset{h\to \infty }{\underleftarrow{\lim }}{\Lambda }_{N_p,h},{\Lambda }_{\{N_p\}}:=\underset{h\to 0^{+}}{\underrightarrow{\lim }}{\Lambda }_{N_p,h}.$$

$$\mathcal{B}:{\mathcal{D}}_{[-1,1]}^{(N_p)}\to {\Lambda }_{(N_p)},\mathcal{B}:{\mathcal{D}}_{[-1,1]}^{\{N_p\}}\to {\Lambda }_{\{N_p\}}$$

$$\Vert \phi {\Vert }_{{\mathcal{S}}_{M_p,h}^n}:=\underset{m\le n}{\max }\underset{p\in \mathbb{N}}{\sup }\underset{x\in \mathbb{R}}{\sup }\frac{h^p|x^p{\phi }^{(m)}(x)|}{M_p}<\infty .$$

$$\Vert \phi {\Vert }_{{\mathcal{S}}_{M_p,h}^n}=\underset{m\le n}{\max }\underset{x\in \mathbb{R}}{\sup }|{\phi }^{(m)}(x)|e^{M(h|x|)},\phi \in {\mathcal{S}}_{M_p,h}^n(\mathbb{R}).$$

$${\mathcal{S}}_{(M_p)}(\mathbb{R}):=\underset{n\to \infty }{\underleftarrow{\lim }}{\mathcal{S}}_{M_p,n}^n(\mathbb{R}),$$

$${\overrightarrow{\mathcal{S}}}_{\{M_p\}}(\mathbb{R}):=\underset{h\to 0^{+}}{\underrightarrow{\lim }}\underset{n\to \infty }{\underleftarrow{\lim }}{\mathcal{S}}_{M_p,h}^n(\mathbb{R}),$$

$${\overleftarrow{\mathcal{S}}}_{\{M_p\}}(\mathbb{R}):=\underset{n\to \infty }{\underleftarrow{\lim }}\underset{h\to 0^{+}}{\underrightarrow{\lim }}{\mathcal{S}}_{M_p,h}^n(\mathbb{R}).$$

$$\mathcal{M}:{\mathcal{S}}_{\ast }(\mathbb{R})\to {\Lambda }_{\ast }$$

$$\forall N\exists L\ge N\forall K\ge L\exists \theta \in (0,1)\exists C>0\forall x\in \mathbb{R}:$$

$$v_L(x)\le C(v_N(x){)}^{1-\theta }(v_K(x){)}^{\theta }.$$

$$\Vert \phi {\Vert }_{{\mathcal{S}}_n^{M_p,h}}:=\underset{m\le n}{\max }\underset{p\in \mathbb{N}}{\sup }\underset{x\in \mathbb{R}}{\sup }\frac{h^p|x^m{\phi }^{(p)}(x)|}{M_p}<\infty .$$

$${\mathcal{S}}^{(M_p)}(\mathbb{R}):=\underset{n\to \infty }{\underleftarrow{\lim }}{\mathcal{S}}_n^{M_p,n}(\mathbb{R}),$$

$${\overrightarrow{\mathcal{S}}}^{\{M_p\}}(\mathbb{R}):=\underset{h\to 0^{+}}{\underrightarrow{\lim }}\underset{n\to \infty }{\underleftarrow{\lim }}{\mathcal{S}}_n^{M_p,h}(\mathbb{R}),$$

$${\overleftarrow{\mathcal{S}}}^{\{M_p\}}(\mathbb{R}):=\underset{n\to \infty }{\underleftarrow{\lim }}\underset{h\to 0^{+}}{\underrightarrow{\lim }}{\mathcal{S}}_n^{M_p,h}(\mathbb{R}).$$

$${\mathcal{S}}_{M_p,h}^n(0,\infty ):=\{\phi \in {\mathcal{S}}_{M_p,h}^n(\mathbb{R})|\mathrm{supp}\phi \subseteq [0,\infty )\},$$

$$\displaystyle\mathcal{S}^{n,0}_{M_{p},h}(\mathbb{R}):=\{\varphi\in\mathcal{S}^{n}_{M_{p},h}(\mathbb{R})\,|\,\varphi^{(m)}(0)=0\mbox{ for all $m=0,\ldots,n$}\},$$

$${\mathcal{S}}_{(M_p)}(0,\infty ):=\underset{n\to \infty }{\underleftarrow{\lim }}{\mathcal{S}}_{M_p,n}^n(0,\infty ),{\mathcal{S}}_{(M_p)}^0(\mathbb{R}):=\underset{n\to \infty }{\underleftarrow{\lim }}{\mathcal{S}}_{M_p,n}^{n,0}(\mathbb{R}),$$

$${\overrightarrow{\mathcal{S}}}_{\{M_p\}}(0,\infty ):=\underset{h\to 0^{+}}{\underrightarrow{\lim }}\underset{n\to \infty }{\underleftarrow{\lim }}{\mathcal{S}}_{M_p,h}^n(0,\infty ),{\overrightarrow{\mathcal{S}}}_{\{M_p\}}^0(\mathbb{R}):=\underset{h\to 0^{+}}{\underrightarrow{\lim }}\underset{n\to \infty }{\underleftarrow{\lim }}{\mathcal{S}}_{M_p,h}^{n,0}(\mathbb{R}),$$

$${\overleftarrow{\mathcal{S}}}_{\{M_p\}}(0,\infty ):=\underset{n\to \infty }{\underleftarrow{\lim }}\underset{h\to 0^{+}}{\underrightarrow{\lim }}{\mathcal{S}}_{M_p,h}^n(0,\infty ),{\overleftarrow{\mathcal{S}}}_{\{M_p\}}^0(\mathbb{R}):=\underset{n\to \infty }{\underleftarrow{\lim }}\underset{h\to 0^{+}}{\underrightarrow{\lim }}{\mathcal{S}}_{M_p,h}^{n,0}(\mathbb{R}).$$

$${\mathcal{S}}_{\ast }(0,\infty )=\{\phi \in {\mathcal{S}}_{\ast }(\mathbb{R})|\mathrm{supp}\phi \subseteq [0,\infty )\},$$

Full text

The author is supported by FWO-Vlaanderen, via the postdoctoral grant 12T0519N

1. Introduction

In 1939, Boas [1] and Pólya [17] independently showed that for every sequence $(a_{p})_{p\in\mathbb{N}}$ of complex numbers there is a function $F$ of bounded variation such that

[TABLE]

A. J. Durán [7] (see also [8]) improved this result in 1989 by showing that for every sequence $(a_{p})_{p\in\mathbb{N}}$ of complex numbers the infinite system of linear equations

[TABLE]

admits a solution $\varphi\in\mathcal{S}(0,\infty)$ (= the space of rapidly decreasing smooth functions with support in $[0,\infty)$). Over the past 20 years, various authors studied the (unrestricted) Stieltjes moment problem (1.1) in the context of Gelfand-Shilov spaces [10]; see [4, 3, 13, 14, 5]. In this article, we provide a complete solution to this problem.

In order to be able to discuss our results, we need to introduce some notation; see Section 2 for unexplained notions concerning weight sequences. Let $(M_{p})_{p\in\mathbb{N}}$ be a weight sequence. We define $\mathcal{S}_{(M_{p})}(0,\infty)$ as the space consisting of all $\varphi\in\mathcal{S}(0,\infty)$ such that

[TABLE]

for all $h>0$ and $n\in\mathbb{N}$. Similarly, we define $\mathcal{S}_{\{M_{p}\}}(0,\infty)$ as the space consisting of all $\varphi\in\mathcal{S}(0,\infty)$ such that there is $h>0$ for which (1.2) holds for all $n\in\mathbb{N}$. $\mathcal{S}_{(M_{p})}(0,\infty)$ and $\mathcal{S}_{\{M_{p}\}}(0,\infty)$ are endowed with their natural Fréchet space and $(LF)$-space topology, respectively. Next, we define $\Lambda_{(M_{p})}$ and $\Lambda_{\{M_{p}\}}$ as the sequence spaces consisting of all $a=(a_{p})_{p\in\mathbb{N}}\in\mathbb{C}^{\mathbb{N}}$ such that

[TABLE]

for all $h>0$ and some $h>0$, respectively. $\Lambda_{(M_{p})}$ and $\Lambda_{\{M_{p}\}}$ are endowed with their natural Fréchet space and $(LB)$-space topology, respectively. If $(M_{p})_{p\in\mathbb{N}}$ satisfies $(\operatorname{dc})$, the Stieltjes moment mapping

[TABLE]

is well-defined and continuous, where $\ast$ stands for either $(M_{p})$ or $\{M_{p}\}$. Jiménez-Garrido, Sanz and the author [5] characterized the surjectivity of $\mathcal{M}:\mathcal{S}_{\{M_{p}\}}(0,\infty)\rightarrow\Lambda_{\{M_{p}\}}$ in the following way; see [14] for earlier work in this direction.

Theorem 1.1**.**

[5, Thm. 3.5]** Let $(M_{p})_{p\in\mathbb{N}}$ be a weight sequence satisfying $\operatorname{(slc)}$ (= $(M_{p}/p!)_{p\in\mathbb{N}}$ satisfies $\operatorname{(lc)}$) and $\operatorname{(dc)}$. If the mapping $\mathcal{M}:\mathcal{S}_{\{M_{p}\}}(0,\infty)\rightarrow\Lambda_{\{M_{p}\}}$ is surjective, then $(M_{p})_{p\in\mathbb{N}}$ satisfies

• $(\gamma_{2})$

$\displaystyle\sup_{p\in\mathbb{Z}_{+}}\frac{(M_{p}/M_{p-1})^{1/2}}{p}\sum_{q=p}^{\infty}\frac{1}{(M_{q}/M_{q-1})^{1/2}}<\infty$.

If, in addition, $(M_{p})_{p\in\mathbb{N}}$ satisfies $(\operatorname{mg})$, $(\gamma_{2})$ implies that $\mathcal{M}:\mathcal{S}_{\{M_{p}\}}(0,\infty)\rightarrow\Lambda_{\{M_{p}\}}$ is surjective.

Condition $(\gamma_{2})$ means that $(M^{1/2}_{p})_{p\in\mathbb{N}}$ is strongly non-quasianalytic [12]. The main goal of this article is to improve and complete Theorem 1.1 in the following three ways: Consider the Beurling case as well; replace $\operatorname{(slc)}$ and $\operatorname{(mg)}$ by the weaker conditions $\operatorname{(lc)}$ and $\operatorname{(dc)}$; characterize the existence of a continuous linear right inverse of $\mathcal{M}:\mathcal{S}_{\ast}(0,\infty)\rightarrow\Lambda_{\ast}$. More precisely, we show the following result; see Theorem 6.1.

Theorem 1.2**.**

Let $(M_{p})_{p\in\mathbb{N}}$ be a weight sequence satisfying $(\operatorname{lc})$ and $(\operatorname{dc})$.

• $(a)$

The following statements are equivalent:

• $(i)$

$\mathcal{M}:\mathcal{S}_{(M_{p})}(0,\infty)\rightarrow\Lambda_{(M_{p})}$* is surjective.*

• $(ii)$

$\mathcal{M}:\mathcal{S}_{(M_{p})}(0,\infty)\rightarrow\Lambda_{(M_{p})}$* has a continuous linear right inverse.*

• $(iii)$

$(M_{p})_{p\in\mathbb{N}}$* satisfies $(\gamma_{2})$.*

• $(b)$

$\mathcal{M}:\mathcal{S}_{\{M_{p}\}}(0,\infty)\rightarrow\Lambda_{\{M_{p}\}}$* is surjective if and only if $(M_{p})_{p\in\mathbb{N}}$ satisfies $(\gamma_{2})$.*

• $(c)$

$\mathcal{M}:\mathcal{S}_{\{M_{p}\}}(0,\infty)\rightarrow\Lambda_{\{M_{p}\}}$* has a continuous linear right inverse if and only if $(M_{p})_{p\in\mathbb{N}}$ satisfies $(\gamma_{2})$ and*

• $(\beta_{2})$

$\displaystyle\forall\varepsilon>0\,\exists n\in\mathbb{Z}_{+}\,:\,\limsup_{p\to\infty}\left(\frac{M_{np}}{M_{p}}\right)^{\frac{1}{p(n-1)}}\frac{M_{np-1}}{M_{np}}\leq\varepsilon$.

Condition $(\beta_{2})$ is due to Petzsche [16] and appears in his characterization of the existence of a continuous linear right inverse of the Borel mapping on spaces of ultradifferentiable functions of Roumieu type. We also give an analogue of Theorem 1.2 for Gelfand-Shilov spaces of type $\mathcal{S}^{\dagger}_{\ast}(0,\infty)$ (cf. [5, Thm. 3.5]); see Theorem 7.2.

In [5], Theorem 1.1 is shown by reducing it to the Borel-Ritt problem [18, 19, 20, 11] in spaces of ultraholomorphic functions on the upper half-plane and then using solutions to this problem from [20, 11]; a technique that goes back to A. L. Durán and Estrada [8]. Up until now, this seems to be the only known method to study the Stieltjes moment problem in Gelfand-Shilov spaces. It also explains why we had to assume $\operatorname{(slc)}$ and $\operatorname{(mg)}$ in Theorem 1.1: These conditions are needed to solve the Borel-Ritt problem in spaces of ultraholomorphic functions [20, Thm. 3.2.1]. We develop here a completely new approach. Namely, we show Theorem 1.2 by reducing it to the Borel problem in spaces of ultradifferentiable functions of class $(N_{p})_{p\in\mathbb{N}}$, where $(N_{p})_{p\in\mathbb{N}}$ denotes the $2$-interpolating sequence associated to $(M_{p})_{p\in\mathbb{N}}$ [19], and then using Petzsche’s classical solution to this problem [16].

As a corollary, we obtain an analogue of Theorem 1.2 for the Borel-Ritt problem in spaces of ultraholomorphic functions on the upper half-plane, thereby improving some results of Schmets and Valdivia [19] and Thilliez [20] in the particular case of the upper half-plane; see Theorem 7.4 and Remark 7.5. Of course, due to the distinct geometry of the upper half-plane, this special case is much simpler to handle than the Borel-Ritt problem in spaces of ultraholomorphic functions on general sectors.

The plan of this article is as follows. In Section 2, we fix the notation, introduce weight sequences and recall Petzsche’s solution to the Borel problem in spaces of ultradifferentiable functions. In Section 3, we define Gelfand-Shilov spaces of type $\mathcal{S}_{\ast}$ and collect several properties of these spaces that will be used later on. Next, in the auxiliary Sections 4 and 5, we present an abstract result about the existence of a continuous linear right inverse and prove a Borel type theorem. These results are used in the proof of Theorem 1.2, which is given in Section 6. Finally, in Section 7, we consider the Stieltjes moment problem in Gelfand-Shilov spaces of type $\mathcal{S}^{\dagger}_{\ast}(0,\infty)$ and the Borel-Ritt problem in spaces of ultraholomorphic functions on the upper half-plane.

2. Preliminaries

2.1. Notation

We set $\mathbb{N}=\{0,1,2,\ldots\}$ and $\mathbb{Z}_{+}=\{1,2,\ldots\}$. The Fréchet space of rapidly decreasing smooth functions on $\mathbb{R}$ is denoted by $\mathcal{S}(\mathbb{R})$. We fix the constants in the Fourier transform as follows

[TABLE]

The $p$-th moment, $p\in\mathbb{N}$, of an element $\varphi\in\mathcal{S}(\mathbb{R})$ is given by

[TABLE]

Notice that $\widehat{\varphi}^{(p)}(0)=i^{p}\mu_{p}(\varphi)$ for all $p\in\mathbb{N}$.

We define the Borel mapping as

[TABLE]

and the Stieltjes moment mapping as

[TABLE]

A lcHs (= locally convex Hausdorff space) $E$ is said to be an $(LF)$-space if there is a sequence $(E_{n})_{n\in\mathbb{N}}$ of Fréchet spaces with $E_{n}\subseteq E_{n+1}$ and continuous inclusion mappings for all $n\in\mathbb{N}$ such that $E=\bigcup_{n\in\mathbb{N}}E_{n}$ and the topology of $E$ coincides with the finest locally convex topology such that all the inclusion mappings $E_{n}\rightarrow E$, $n\in\mathbb{N}$, are continuous. We write $E=\varinjlim_{n\in\mathbb{N}}E_{n}$. If the sequence $(E_{n})_{n\in\mathbb{N}}$ consists of Banach spaces, $E$ is called an $(LB)$-space. Finally, a lcHs is said to be a (PLB)-space if it can be written as the projective limit of a countable spectrum of $(LB)$-spaces.

2.2. Weight sequences

A sequence $(M_{p})_{p\in\mathbb{N}}$ of positive numbers is called a weight sequence if $M_{0}=1$ and $m_{p}:=M_{p}/M_{p-1}\rightarrow\infty$ as $p\to\infty$. The associated function of a weight sequence $(M_{p})_{p\in\mathbb{N}}$ is defined as $M(0):=0$ and

[TABLE]

We will make use of the following conditions on weight sequences:

1. $(\operatorname{lc})$

(log-convexity) $M_{p}^{2}\leq M_{p-1}M_{p+1}$, $p\in\mathbb{Z}_{+}$. 2. $(\operatorname{dc})$

(derivation-closedness) $M_{p+1}\leq C_{0}H^{p+1}M_{p}$, $p\in\mathbb{N}$, for some $C_{0},H\geq 1$. 3. $(\operatorname{mg})$

(moderate growth) $M_{p+q}\leq C_{0}H^{p+q}M_{p}M_{q}$, $p,q\in\mathbb{N}$, for some $C_{0},H\geq 1$. 4. $(\gamma)$

(non-quasianalyticity) $\displaystyle\sum_{p=1}^{\infty}\frac{1}{m_{p}}<\infty$. 5. $(\gamma_{1})$

(strong non-quasianalyticity) $\displaystyle\sup_{p\in\mathbb{Z}_{+}}\frac{m_{p}}{p}\sum_{q=p}^{\infty}\frac{1}{m_{q}}<\infty$. 6. $(\gamma_{2})$

$\displaystyle\sup_{p\in\mathbb{Z}_{+}}\frac{m_{p}^{1/2}}{p}\sum_{q=p}^{\infty}\frac{1}{m_{q}^{1/2}}<\infty$. 7. $(\beta_{2})$

$\displaystyle\forall\varepsilon>0\,\exists n\in\mathbb{Z}_{+}\,:\,\limsup_{p\to\infty}\left(\frac{M_{np}}{M_{p}}\right)^{\frac{1}{p(n-1)}}\frac{1}{m_{np}}\leq\varepsilon$.

Clearly, $(\operatorname{mg})\Rightarrow(\operatorname{dc})$ and $(\gamma_{1})\Rightarrow(\gamma)$. Moreover, if $(M_{p})_{p\in\mathbb{N}}$ satisfies $(\operatorname{lc})$, then $(\gamma_{2})\Rightarrow(\gamma_{1})$. The conditions $(\operatorname{lc})$, $(\operatorname{dc})$, $(\operatorname{mg})$, $(\gamma)$ and $(\gamma_{1})$ are standard in the theory of ultradifferentiable functions and their meaning is well explained in the classical work of Komatsu [12]. Conditions $(\gamma_{1})$ and $(\gamma_{2})$ are particular instances of

1. $(\gamma_{r})$

$\displaystyle\sup_{p\in\mathbb{Z}_{+}}\frac{m^{1/r}_{p}}{p}\sum_{q=p}^{\infty}\frac{1}{m^{1/r}_{q}}<\infty,\qquad r>0$.

Condition $(\gamma_{r})$ means that $(M^{1/r}_{p})_{p\in\mathbb{N}}$ satisfies $(\gamma_{1})$. These conditions, which were introduced by Schmets and Valdivia [19] for $r\in\mathbb{N}$ and by Thilliez [20] for arbitrary $r>0$, play an important role in the study of the Borel-Ritt problem in spaces of ultraholomorphic functions [18, 19, 20, 11]. Condition $(\beta_{2})$ is due to Petzsche [16] and appears in his characterization of the existence of a continuous linear right inverse of the Borel mapping on spaces of ultradifferentiable functions of Roumieu type; see Theorem 2.4 below.

Remark 2.1*.*

Consider

• $(\beta^{0}_{2})$

$\displaystyle\exists n\in\mathbb{Z}_{+}\,:\,\lim_{p\to\infty}\frac{m_{np}}{m_{p}}=\infty$.

• $(\beta^{1}_{2})$

$\displaystyle\lim_{p\to\infty}\frac{M_{p}^{1/p}}{m_{p}}=0$.

Petzsche has shown that $(\beta^{0}_{2})\Rightarrow(\beta_{2})\Rightarrow(\beta^{1}_{2})$ [16, Prop. 1.5(b) and Prop. 1.6(a)] and that the converse implications are false in general [16, Example 1.8]. However, $(\beta^{0}_{2})$ and $(\beta_{2})$ are equivalent within the class of weight sequences $(M_{p})_{p\in\mathbb{N}}$ satisfying the following mild regularity condition: There is $n\in\mathbb{Z}_{+}$ such that the set of finite limit points of the set $\{m_{n^{l}}/m_{n^{l-1}}\,|\,l\in\mathbb{Z}_{+}\}$ is bounded [16, Prop. 1.6(b)].

Example 2.2**.**

$(i)$ The Gevrey sequence $(p!^{\alpha})_{p\in\mathbb{N}}$, $\alpha>0$, satisfies $(\operatorname{lc})$ and $(\operatorname{mg})$; it satisfies $(\gamma_{r})$ if and only if $\alpha>r$; it does not satisfy $(\beta^{1}_{2})$ and, thus, also not $(\beta_{2})$.

$(ii)$ The $q$-Gevrey sequence $(q^{p^{2}})_{p\in\mathbb{N}}$, $q>1$, satisfies $(\operatorname{lc})$ and $(\operatorname{dc})$ but not $(\operatorname{mg})$; it satisfies $(\gamma_{r})>0$ for all $r>0$; it satisfies $(\beta^{0}_{2})$ and, thus, also $(\beta_{2})$.

Following [19], we define the $2$-interpolating sequence $(N_{p})_{p\in\mathbb{N}}$ associated to a weight sequence $(M_{p})_{p\in\mathbb{N}}$ as

[TABLE]

Lemma 2.3**.**

[19, Lemma 2.3]** Let $(M_{p})_{p\in\mathbb{N}}$ be a weight sequence satisfying $(\operatorname{lc})$. Denote by $(N_{p})_{p\in\mathbb{N}}$ its $2$-interpolating sequence. Then, $(N_{p})_{p\in\mathbb{N}}$ is a weight sequence satisfying $(\operatorname{lc})$. Moreover, the following statements hold:

• $(a)$

$(M_{p})_{p\in\mathbb{N}}$* satisfies $(\operatorname{dc})$ if and only if $(N_{p})_{p\in\mathbb{N}}$ does so.*

• $(b)$

$(M_{p})_{p\in\mathbb{N}}$* satisfies $(\gamma_{2})$ if and only if $(N_{p})_{p\in\mathbb{N}}$ satisfies $(\gamma_{1})$.*

• $(c)$

$(M_{p})_{p\in\mathbb{N}}$* satisfies $(\beta_{2})$ if and only if $(N_{p})_{p\in\mathbb{N}}$ does so.*

2.3. The Borel problem in spaces of ultradifferentiable functions

Let $(N_{p})_{p\in\mathbb{N}}$ be a weight sequence. For $h>0$ we define $\mathcal{D}^{N_{p},h}_{[-1,1]}$ as the Banach space consisting of all $\varphi\in C^{\infty}(\mathbb{R})$ with $\operatorname{supp}\varphi\subseteq[-1,1]$ such that

[TABLE]

We set

[TABLE]

$\mathcal{D}^{(N_{p})}_{[-1,1]}$ is a Fréchet space, while $\mathcal{D}^{\{N_{p}\}}_{[-1,1]}$ is an $(LB)$-space. If $(N_{p})_{p\in\mathbb{N}}$ satisfies $(\operatorname{lc})$, the spaces $\mathcal{D}^{(N_{p})}_{[-1,1]}$ and $\mathcal{D}^{\{N_{p}\}}_{[-1,1]}$ are non-trivial if and only if $(N_{p})_{p\in\mathbb{N}}$ satisfies $(\gamma)$, as follows from the Denjoy-Carleman theorem.

For $h>0$ we define $\Lambda_{N_{p},h}$ as the Banach space consisting of all sequences $a=(a_{p})_{p\in\mathbb{N}}\in\mathbb{C}^{\mathbb{N}}$ such that

[TABLE]

We set

[TABLE]

$\Lambda_{(N_{p})}$ is a Fréchet space, while $\Lambda_{\{N_{p}\}}$ is an $(LB)$-space. The mappings

[TABLE]

are well-defined and continuous. Petzsche characterized the surjectivity and the existence of a continuous linear right inverse of these mappings in the following way.

Theorem 2.4**.**

Let $(N_{p})_{p\in\mathbb{N}}$ be a weight sequence satisfying $(\operatorname{lc})$ and $(\gamma)$.

• $(a)$

([16, Thm. 3.4])* The following statements are equivalent:*

• $(i)$

$(N_{p})_{p\in\mathbb{N}}$* satisfies $(\gamma_{1})$.*

• $(ii)$

$\mathcal{B}:\mathcal{D}^{(N_{p})}_{[-1,1]}\rightarrow\Lambda_{(N_{p})}$* has a continuous linear right inverse.*

• $(iii)$

$\mathcal{B}:\mathcal{D}^{(N_{p})}_{[-1,1]}\rightarrow\Lambda_{(N_{p})}$* is surjective.*

• $(b)$

([16, Thm. 3.5])* $(N_{p})_{p\in\mathbb{N}}$ satisfies $(\gamma_{1})$ if and only if $\mathcal{B}:\mathcal{D}^{\{N_{p}\}}_{[-1,1]}\rightarrow\Lambda_{\{N_{p}\}}$ is surjective.*

• $(c)$

([16, Thm. 3.1(a)])111As pointed out in [19, p. 223], the statement of [16, Thm. 3.1(a)] contains a mistake, namely, one should read “$(\gamma_{1})$ and $(\beta_{2})$” instead of “$(\beta_{2})$”. $(N_{p})_{p\in\mathbb{N}}$ satisfies $(\gamma_{1})$ and $(\beta_{2})$ if and only if $\mathcal{B}:\mathcal{D}^{\{N_{p}\}}_{[-1,1]}\rightarrow\Lambda_{\{N_{p}\}}$ has a continuous linear right inverse.

3. Gelfand-Shilov spaces of type $\mathcal{S}_{\ast}$

Let $(M_{p})_{p\in\mathbb{N}}$ be a weight sequence. For $n\in\mathbb{N}$ and $h>0$ we write $\mathcal{S}^{n}_{M_{p},h}(\mathbb{R})$ for the Banach space consisting of all $\varphi\in C^{n}(\mathbb{R})$ such that

[TABLE]

Notice that

[TABLE]

We set

[TABLE]

$\mathcal{S}_{(M_{p})}(\mathbb{R})$ is a Fréchet space, $\overrightarrow{\mathcal{S}}_{\{M_{p}\}}(\mathbb{R})$ is an $(LF)$-space, while $\overleftarrow{\mathcal{S}}_{\{M_{p}\}}(\mathbb{R})$ is a $(PLB)$-space. In the sequel, we shall sometimes use $\mathcal{S}_{\ast}(\mathbb{R})$ as a common notation for $\mathcal{S}_{(M_{p})}(\mathbb{R})$, $\overrightarrow{\mathcal{S}}_{\{M_{p}\}}(\mathbb{R})$ and $\overleftarrow{\mathcal{S}}_{\{M_{p}\}}(\mathbb{R})$; a similar convention will be used for other spaces. If $(M_{p})_{p\in\mathbb{N}}$ satisfies $(\operatorname{dc})$, the mapping

[TABLE]

is well-defined and continuous. The following result will be used later on.

Proposition 3.1**.**

The $(LF)$-space $\overrightarrow{\mathcal{S}}_{\{M_{p}\}}(\mathbb{R})$ is complete.

Proof.

In the notation of [6], we have that $\overrightarrow{\mathcal{S}}_{\{M_{p}\}}(\mathbb{R})=\mathcal{B}_{\mathcal{V}}(\mathbb{R})$, where $\mathcal{V}=(v_{N})_{N\in\mathbb{N}}$ with $v_{N}=e^{M(|\,\cdot\,|/N)}$ for $N\in\mathbb{N}$. By [6, Thm. 3.4], it suffices to show that $\mathcal{V}$ satisfies $(\Omega)$, that is,

[TABLE]

The latter follows from the fact that the function $t\rightarrow M(e^{t})$ is increasing and convex on $[0,\infty)$. ∎

Next, we discuss the Fourier transform on $\mathcal{S}_{\ast}(\mathbb{R})$. For $n\in\mathbb{N}$ and $h>0$ we write $\mathcal{S}^{M_{p},h}_{n}(\mathbb{R})$ for the Banach space consisting of all $\varphi\in C^{\infty}(\mathbb{R})$ such that

[TABLE]

We set

[TABLE]

$\mathcal{S}^{(M_{p})}(\mathbb{R})$ is a Fréchet space, $\overrightarrow{\mathcal{S}}^{\{M_{p}\}}(\mathbb{R})$ is an $(LF)$-space, while $\overleftarrow{\mathcal{S}}^{\{M_{p}\}}(\mathbb{R})$ is a $(PLB)$-space. If $(M_{p})_{p\in\mathbb{N}}$ satisfies $(\operatorname{lc})$ and $(\operatorname{dc})$, the Fourier transform is a topological isomorphism from $\mathcal{S}_{\ast}(\mathbb{R})$ onto $\mathcal{S}^{\ast}(\mathbb{R})$ (cf. [10, Sect. IV.6]).

We now introduce Gelfand-Shilov spaces of type $\mathcal{S}_{\ast}(0,\infty)$. Let $n\in\mathbb{N}$ and $h>0$. We define the following closed subspaces of $\mathcal{S}^{n}_{M_{p},h}(\mathbb{R})$

[TABLE]

and endow them with the norm $\|\,\cdot\,\|_{\mathcal{S}^{n}_{M_{p},h}}$. Hence, they become Banach spaces. We set

[TABLE]

Notice that $\overrightarrow{\mathcal{S}}_{\{M_{p}\}}(0,\infty)$ was denoted by $\mathcal{S}_{\{M_{p}\}}(0,\infty)$ in the introduction. $\mathcal{S}_{(M_{p})}(0,\infty)$ and $\mathcal{S}^{0}_{(M_{p})}(\mathbb{R})$ are Fréchet spaces, $\overrightarrow{\mathcal{S}}_{\{M_{p}\}}(0,\infty)$ and $\overrightarrow{\mathcal{S}}^{0}_{\{M_{p}\}}(\mathbb{R})$ are $(LF)$-spaces, while $\overleftarrow{\mathcal{S}}_{\{M_{p}\}}(0,\infty)$ and $\overleftarrow{\mathcal{S}}^{0}_{\{M_{p}\}}(\mathbb{R})$ are $(PLB)$-spaces. We have that

[TABLE]

as sets.

Lemma 3.2**.**

Let $\mathcal{S}_{\ast}(\mathbb{R})=\mathcal{S}_{(M_{p})}(\mathbb{R})$ or $\mathcal{S}_{\ast}(\mathbb{R})=\overrightarrow{\mathcal{S}}_{\{M_{p}\}}(\mathbb{R})$. Then, the equalities (3.1) and (3.2) hold topologically if the spaces at the right-hand side are endowed with the relative topology induced by $\mathcal{S}_{\ast}(\mathbb{R})$.

We need some preparation for the proof of Lemma 3.2. Let $E=\varinjlim_{n\in\mathbb{N}}E_{n}$ be an $(LF)$-space. A subspace $L$ of $E$ is called a limit subspace of E if $L=\varinjlim_{n\in\mathbb{N}}L\cap E_{n}$ topologically, where $L$ is endowed with the relative topology induced by $E$ and $L\cap E_{n}$, $n\in\mathbb{N}$, is endowed with the relative topology induced by $E_{n}$. The following result is a consequence of [21, Prop. 1.2] and the fact that every Fréchet space is an acyclic $(LF)$-space; we refer to [21] for the definition of an acyclic $(LF)$-space.

Lemma 3.3**.**

(cf. [21, Prop. 1.2])* Let $E$ be an $(LF)$-space, let $F$ be a Fréchet space and let $T:E\rightarrow F$ be a surjective continuous linear mapping. Then, $\operatorname{ker}T$ is a limit subspace of $E$.*

Lemma 3.4**.**

Let $E$ be an $(LF)$-space. Every complemented subspace of $E$ is a limit subspace of $E$.

Proof.

Since $(LF)$-spaces are webbed and ultrabornological [15, Remark 24.36] and the class of ultrabornological lcHs is closed under taking complemented subspaces, this follows from De Wilde’s open mapping theorem [15, Thm. 24.30]. ∎

Proof of Lemma 3.2.

$\mathcal{S}_{\ast}(\mathbb{R})=\mathcal{S}_{(M_{p})}(\mathbb{R})$: Obvious.

$\mathcal{S}_{\ast}(\mathbb{R})=\overrightarrow{\mathcal{S}}_{\{M_{p}\}}(\mathbb{R})$: It suffices to show that $\overrightarrow{\mathcal{S}}_{\{M_{p}\}}(0,\infty)$ and $\overrightarrow{\mathcal{S}}^{0}_{\{M_{p}\}}(\mathbb{R})$ are limit subspaces of $\overrightarrow{\mathcal{S}}_{\{M_{p}\}}(\mathbb{R})$. We first consider $\overrightarrow{\mathcal{S}}^{0}_{\{M_{p}\}}(\mathbb{R})$. By Borel’s theorem, the continuous linear mapping $\mathcal{B}:\overrightarrow{\mathcal{S}}_{\{M_{p}\}}(\mathbb{R})\rightarrow\mathbb{C}^{\mathbb{N}}$ is surjective. Clearly, $\overrightarrow{\mathcal{S}}^{0}_{\{M_{p}\}}(\mathbb{R})=\ker\mathcal{B}$. Hence, the result follows from Lemma 3.3. Next, we deal with $\overrightarrow{\mathcal{S}}_{\{M_{p}\}}(0,\infty)$. Since $\overrightarrow{\mathcal{S}}_{\{M_{p}\}}(0,\infty)$ is a complemented subspace of $\overrightarrow{\mathcal{S}}^{0}_{\{M_{p}\}}(\mathbb{R})$, Lemma 3.4 yields that $\overrightarrow{\mathcal{S}}_{\{M_{p}\}}(0,\infty)$ is a limit subspace of $\overrightarrow{\mathcal{S}}^{0}_{\{M_{p}\}}(\mathbb{R})$. Consequently, as we already have shown that $\overrightarrow{\mathcal{S}}^{0}_{\{M_{p}\}}(\mathbb{R})$ is a limit subspace of $\overrightarrow{\mathcal{S}}_{\{M_{p}\}}(\mathbb{R})$, $\overrightarrow{\mathcal{S}}_{\{M_{p}\}}(0,\infty)$ is a limit subspace of $\overrightarrow{\mathcal{S}}_{\{M_{p}\}}(\mathbb{R})$. ∎

Finally, we present two technical lemmas that will play an important role later on.

Lemma 3.5**.**

Let $(N_{p})_{p\in\mathbb{N}}$ be a weight sequence satisfying $(\operatorname{lc})$ and $(\operatorname{dc})$.

• $(a)$

$T:\mathcal{S}_{(N_{p})}(0,\infty)\rightarrow\mathcal{S}_{(N_{p})}(0,\infty)$* and $T:\overrightarrow{\mathcal{S}}_{\{N_{p}\}}(0,\infty)\rightarrow\overrightarrow{\mathcal{S}}_{\{N_{p}\}}(0,\infty)$ are well-defined continuous mappings, where*

[TABLE]

• $(b)$

$T:\mathcal{S}_{(N_{p})}(0,\infty)\rightarrow\mathcal{S}_{(N_{p})}(0,\infty)$* and $T:\overleftarrow{\mathcal{S}}_{\{N_{p}\}}(0,\infty)\rightarrow\overleftarrow{\mathcal{S}}_{\{N_{p}\}}(0,\infty)$ are well-defined continuous mappings, where*

[TABLE]

Proof.

We start by recalling the following consequence of Taylor’s theorem: Let $\varphi\in C^{n}([0,1])$, $n\in\mathbb{N}$, be such that $\varphi^{(m)}(0)=0$ for all $m=0,\ldots,n$. Then,

[TABLE]

for all $j,k\in\mathbb{N}\mbox{ with }j+k\leq n$.

$(a)$ It suffices to show that $T:\mathcal{S}^{n+2}_{N_{p},h}(0,\infty)\rightarrow\mathcal{S}^{n}_{N_{p},h}(0,\infty)$ is well-defined and continuous for all $n\in\mathbb{N}$ and $h>0$. Let $\varphi\in\mathcal{S}^{n+2}_{N_{p},h}(0,\infty)$ be arbitrary. It holds that

[TABLE]

for all $m\leq n$. Hence, (3.3) yields that $T(\varphi)\in C^{n}(\mathbb{R})$ with $\operatorname{supp}\varphi\subseteq[0,\infty)$ and

[TABLE]

$(b)$ It suffices to show that $T:\mathcal{S}^{n}_{N_{p},hH}(0,\infty)\rightarrow\mathcal{S}^{n}_{N_{p},h}(0,\infty)$ is well-defined and continuous for all $n\in\mathbb{N}$ and $h>0$. Let $\varphi\in\mathcal{S}^{n}_{N_{p},hH}(0,\infty)$ be arbitrary. It holds that $T(\varphi)\in C^{n}(\mathbb{R})$ with $\operatorname{supp}\varphi\subseteq[0,\infty)$ and

[TABLE]

∎

Lemma 3.6**.**

Let $(M_{p})_{p\in\mathbb{N}}$ be a weight sequence satisfying $(\operatorname{lc})$ and $(\operatorname{dc})$. Denote by $(N_{p})_{p\in\mathbb{N}}$ its $2$-interpolating sequence.

• $(a)$

$T:\mathcal{S}_{(N_{p})}(0,\infty)\rightarrow\mathcal{S}_{(M_{p})}(0,\infty)$* and $T:\overrightarrow{\mathcal{S}}_{\{N_{p}\}}(0,\infty)\rightarrow\overrightarrow{\mathcal{S}}_{\{M_{p}\}}(0,\infty)$ are well-defined continuous mappings, where*

[TABLE]

• $(b)$

$T:\mathcal{S}_{(M_{p})}(0,\infty)\rightarrow\mathcal{S}_{(N_{p})}(0,\infty)$* and $T:\overleftarrow{\mathcal{S}}_{\{M_{p}\}}(0,\infty)\rightarrow\overleftarrow{\mathcal{S}}_{\{N_{p}\}}(0,\infty)$ are well-defined continuous mappings, where*

[TABLE]

Proof.

$(a)$ It suffices to show that $T:\mathcal{S}^{2n+1}_{N_{p},h^{1/2}}(0,\infty)\rightarrow\mathcal{S}^{n}_{M_{p},h}(0,\infty)$ is well-defined and continuous for all $n\in\mathbb{N}$ and $h>0$. Let $\varphi\in\mathcal{S}^{2n+1}_{N_{p},h^{1/2}}(0,\infty)$ be arbitrary. We set $I_{0}=\{0\}$ and

[TABLE]

Faà di Bruno’s formula implies that

[TABLE]

for all $m\leq n$, where $a_{\alpha}$ are real constants. Hence, (3.3) yields that $T(\varphi)\in C^{n}(\mathbb{R})$ with $\operatorname{supp}\varphi\subseteq[0,\infty)$. Since $M_{p}=N_{2p}$ for all $p\in\mathbb{N}$, it holds that

[TABLE]

$(b)$ Lemma 2.3$(a)$ yields that $(N_{p})_{p\in\mathbb{N}}$ satisfies $(\operatorname{dc})$. We may assume without loss of generality that the constants $C_{0}$ and $H$ occuring in $(\operatorname{dc})$ are the same for $(M_{p})_{p\in\mathbb{N}}$ and $(N_{p})_{p\in\mathbb{N}}$. It suffices to show that $T:\mathcal{S}^{n}_{M_{p},H^{2n+1}h^{2}}(0,\infty)\rightarrow\mathcal{S}^{n}_{N_{p},h}(0,\infty)$ is well-defined and continuous for all $n\in\mathbb{N}$ and $h>0$. Set $l=H^{2n+1}h^{2}$. Let $\varphi\in\mathcal{S}^{n}_{M_{p},l}(0,\infty)$ be arbitrary. Clearly, $T(\varphi)\in C^{n}(\mathbb{R})$ with $\operatorname{supp}\varphi\subseteq[0,\infty)$. Faà di Bruno’s formula implies that

[TABLE]

for all $m\leq n$, where $a_{j}$ are positive constants. It holds that $N_{p+q}\leq C_{0}^{q}H^{q(q-1)/2}H^{pq}N_{p}$ for all $p,q\in\mathbb{N}$. Moreover, there is $C_{1}>0$ such that $M_{\lceil p/2\rceil}\leq C_{1}H^{\lceil p/2\rceil}N_{p}$ for all $p\in\mathbb{N}$. Therefore,

[TABLE]

∎

4. A functional analytic tool

In this section, we show an abstract result about the existence of a continuous linear right inverse that it is tailor-made to prove Proposition 6.3 below. We start with the following simple observation.

Lemma 4.1**.**

Let $E$, $F$ and $G$ be vector spaces and let $T:E\rightarrow F$ and $S:E\rightarrow G$ be linear mappings. If both $T:E\rightarrow F$ and $S_{|\ker T}:\ker T\rightarrow G$ are surjective, then $T_{|\ker S}:\ker S\rightarrow F$ is also surjective.

Proof.

Let $x\in F$ be arbitrary. Choose $y\in E$ such that $T(y)=x$ and $z\in\ker T$ such that $S(z)=S(y)$. Then, $y-z\in\ker S$ and $T(y-z)=T(y)=x$. ∎

Now suppose that $E$, $F$ and $G$ are lcHs and that $T$ and $S$ are continuous linear mappings. If both $T$ and $S_{|\ker T}$ have a continuous linear right inverse, it is clear from the proof of Lemma 4.1 that also $T_{|\ker S}$ has a continuous linear right inverse. Our goal is to show that, under suitable extra conditions on $F$ and $T$, $T_{|\ker S}$ has a continuous linear right inverse if one merely assumes that $S_{|\ker T}$ lifts bounded sets. We need some preparation to formulate and prove this result.

Let $E$ and $F$ be lcHs. We denote by $\operatorname{csn}(E)$ the set consisting of all continuous seminorms on $E$ and by $L(E,F)$ the space consisting of all continuous linear mappings from $E$ to $F$. Let $(x_{n})_{n\in\mathbb{N}}\subset F$ and $(x^{\prime}_{n})_{n\in\mathbb{N}}\subset F^{\prime}$. The pair $((x_{n})_{n\in\mathbb{N}},(x^{\prime}_{n})_{n\in\mathbb{N}})$ is said to be a Schauder frame (in $F$) [2, Def. 1.1] if

[TABLE]

for all $x\in F$. A Schauder frame $((x_{n})_{n\in\mathbb{N}},(x^{\prime}_{n})_{n\in\mathbb{N}})$ is called absolute if for all $p\in\operatorname{csn}(F)$ there is $q\in\operatorname{csn}(F)$ such that

[TABLE]

for all $x\in F$. Every absolute Schauder basis [15, p. 340] canonically determines a Schauder frame. We need the following lemma.

Lemma 4.2**.**

Let $E$ and $F$ be lcHs and let $T\in L(E,F)$. Suppose that $E$ is sequentially complete and that $F$ possesses an absolute Schauder frame $((x_{n})_{n\in\mathbb{N}},(x^{\prime}_{n})_{n\in\mathbb{N}})$. If there is a sequence $(y_{n})_{n\in\mathbb{N}}\subset E$ such that $T(y_{n})=x_{n}$ for all $n\in\mathbb{N}$ and for all $p\in\operatorname{csn}(E)$ there is $q\in\operatorname{csn}(F)$ such that $p(y_{n})\leq q(x_{n})$ for all $n\in\mathbb{N}$, then $T$ has a continuous linear right inverse.

Proof.

For each $x\in F$ the sequence $\left(\sum_{n=0}^{N}\langle x^{\prime}_{n},x\rangle y_{n}\right)_{N\in\mathbb{N}}$ is Cauchy in $E$. Since $E$ is sequentially complete, we have that

[TABLE]

Then, $R:F\rightarrow E$ is a continuous linear right inverse of $T$. ∎

Proposition 4.3**.**

Let $E$, $F$ and $G$ be lcHs and let $T\in L(E,F)$ and $S\in L(E,G)$. Endow $\ker T$ and $\ker S$ with the relative topology induced by $E$. Suppose that the following conditions are satisfied:

• $(1)$

$E$* is sequentially complete.*

• $(2)$

$F$* possesses an absolute Schauder frame.*

• $(3)$

$S_{|\ker T}:\ker T\rightarrow G$* lifts bounded sets, that is, for every $B\subset G$ bounded there is $A\subset\ker T$ bounded such that $S(A)=B$.*

• $(4)$

There is a lcHs $E_{0}$ with the following properties:

• $(4.1)$

$E_{0}\subset E$* with continuous inclusion mapping.*

• $(4.2)$

$T_{|E_{0}}:E_{0}\rightarrow F$* has a continuous linear right inverse.*

• $(4.3)$

$S_{|E_{0}}:E_{0}\rightarrow G$* is locally bounded, that is, there is a neighbourhood $U$ of [math] in $E_{0}$ such that $S(U)$ is bounded in $G$.*

Then, $T_{|\ker S}:\ker S\rightarrow F$ has a continuous linear right inverse.

Proof.

Suppose that $((x_{n})_{n\in\mathbb{N}},(x^{\prime}_{n})_{n\in\mathbb{N}})$ is an absolute Schauder frame in $F$. Since $\ker S$ is sequentially complete (as a closed subspace of $E$), it suffices to construct a sequence $(y_{n})_{n\in\mathbb{N}}\subset\ker S$ satisfying the assumptions of Lemma 4.2. By $(4.2)$, there is a sequence $(y_{0,n})_{n\in\mathbb{N}}\subset E_{0}$ such that $T(y_{0,n})=x_{n}$ for all $n\in\mathbb{N}$ and for all $p\in\operatorname{csn}(E_{0})$ there is $q\in\operatorname{csn}(F)$ such that $p(y_{0,n})\leq q(x_{n})$ for all $n\in\mathbb{N}$. $(4.3)$ means that there is $p_{0}\in\operatorname{csn}(E_{0})$ such that $S_{|E_{0}}:(E_{0},p_{0})\rightarrow G$ is continuous, where $(E_{0},p_{0})$ stands for the vector space $E_{0}$ endowed with the topology generated by the single seminorm $p_{0}$. In particular, $S(y)=0$ for all $y\in E_{0}$ with $p_{0}(y)=0$. We set

[TABLE]

for $n\in\mathbb{N}$. Since the sequence $(s_{n})_{n\in\mathbb{N}}$ is bounded in $G$, $(3)$ yields that there is a bounded sequence $(z_{n})_{n\in\mathbb{N}}\subset\ker T$ such that $S(z_{n})=s_{n}$ for all $n\in\mathbb{N}$. Set $y_{n}=y_{0,n}-p_{0}(y_{0,n})z_{n}$ for all $n\in\mathbb{N}$. Then, $y_{n}\in\ker S$ and $T(y_{n})=T(y_{0,n})=x_{n}$ for all $n\in\mathbb{N}$. Finally, let $p\in\operatorname{csn}(E)$ be arbitrary. Since the sequence $(z_{n})_{n\in\mathbb{N}}$ is bounded in $E$, there is $C>0$ such that

[TABLE]

for all $n\in\mathbb{N}$, where $p^{\prime}=(1+C)\max\{p_{|E_{0}},p_{0}\}$. $(4.1)$ yields that $p^{\prime}\in\operatorname{csn}(E_{0})$. Hence, there is $q\in\operatorname{csn}(F)$ such that $p(y_{n})\leq p^{\prime}(y_{0,n})\leq q(x_{n})$ for all $n\in\mathbb{N}$. ∎

Remark 4.4*.*

If $E$ is an $(FS)$-space (= Fréchet-Schwartz space) and $G$ is a Fréchet space, then condition $(3)$ in Proposition 4.3 may be relaxed to “$S_{|\ker T}:\ker T\rightarrow G$ is surjective”. Indeed, as a closed subspace of an $(FS)$-space is again an $(FS)$-space, $\ker(S_{|\ker T})$ is an $(FS)$-space. Since every $(FS)$-space is quasinormable, the result follows from the fact that a surjective continuous linear mapping $Q:X\rightarrow Y$ between two Fréchet spaces $X$ and $Y$ lifts bounded sets if $\ker Q$ is quasinormable [15, Lemma 26.13].

5. The Borel problem in $\mathcal{S}_{(N_{p}),0}(\mathbb{R})$

Given a weight sequence $(N_{p})_{p\in\mathbb{N}}$, we define the following closed subspace of $\mathcal{S}_{(N_{p})}(\mathbb{R})$

[TABLE]

and endow it with the relative topology induced by $\mathcal{S}_{(N_{p})}(\mathbb{R})$. Hence, it becomes a Fréchet space. The goal of this section is to show the following result; it will be used in the proof of Proposition 6.3 below.

Proposition 5.1**.**

Let $(N_{p})_{p\in\mathbb{N}}$ be a weight sequence satisfying $(\operatorname{lc})$, $(\operatorname{dc})$ and $(\gamma_{1})$. Then, $\mathcal{B}:\mathcal{S}_{(N_{p}),0}(\mathbb{R})\rightarrow\mathbb{C}^{\mathbb{N}}$ is surjective.

If $(N_{p})_{p\in\mathbb{N}}$ satisfies $\operatorname{(lc)}$ and $\operatorname{(dc)}$, [12, Prop. 3.4] implies that $\mathcal{S}_{(N_{p})}(\mathbb{R})$ is an $(FS)$-space. Hence, Proposition 5.1 can be strengthened as follows (cf. Remark 4.4).

Proposition 5.2**.**

Let $(N_{p})_{p\in\mathbb{N}}$ be a weight sequence satisfying $(\operatorname{lc})$, $(\operatorname{dc})$ and $(\gamma_{1})$. Then, $\mathcal{B}:\mathcal{S}_{(N_{p}),0}(\mathbb{R})\rightarrow\mathbb{C}^{\mathbb{N}}$ lifts bounded sets.

The proof of Proposition 5.1 is based on the following variant of Eidelheit’s theorem.

Proposition 5.3**.**

Let $E$ be a Fréchet space and let $(x^{\prime}_{n})_{n\in\mathbb{N}}\subset E^{\prime}$. Let $F$ be a closed subspace of $E$ and set

[TABLE]

The mapping $F\rightarrow\mathbb{C}^{\mathbb{N}}:x\rightarrow(\langle x^{\prime}_{n},x\rangle)_{n\in\mathbb{N}}$ is surjective if and only if

• $(1)$

For all $N\in\mathbb{N}$ and $c_{0},\ldots,c_{N}\in\mathbb{C}$ it holds that

[TABLE]

implies that $c_{0}=\cdots=c_{N}=0$.

• $(2)$

For every $B\subset E^{\prime}$ equicontinuous there is $\nu\in\mathbb{N}$ such that for all $N\geq\nu$ and $c_{0},\ldots,c_{N}\in\mathbb{C}$ it holds that

[TABLE]

implies that $c_{\nu}=\cdots=c_{N}=0$.

Proof.

This is a consequence of the classical theorem of Eidelheit [15, Thm. 26.27] and the Hahn-Banach theorem. ∎

We shall prove Proposition 5.1 via Proposition 5.3 with $E=\mathcal{S}_{(N_{p})}(\mathbb{R})$, $F=\mathcal{S}_{(N_{p}),0}(\mathbb{R})$ and $(x^{\prime}_{n})_{n\in\mathbb{N}}=((-1)^{n}\delta^{(n)})_{n\in\mathbb{N}}$). To this end, we first give an explicit description of the space $(\mathcal{S}_{(N_{p}),0}(\mathbb{R}))^{\perp}$ and the equicontinuous subsets of $\mathcal{S}^{\prime}_{(N_{p})}(\mathbb{R})$. We need some preparation.

Let $(N_{p})_{p\in\mathbb{N}}$ be a weight sequence. An entire function $P(z)=\sum_{p=0}^{\infty}b_{p}z^{p}$, $b_{p}\in\mathbb{C}$, is said to be an ultrapolynomial of class $(N_{p})$ if

[TABLE]

for some $h>0$. If $(N_{p})_{p\in\mathbb{N}}$ satisfies $(\operatorname{lc})$, an entire function $P$ is an ultrapolynomial of class $(N_{p})$ if and only if

[TABLE]

for some $h>0$.

Lemma 5.4**.**

Let $(N_{p})_{p\in\mathbb{N}}$ be a weight sequence satisfying $(\operatorname{lc})$, $(\operatorname{dc})$ and $(\gamma_{1})$. Then, $f\in\mathcal{S}^{\prime}_{(N_{p})}(\mathbb{R})$ belongs to $(\mathcal{S}_{(N_{p}),0}(\mathbb{R}))^{\perp}$ if and only if there is an ultrapolynomial $P$ of class $(N_{p})$ such that $f=P$ in $\mathcal{S}^{\prime}_{(N_{p})}(\mathbb{R})$.

Proof.

Since $\mathcal{D}^{(N_{p})}_{[-1,1]}\subseteq\mathcal{S}^{(N_{p})}(\mathbb{R})$, Theorem 2.4$(a)$ implies that $\mathcal{B}:\mathcal{S}^{(N_{p})}(\mathbb{R})\rightarrow\Lambda_{(N_{p})}$ is surjective. By taking the Fourier transform, we obtain that $\mathcal{M}:\mathcal{S}_{(N_{p})}(\mathbb{R})\rightarrow\Lambda_{(N_{p})}$ is surjective. Consequently, the sequence

[TABLE]

is exact, where $\iota:\mathcal{S}_{(N_{p}),0}(\mathbb{R})\rightarrow\mathcal{S}_{(N_{p})}(\mathbb{R})$ denotes the inclusion mapping. Therefore, its dual sequence

[TABLE]

is also exact [15, Prop. 26.4]. In particular, $\operatorname{im}\mathcal{M}^{t}=\ker\iota^{t}$. It is clear that $\ker\iota^{t}=(\mathcal{S}_{(N_{p}),0}(\mathbb{R}))^{\perp}$. On the other hand, $\Lambda^{\prime}_{(N_{p})}$ may be identified with the space consisting of all sequences $b=(b_{p})_{p\in\mathbb{N}}\in\mathbb{C}^{\mathbb{N}}$ satisfying (5.1) for some $h>0$ and, under this identification, the duality is given by

[TABLE]

Hence, $\operatorname{im}\mathcal{M}^{t}$ coincides with the subspace of $\mathcal{S}^{\prime}_{(N_{p})}(\mathbb{R})$ consisting of all ultrapolynomials of class $(N_{p})$. ∎

The next result follows from the structural theorem for general Gelfand-Shilov spaces [10, p. 113] and [12, Prop. 3.4].

Lemma 5.5**.**

Let $(N_{p})_{p\in\mathbb{N}}$ be a weight sequence satisfying $(\operatorname{lc})$ and $(\operatorname{dc})$. For every $B\subset\mathcal{S}^{\prime}_{(N_{p})}(\mathbb{R})$ equicontinuous there are $\nu\in\mathbb{N}$ and $C,h>0$ such that for all $f\in B$ there are measurable functions $g_{0},\ldots,g_{\nu}$ such that

[TABLE]

and

[TABLE]

for all $n=0,\ldots,\nu$.

Finally, we will also use the following well-known fact from distribution theory.

Lemma 5.6**.**

Let $\nu\in\mathbb{N}$. For all $N\geq\nu$, $c_{0},\ldots,c_{N}\in\mathbb{C}$ and $g_{0},\ldots,g_{\nu}\in L^{1}_{\operatorname{loc}}(\mathbb{R})$ it holds that

[TABLE]

implies that $c_{\nu}=\cdots=c_{N}=0$.

Proof of Proposition 5.1.

We use Proposition 5.3 with $E=\mathcal{S}_{(N_{p})}(\mathbb{R})$, $F=\mathcal{S}_{(N_{p}),0}(\mathbb{R})$ and $(x^{\prime}_{n})_{n\in\mathbb{N}}=((-1)^{n}\delta^{(n)})_{n\in\mathbb{N}}$. In view of Lemma 5.4, condition $(1)$ follows from Lemma 5.6 (with $\nu=0$), while condition $(2)$ follows from Lemmas 5.5 and 5.6. ∎

6. The Stieltjes moment problem in $\mathcal{S}_{\ast}(0,\infty)$

We are ready to characterize the surjectivity and the existence of a continuous linear right inverse of $\mathcal{M}:\mathcal{S}_{\ast}(0,\infty)\rightarrow\Lambda_{\ast}$ (cf. Theorem 1.2).

Theorem 6.1**.**

Let $(M_{p})_{p\in\mathbb{N}}$ be a weight sequence satisfying $(\operatorname{lc})$ and $(\operatorname{dc})$.

• $(a)$

The following statements are equivalent:

• $(i)$

$(M_{p})_{p\in\mathbb{N}}$* satisfies $(\gamma_{2})$.*

• $(ii)$

$\mathcal{M}:\mathcal{S}_{(M_{p})}(0,\infty)\rightarrow\Lambda_{(M_{p})}$* has a continuous linear right inverse.*

• $(iii)$

$\mathcal{M}:\mathcal{S}_{(M_{p})}(0,\infty)\rightarrow\Lambda_{(M_{p})}$* is surjective.*

• $(b)$

The following statements are equivalent:

• $(i)$

$(M_{p})_{p\in\mathbb{N}}$* satisfies $(\gamma_{2})$.*

• $(ii)$

$\mathcal{M}:\overrightarrow{\mathcal{S}}_{\{M_{p}\}}(0,\infty)\rightarrow\Lambda_{\{M_{p}\}}$* is surjective.*

• $(iii)$

$\mathcal{M}:\overleftarrow{\mathcal{S}}_{\{M_{p}\}}(0,\infty)\rightarrow\Lambda_{\{M_{p}\}}$* is surjective.*

• $(c)$

The following statements are equivalent:

• $(i)$

$(M_{p})_{p\in\mathbb{N}}$* satisfies $(\gamma_{2})$ and $(\beta_{2})$.*

• $(ii)$

$\mathcal{M}:\overrightarrow{\mathcal{S}}_{\{M_{p}\}}(0,\infty)\rightarrow\Lambda_{\{M_{p}\}}$* has a continuous linear right inverse.*

• $(iii)$

$\mathcal{M}:\overleftarrow{\mathcal{S}}_{\{M_{p}\}}(0,\infty)\rightarrow\Lambda_{\{M_{p}\}}$* has a continuous linear right inverse.*

In view of Lemma 2.3, Theorem 6.1 is a consequence of the following two results.

Proposition 6.2**.**

Let $(M_{p})_{p\in\mathbb{N}}$ be a weight sequence satisfying $(\operatorname{lc})$ and $(\operatorname{dc})$. Denote by $(N_{p})_{p\in\mathbb{N}}$ its $2$-interpolating sequence.

• $(a)$

$\mathcal{M}:\mathcal{S}_{(M_{p})}(0,\infty)\rightarrow\Lambda_{(M_{p})}$* is surjective (has a continuous linear right inverse) if and only if $\mathcal{M}:\mathcal{S}_{(N_{p})}^{0}(\mathbb{R})\rightarrow\Lambda_{(N_{p})}$ is surjective (has a continuous linear right inverse).*

• $(b)$

If $\mathcal{M}:\overrightarrow{\mathcal{S}}_{\{N_{p}\}}^{0}(\mathbb{R})\rightarrow\Lambda_{\{N_{p}\}}$ is surjective (has a continuous linear right inverse), $\mathcal{M}:\overrightarrow{\mathcal{S}}_{\{M_{p}\}}(0,\infty)\rightarrow\Lambda_{\{M_{p}\}}$ is surjective (has a continuous linear right inverse) as well.

• $(c)$

If $\mathcal{M}:\overleftarrow{\mathcal{S}}_{\{M_{p}\}}(0,\infty)\rightarrow\Lambda_{\{M_{p}\}}$ is surjective (has a continuous linear right inverse), $\mathcal{M}:\overleftarrow{\mathcal{S}}_{\{N_{p}\}}^{0}(\mathbb{R})\rightarrow\Lambda_{\{N_{p}\}}$ is surjective (has a continuous linear right inverse) as well.

Proposition 6.3**.**

Let $(N_{p})_{p\in\mathbb{N}}$ be a weight sequence satisfying $(\operatorname{lc})$ and $(\operatorname{dc})$.

• $(a)$

The following statements are equivalent:

• $(i)$

$(N_{p})_{p\in\mathbb{N}}$* satisfies $(\gamma_{1})$.*

• $(ii)$

$\mathcal{M}:\mathcal{S}_{(N_{p})}^{0}(\mathbb{R})\rightarrow\Lambda_{(N_{p})}$* has a continuous linear right inverse.*

• $(iii)$

$\mathcal{M}:\mathcal{S}_{(N_{p})}^{0}(\mathbb{R})\rightarrow\Lambda_{(N_{p})}$* is surjective.*

• $(b)$

The following statements are equivalent:

• $(i)$

$(N_{p})_{p\in\mathbb{N}}$* satisfies $(\gamma_{1})$.*

• $(ii)$

$\mathcal{M}:\overrightarrow{\mathcal{S}}_{\{N_{p}\}}^{0}(\mathbb{R})\rightarrow\Lambda_{\{N_{p}\}}$* is surjective.*

• $(iii)$

$\mathcal{M}:\overleftarrow{\mathcal{S}}_{\{N_{p}\}}^{0}(\mathbb{R})\rightarrow\Lambda_{\{N_{p}\}}$* is surjective.*

• $(c)$

The following statements are equivalent:

• $(i)$

$(N_{p})_{p\in\mathbb{N}}$* satisfies $(\gamma_{1})$ and $(\beta_{2})$.*

• $(ii)$

$\mathcal{M}:\overrightarrow{\mathcal{S}}_{\{N_{p}\}}^{0}(\mathbb{R})\rightarrow\Lambda_{\{N_{p}\}}$* has a continuous linear right inverse.*

• $(iii)$

$\mathcal{M}:\overleftarrow{\mathcal{S}}^{0}_{\{N_{p}\}}(\mathbb{R})\rightarrow\Lambda_{\{N_{p}\}}$* has a continuous linear right inverse.*

The rest of this section is devoted to the proofs of the above two results.

Proof of Proposition 6.2.

We only show that $\mathcal{M}:\mathcal{S}_{(M_{p})}(0,\infty)\rightarrow\Lambda_{(M_{p})}$ has a continuous linear right inverse if and only if $\mathcal{M}:\mathcal{S}_{(N_{p})}^{0}(\mathbb{R})\rightarrow\Lambda_{(N_{p})}$ does so; all other statements follow from a similar argument. We start with the direct implication. The proof is divided into two steps.

STEP I: $\mathcal{M}_{e}:\mathcal{S}_{(N_{p})}(0,\infty)\rightarrow\Lambda_{(M_{p})}:\varphi\rightarrow(\mu_{2p}(\varphi))_{p\in\mathbb{N}}$ and $\mathcal{M}_{o}:\mathcal{S}_{(N_{p})}(0,\infty)\rightarrow\Lambda_{(M_{p})}:\varphi\rightarrow(\mu_{2p+1}(\varphi))_{p\in\mathbb{N}}$ have a continuous linear right inverse. Let $R:\Lambda_{(M_{p})}\rightarrow\mathcal{S}_{(M_{p})}(0,\infty)$ be a continuous linear right inverse of $\mathcal{M}:\mathcal{S}_{(M_{p})}(0,\infty)\rightarrow\Lambda_{(M_{p})}$. Lemmas 3.5$(b)$ and 3.6$(b)$ imply that the mapping

[TABLE]

is well-defined and continuous. We claim that $T_{e}\circ R:\Lambda_{(M_{p})}\rightarrow\mathcal{S}_{(N_{p})}(0,\infty)$ is a continuous linear right inverse of $\mathcal{M}_{e}:\mathcal{S}_{(N_{p})}(0,\infty)\rightarrow\Lambda_{(M_{p})}$. Let $a=(a_{p})_{p\in\mathbb{N}}\in\Lambda_{(M_{p})}$ be arbitrary. Then,

[TABLE]

for all $p\in\mathbb{N}$. Likewise, Lemma 3.6$(b)$ yields that the mapping

[TABLE]

is well-defined and continuous. We claim that $T_{o}\circ R:\Lambda_{(M_{p})}\rightarrow\mathcal{S}_{(N_{p})}(0,\infty)$ is a continuous linear right inverse of $\mathcal{M}_{o}:\mathcal{S}_{(N_{p})}(0,\infty)\rightarrow\Lambda_{(M_{p})}$. Let $a=(a_{p})_{p\in\mathbb{N}}\in\Lambda_{(M_{p})}$ be arbitrary. Then,

[TABLE]

for all $p\in\mathbb{N}$.

STEP II: $\mathcal{M}:\mathcal{S}_{(N_{p})}^{0}(\mathbb{R})\rightarrow\Lambda_{(N_{p})}$ has a continuous linear right inverse. Let $R_{e}:\Lambda_{(M_{p})}\rightarrow\mathcal{S}_{(N_{p})}(0,\infty)$ and $R_{o}:\Lambda_{(M_{p})}\rightarrow\mathcal{S}_{(N_{p})}(0,\infty)$ be continuous linear right inverses of $\mathcal{M}_{e}:\mathcal{S}_{(N_{p})}(0,\infty)\rightarrow\Lambda_{(M_{p})}$ and $\mathcal{M}_{o}:\mathcal{S}_{(N_{p})}(0,\infty)\rightarrow\Lambda_{(M_{p})}$, respectively. Consider the continuous mappings

[TABLE]

and

[TABLE]

Let $a=(a_{p})_{p\in\mathbb{N}}\in\Lambda_{(N_{p})}$ be arbitrary. Then,

[TABLE]

for all $p\in\mathbb{N}$. Hence, $S_{e}\circ R_{e}\circ T_{e}+S_{o}\circ R_{o}\circ T_{o}$ is a continuous linear right inverse of $\mathcal{M}:\mathcal{S}_{(N_{p})}^{0}(\mathbb{R})\rightarrow\Lambda_{(N_{p})}$.

Next, we show the converse implication. Again, we divide the proof into two steps.

STEP I: $\mathcal{M}_{e}:\mathcal{S}_{(N_{p})}(0,\infty)\rightarrow\Lambda_{(M_{p})}:\varphi\rightarrow(\mu_{2p}(\varphi))_{p\in\mathbb{N}}$ has a continuous linear right inverse. Let $R:\Lambda_{(N_{p})}\rightarrow\mathcal{S}_{(N_{p})}^{0}(\mathbb{R})$ be a continuous linear right inverse of $\mathcal{M}:\mathcal{S}_{(N_{p})}^{0}(\mathbb{R})\rightarrow\Lambda_{(N_{p})}$. Consider the continuous mappings $T:\Lambda_{(M_{p})}\rightarrow\Lambda_{(N_{p})}$ given by $T((a_{p})_{p\in\mathbb{N}})=(b_{p})_{p\in\mathbb{N}}$, where

[TABLE]

and

[TABLE]

We claim that $S\circ R\circ T:\Lambda_{(M_{p})}\rightarrow\mathcal{S}_{(N_{p})}(0,\infty)$ is a continuous linear right inverse of $\mathcal{M}_{e}:\mathcal{S}_{(N_{p})}(0,\infty)\rightarrow\Lambda_{(M_{p})}$. Let $a=(a_{p})_{p\in\mathbb{N}}\in\Lambda_{(M_{p})}$ be arbitrary. Then,

[TABLE]

for all $p\in\mathbb{N}$.

STEP II: $\mathcal{M}:\mathcal{S}_{(M_{p})}(0,\infty)\rightarrow\Lambda_{(M_{p})}$ has a continuous linear right inverse. Let $R_{e}:\Lambda_{(M_{p})}\rightarrow\mathcal{S}_{(N_{p})}(0,\infty)$ be a continuous linear right inverse of $\mathcal{M}_{\operatorname{e}}:\mathcal{S}_{(N_{p})}(0,\infty)\rightarrow\Lambda_{(M_{p})}$. Lemmas 3.5$(a)$ and 3.6$(a)$ imply that the mapping

[TABLE]

is well-defined and continuous. We claim that $T\circ R_{e}:\Lambda_{(M_{p})}\rightarrow\mathcal{S}_{(M_{p})}(0,\infty)$ is a continuous linear right inverse of $\mathcal{M}:\mathcal{S}_{(M_{p})}(0,\infty)\rightarrow\Lambda_{(M_{p})}$. Let $a=(a_{p})_{p\in\mathbb{N}}\in\Lambda_{(M_{p})}$ be arbitrary. Then,

[TABLE]

for all $p\in\mathbb{N}$. ∎

Proof of Proposition 6.3.

$(a)$ $(i)\Rightarrow(ii)$ We apply Proposition 4.3 with $E=\mathcal{S}_{(N_{p})}(\mathbb{R})$, $F=\Lambda_{(N_{p})}$, $G=\mathbb{C}^{\mathbb{N}}$, $T=\mathcal{M}:\mathcal{S}_{(N_{p})}(\mathbb{R})\rightarrow\Lambda_{(N_{p})}$ and $S=\mathcal{B}:\mathcal{S}_{(N_{p})}(\mathbb{R})\rightarrow\mathbb{C}^{\mathbb{N}}$. Lemma 3.2 yields that the topology induced by $E=\mathcal{S}_{(N_{p})}(\mathbb{R})$ on $\ker S=\mathcal{S}^{0}_{(N_{p})}(\mathbb{R})$ coincides with the original topology of $\mathcal{S}^{0}_{(N_{p})}(\mathbb{R})$. We now verify conditions $(1)$-$(4)$: $(1)$ Obvious. $(2)$ The sequence of standard unit vectors is an absolute Schauder basis in $\Lambda_{(N_{p})}$. $(3)$ This has been shown in Proposition 5.2. $(4)$ Set $E_{0}=\mathcal{F}^{-1}\left(\mathcal{D}^{(N_{p})}_{[-1,1]}\right)$ and endow it with the topology generated by the system of seminorms $\left\{p\circ\mathcal{F}\,|\,p\in\operatorname{csn}(\mathcal{D}^{(N_{p})}_{[-1,1]})\right\}$. Notice that $\mathcal{D}^{(N_{p})}_{[-1,1]}\subset\mathcal{S}^{(N_{p})}(\mathbb{R})$ with continuous inclusion mapping and recall that $\mathcal{F}:\mathcal{S}_{(N_{p})}(\mathbb{R})\rightarrow\mathcal{S}^{(N_{p})}(\mathbb{R})$ is a topological isomorphism. Hence, $(4.1)$ is clear, while $(4.2)$ follows from Theorem 2.4$(a)$. Finally,

[TABLE]

for all $\varphi\in E_{0}$, whence $(4.3)$ holds.

$(ii)\Rightarrow(iii)$ Trivial.

$(iii)\Rightarrow(i)$ In particular, $\mathcal{M}:\mathcal{S}_{(N_{p})}(\mathbb{R})\rightarrow\Lambda_{(N_{p})}$ is surjective. By taking the Fourier transform, we obtain that $\mathcal{B}:\mathcal{S}^{(N_{p})}(\mathbb{R})\rightarrow\Lambda_{(N_{p})}$ is surjective. Choose $\varphi\in\mathcal{S}^{(N_{p})}(\mathbb{R})$ such that $\varphi^{(p)}(0)=\delta_{0,p}$ for all $p\in\mathbb{N}$. Set $\psi=\varphi-1$. Then,

[TABLE]

for all $h>0$ and $\psi^{(p)}(0)=0$ for all $p\in\mathbb{N}$. Since $\lim_{|x|\to\infty}\varphi(x)=0$, $\psi$ is not identically zero. Hence, the Denjoy-Carleman theorem implies that $(N_{p})_{p\in\mathbb{N}}$ satisfies $(\gamma)$. By Theorem 2.4$(a)$, it therefore suffices to show that the mapping $\mathcal{B}:\mathcal{D}^{(N_{p})}_{[-1,1]}\rightarrow\Lambda_{(N_{p})}$ is surjective. Let $a\in\Lambda_{(N_{p})}$ be arbitrary and choose $\varphi\in\mathcal{S}^{(N_{p})}(\mathbb{R})$ such that $\mathcal{B}(\varphi)=a$. Pick $\psi\in\mathcal{D}^{(N_{p})}_{[-1,1]}$ such that $\psi\equiv 1$ in a neighbourhood of 0. Then, $\varphi\psi\in\mathcal{D}^{(N_{p})}_{[-1,1]}$ and $\mathcal{B}(\varphi\psi)=a$.

$(b)$ $(i)\Rightarrow(ii)$ We apply Lemma 4.1 with $E=\overrightarrow{\mathcal{S}}_{\{N_{p}\}}(\mathbb{R})$, $F=\Lambda_{\{N_{p}\}}$, $G=\mathbb{C}^{\mathbb{N}}$, $T=\mathcal{M}:\overrightarrow{\mathcal{S}}_{\{N_{p}\}}(\mathbb{R})\rightarrow\Lambda_{\{N_{p}\}}$ and $S=\mathcal{B}:\overrightarrow{\mathcal{S}}_{\{N_{p}\}}(\mathbb{R})\rightarrow\mathbb{C}^{\mathbb{N}}$. Theorem 2.4$(b)$ and the inclusion $\mathcal{D}^{\{N_{p}\}}_{[-1,1]}\subset\overrightarrow{\mathcal{S}}^{\{N_{p}\}}(\mathbb{R})$ yield that $\mathcal{B}:\overrightarrow{\mathcal{S}}^{\{N_{p}\}}(\mathbb{R})\rightarrow\Lambda_{\{N_{p}\}}$ is surjective. By taking the Fourier transform, we obtain that $T$ is surjective. $S_{|\ker T}$ is surjective because of Proposition 5.1 and the inclusion $\mathcal{S}_{(N_{p})}(\mathbb{R})\subset\overrightarrow{\mathcal{S}}_{\{N_{p}\}}(\mathbb{R})$.

$(ii)\Rightarrow(iii)$ This follows from the continuous inclusion $\overrightarrow{\mathcal{S}}_{\{N_{p}\}}(\mathbb{R})\subset\overleftarrow{\mathcal{S}}_{\{N_{p}\}}(\mathbb{R})$.

$(iii)\Rightarrow(i)$ This can be shown in a similar way as $(iii)\Rightarrow(i)$ from part $(a)$.

$(c)$ $(i)\Rightarrow(ii)$ We apply Proposition 4.3 with $E=\overrightarrow{\mathcal{S}}_{\{N_{p}\}}(\mathbb{R})$, $F=\Lambda_{\{N_{p}\}}$, $G=\mathbb{C}^{\mathbb{N}}$, $T=\mathcal{M}:\overrightarrow{\mathcal{S}}_{\{N_{p}\}}(\mathbb{R})\rightarrow\Lambda_{\{N_{p}\}}$ and $S=\mathcal{B}:\overrightarrow{\mathcal{S}}_{\{N_{p}\}}(\mathbb{R})\rightarrow\mathbb{C}^{\mathbb{N}}$. Lemma 3.2 yields that the topology induced by $E=\overrightarrow{\mathcal{S}}_{\{N_{p}\}}(\mathbb{R})$ on $\ker S=\overrightarrow{\mathcal{S}}^{0}_{\{N_{p}\}}(\mathbb{R})$ coincides with the original topology of $\overrightarrow{\mathcal{S}}^{0}_{\{N_{p}\}}(\mathbb{R})$. We now verify conditions $(1)$-$(4)$: $(1)$ $\overrightarrow{\mathcal{S}}_{\{N_{p}\}}(\mathbb{R})$ is complete by Proposition 3.1. $(2)$ The sequence of standard unit vectors is an absolute Schauder basis in $\Lambda_{\{N_{p}\}}$. $(3)$ This follows from Proposition 5.2 and the continuous inclusion $\mathcal{S}_{(N_{p})}(\mathbb{R})\subset\overrightarrow{\mathcal{S}}_{\{N_{p}\}}(\mathbb{R})$. $(4)$ Set $E_{0}=\mathcal{F}^{-1}\left(\mathcal{D}^{\{N_{p}\}}_{[-1,1]}\right)$ and endow it with the topology generated by the system of seminorms $\left\{p\circ\mathcal{F}\,|\,p\in\operatorname{csn}(\mathcal{D}^{\{N_{p}\}}_{[-1,1]})\right\}$. Notice that $\mathcal{D}^{\{N_{p}\}}_{[-1,1]}\subset\overrightarrow{\mathcal{S}}^{\{N_{p}\}}(\mathbb{R})$ with continuous inclusion mapping and recall that $\mathcal{F}:\overrightarrow{\mathcal{S}}_{\{N_{p}\}}(\mathbb{R})\rightarrow\overrightarrow{\mathcal{S}}^{\{N_{p}\}}(\mathbb{R})$ is a topological isomorphism. Hence, $(4.1)$ is clear, while $(4.2)$ follows from Theorem 2.4$(c)$. Finally,

[TABLE]

for all $\varphi\in E_{0}$, whence $(4.3)$ holds.

$(ii)\Rightarrow(iii)$ This follows from the inclusion $\overrightarrow{\mathcal{S}}_{\{N_{p}\}}(\mathbb{R})\subset\overleftarrow{\mathcal{S}}_{\{N_{p}\}}(\mathbb{R})$.

$(iii)\Rightarrow(i)$ This can be shown in a similar way as $(iii)\Rightarrow(i)$ from part $(a)$. ∎

7. The Stieltjes moment problem in $\mathcal{S}^{\dagger}_{\ast}(0,\infty)$ and the Borel-Ritt problem in spaces of ultraholomorphic functions on the upper half-plane

In this final section, we show an analogue of Theorem 7.2 both for the Stieltjes moment problem in Gelfand-Shilov spaces of type $\mathcal{S}^{\dagger}_{\ast}(0,\infty)$ and the Borel-Ritt problem in spaces of ultraholomorphic functions on the upper half-plane $\mathbb{H}=\{z\in\mathbb{C}\,|\,\Im mz>0\}$.

We start by introducing Gelfand-Shilov spaces of type $\mathcal{S}^{\dagger}_{\ast}$. Let $(M_{p})_{p\in\mathbb{N}}$ and $(A_{p})_{p\in\mathbb{N}}$ be two weight sequences. For $h>0$ we write $\mathcal{S}^{A_{p},h}_{M_{p},h}(\mathbb{R})$ for the Banach space consisting of all $\varphi\in C^{\infty}(\mathbb{R})$ such that

[TABLE]

We set

[TABLE]

$\mathcal{S}^{(A_{p})}_{(M_{p})}(\mathbb{R})$ is a Fréchet space, while $\mathcal{S}^{\{A_{p}\}}_{\{M_{p}\}}(\mathbb{R})$ is an $(LB)$-space. Similarly as before, we will sometimes use $\mathcal{S}^{\dagger}_{\ast}(\mathbb{R})$ as a common notation for $\mathcal{S}^{(A_{p})}_{(M_{p})}(\mathbb{R})$ and $\mathcal{S}^{\{A_{p}\}}_{\{M_{p}\}}(\mathbb{R})$. If both $(M_{p})_{p\in\mathbb{N}}$ and $(A_{p})_{p\in\mathbb{N}}$ satisfy $(\operatorname{lc})$, $(\operatorname{dc})$ and $(\gamma)$, the Fourier transform is a topological isomorphism from $\mathcal{S}^{\dagger}_{\ast}(\mathbb{R})$ onto $\mathcal{S}_{\dagger}^{\ast}(\mathbb{R})$ (cf. [10, Sect. IV.6] and [12, Lemma 4.1]).

Let $h>0$. We define the following closed subspace of $\mathcal{S}^{A_{p},h}_{M_{p},h}(\mathbb{R})$

[TABLE]

and endow it with the norm $\|\,\cdot\,\|_{\mathcal{S}^{A_{p},h}_{M_{p},h}}$. Hence, it becomes a Banach space. We set

[TABLE]

$\mathcal{S}^{(A_{p})}_{(M_{p})}(0,\infty)$ is a Fréchet space, while $\mathcal{S}^{\{A_{p}\}}_{\{M_{p}\}}(0,\infty)$ is an $(LB)$-space. Notice that

[TABLE]

as sets. In the Beurling case, it is clear that (7.1) also holds topologically if we endow the space at the right-hand side with the relative topology induced by $\mathcal{S}^{(A_{p})}_{(M_{p})}(\mathbb{R})$. If $(M_{p})_{p\in\mathbb{N}}$ satisfies $\operatorname{(lc)}$ and $\operatorname{(dc)}$, the corresponding statement also holds in the Roumieu case. By [12, Prop. 3.4], these assumptions imply that $\mathcal{S}^{\{A_{p}\}}_{\{M_{p}\}}(\mathbb{R})$ is a $(DFS)$-space, whence the result follows from the fact that a closed subspace of a $(DFS)$-space is again a $(DFS)$-space and De Wilde’s open mapping theorem.

The image of $\mathcal{S}^{\dagger}_{\ast}(0,\infty)$ under the Fourier transform can be described as follows.

Lemma 7.1**.**

(cf. [3, Prop. 2.1])* Let $(M_{p})_{p\in\mathbb{N}}$ and $(A_{p})_{p\in\mathbb{N}}$ be two weight sequences satisfying $(\operatorname{lc})$, $(\operatorname{dc})$ and $(\gamma)$. Let $\psi\in\mathcal{S}_{\dagger}^{\ast}(\mathbb{R})$. Then, $\psi\in\mathcal{F}(\mathcal{S}^{\dagger}_{\ast}(0,\infty))$ if and only if there is $\Psi:\overline{\mathbb{H}}\rightarrow\mathbb{C}$ satisfying the following conditions:*

• $(i)$

$\Psi_{|\mathbb{R}}=\psi$.

• $(ii)$

$\Psi$* is continuous on $\overline{\mathbb{H}}$ and holomorphic on $\mathbb{H}$.*

• $(iii)$

$\lim_{z\in\overline{\mathbb{H}},z\to\infty}\Psi(z)=0$.

Next, we define spaces of ultraholomorphic functions on $\mathbb{H}$. Given an open subset $\Omega\subseteq\mathbb{C}$, we denote by $\mathcal{O}(\Omega)$ the space of holomorphic functions on $\Omega$. Let $(M_{p})_{p\in\mathbb{N}}$ be a weight sequence. For $h>0$ we write $\mathcal{A}^{M_{p},h}(\mathbb{H})$ for the Banach space consisting of all $f\in\mathcal{O}(\mathbb{H})$ such that

[TABLE]

We set

[TABLE]

$\mathcal{A}^{(M_{p})}(\mathbb{H})$ is a Fréchet space, while $\mathcal{A}^{\{M_{p}\}}(\mathbb{H})$ is an $(LB)$-space. Let $f\in\mathcal{A}^{*}(\mathbb{H})$ be arbitrary. Since $f$ and all its derivatives are Lipschitz on $\mathbb{H}$, it holds that

[TABLE]

exists for all $p\in\mathbb{N}$. Moreover, $f_{0}\in C^{\infty}(\mathbb{R})$ and $f_{0}^{(p)}=f_{p}$ for all $p\in\mathbb{N}$. From now on, we simply write $f_{0}=f$. The asymptotic Borel mapping

[TABLE]

is well-defined and continuous.

We are ready to prove the two main results of this section.

Theorem 7.2**.**

Let $(M_{p})_{p\in\mathbb{N}}$ be a weight sequence satisfying $(\operatorname{lc})$ and $(\operatorname{dc})$, and let $(A_{p})_{p\in\mathbb{N}}$ be a weight sequence satisfying $(\operatorname{lc})$ and $(\gamma)$.

• $(a)$

The following statements are equivalent:

• $(i)$

$(M_{p})_{p\in\mathbb{N}}$* satisfies $(\gamma_{2})$.*

• $(ii)$

$\mathcal{M}:\mathcal{S}^{(A_{p})}_{(M_{p})}(0,\infty)\rightarrow\Lambda_{(M_{p})}$* has a continuous linear right inverse.*

• $(iii)$

$\mathcal{M}:\mathcal{S}^{(A_{p})}_{(M_{p})}(0,\infty)\rightarrow\Lambda_{(M_{p})}$* is surjective.*

• $(b)$

$(M_{p})_{p\in\mathbb{N}}$* satisfies $(\gamma_{2})$ if and only if $\mathcal{M}:\mathcal{S}^{\{A_{p}\}}_{\{M_{p}\}}(0,\infty)\rightarrow\Lambda_{\{M_{p}\}}$ is surjective.*

• $(c)$

$(M_{p})_{p\in\mathbb{N}}$* satisfies $(\gamma_{2})$ and $(\beta_{2})$ if and only if $\mathcal{M}:\mathcal{S}^{\{A_{p}\}}_{\{M_{p}\}}(0,\infty)\rightarrow\Lambda_{\{M_{p}\}}$ has a continuous linear right inverse.*

Remark 7.3*.*

In [5, Thm. 3.5], the direct implication of Theorem 7.2$(b)$ was shown under the assumptions $\operatorname{(slc)}$ (= $(M_{p}/p!)_{p\in\mathbb{N}}$ satisfies $\operatorname{(lc)}$) and $\operatorname{(mg)}$, while the converse implication was shown under the assumptions $\operatorname{(slc)}$ and $\operatorname{(dc)}$.

Theorem 7.4**.**

Let $(M_{p})_{p\in\mathbb{N}}$ be a weight sequence satisfying $(\operatorname{lc})$ and $(\operatorname{dc})$.

• $(a)$

The following statements are equivalent:

• $(i)$

$M_{p}$* satisfies $(\gamma_{2})$.*

• $(ii)$

$\mathcal{B}:\mathcal{A}^{(M_{p})}(\mathbb{H})\rightarrow\Lambda_{(M_{p})}$* has a continuous linear right inverse.*

• $(iii)$

$\mathcal{B}:\mathcal{A}^{(M_{p})}(\mathbb{H})\rightarrow\Lambda_{(M_{p})}$* is surjective.*

• $(b)$

$(M_{p})_{p\in\mathbb{N}}$* satisfies $(\gamma_{2})$ if and only if $\mathcal{B}:\mathcal{A}^{\{M_{p}\}}(\mathbb{H})\rightarrow\Lambda_{\{M_{p}\}}$ is surjective.*

• $(c)$

$(M_{p})_{p\in\mathbb{N}}$* satisfies $(\gamma_{2})$ and $(\beta_{2})$ if and only if $\mathcal{B}:\mathcal{A}^{\{M_{p}\}}(\mathbb{H})\rightarrow\Lambda_{\{M_{p}\}}$ has a continuous linear right inverse.*

Remark 7.5*.*

Theorem 7.4 improves various results from [19, 20, 11] in the special case of the upper half-plane: The implication $(i)\Rightarrow(iii)$ from Theorem 7.2$(a)$ was shown in [20, Cor. 3.4.1] under the assumptions $\operatorname{(slc)}$ and $\operatorname{(mg)}$; the existence of a continuous linear right inverse of $\mathcal{B}:\mathcal{A}^{(M_{p})}(\mathbb{H})\rightarrow\Lambda_{(M_{p})}$ was shown in [19, Thm. 4.4 and Thm. 4.5] under the assumptions $\operatorname{(lc)}$ and $(\gamma_{3})$; the direct implication of Theorem 7.4$(b)$ was shown in [20, Thm. 3.2.1] under the assumptions $\operatorname{(slc)}$ and $\operatorname{(mg)}$, while the converse implication was shown in [11, Thm. 4.14] under the assumptions $\operatorname{(slc)}$ and $\operatorname{(dc)}$; the existence of a continuous linear right inverse of $\mathcal{B}:\mathcal{A}^{\{M_{p}\}}(\mathbb{H})\rightarrow\Lambda_{\{M_{p}\}}$ was shown in [19, Thm. 5.4 and Thm. 5.5] under the assumptions $\operatorname{(lc)}$, $(\gamma_{3})$ and $(\beta_{2})$.

In view of Theorem 6.1, Theorems 7.2 and 7.4 are both consequences of the following result; it is essentially shown in [5, Sect. 3], but we repeat the argument here for the sake of completeness.

Proposition 7.6**.**

Let $(M_{p})_{p\in\mathbb{N}}$ be a weight sequence satisfying $(\operatorname{lc})$ and $(\operatorname{dc})$, and let $(A_{p})_{p\in\mathbb{N}}$ be a weight sequence satisfying $(\operatorname{lc})$ and $(\gamma)$.

• $(a)$

The following statements are equivalent:

• $(i)$

$\mathcal{M}:\mathcal{S}^{(A_{p})}_{(M_{p})}(0,\infty)\rightarrow\Lambda_{(M_{p})}$* is surjective (has a continuous linear right inverse).*

• $(ii)$

$\mathcal{M}:\mathcal{S}_{(M_{p})}(0,\infty)\rightarrow\Lambda_{(M_{p})}$* is surjective (has a continuous linear right inverse).*

• $(iii)$

$\mathcal{B}:\mathcal{A}^{(M_{p})}(\mathbb{H})\rightarrow\Lambda_{(M_{p})}$* is surjective (has a continuous linear right inverse).*

• $(b)$

The following statements are equivalent:

• $(i)$

$\mathcal{M}:\mathcal{S}^{\{A_{p}\}}_{\{M_{p}\}}(0,\infty)\rightarrow\Lambda_{\{M_{p}\}}$* is surjective (has a continuous linear right inverse).*

• $(ii)$

$\mathcal{M}:\overrightarrow{\mathcal{S}}_{\{M_{p}\}}(0,\infty)\rightarrow\Lambda_{\{M_{p}\}}$* is surjective (has a continuous linear right inverse).*

• $(iii)$

$\mathcal{B}:\mathcal{A}^{\{M_{p}\}}(\mathbb{H})\rightarrow\Lambda_{\{M_{p}\}}$* is surjective (has a continuous linear right inverse).*

Proof.

We only show the equivalences about the existence of a continuous linear right inverse stated in $(a)$; all other cases follow from a similar argument.

$(i)\Rightarrow(ii)$ This follows from the continuous inclusion $\mathcal{S}^{(A_{p})}_{(M_{p})}(0,\infty)\subset\mathcal{S}_{(M_{p})}(0,\infty)$.

$(ii)\Rightarrow(iii)$ We define the Laplace transform of an element $\varphi\in\mathcal{S}_{(M_{p})}(0,\infty)$ as

[TABLE]

$\mathcal{L}:\mathcal{S}_{(M_{p})}(0,\infty)\rightarrow\mathcal{A}^{(M_{p})}(\mathbb{H})$ is well-defined and continuous, and $\mathcal{L}(\varphi)^{(p)}(0)=i^{p}\mu_{p}(\varphi)$ for all $p\in\mathbb{N}$. Let $R:\Lambda_{(M_{p})}\rightarrow\mathcal{S}_{(M_{p})}(0,\infty)$ be a continuous linear right inverse of $\mathcal{M}:\mathcal{S}_{(M_{p})}(0,\infty)\rightarrow\Lambda_{(M_{p})}$. Consider the continuous mapping

[TABLE]

Then, $\mathcal{L}\circ R\circ T:\Lambda_{(M_{p})}\rightarrow\mathcal{A}^{(M_{p})}(\mathbb{H})$ is a continuous linear right inverse of $\mathcal{B}:\mathcal{A}^{(M_{p})}(\mathbb{H})\rightarrow\Lambda_{(M_{p})}$.

$(iii)\Rightarrow(i)$ We start by making some preliminary observations about the weight sequences $(M_{p})_{p\in\mathbb{N}}$ and $(A_{p})_{p\in\mathbb{N}}$. By [5, Lemma 2.2], we may assume without loss of generality that $(A_{p})_{p\in\mathbb{N}}$ satisfies $(\operatorname{dc})$. Next, choose $f\in\mathcal{A}^{(M_{p})}(\mathbb{H})$ such that $f^{(p)}(0)=\delta_{1,p}$ for all $p\in\mathbb{N}$. Set $\varphi(x)=f(x)-x$ for $x\in\mathbb{R}$. Then,

[TABLE]

for all $h>0$ and $R>0$, and $\varphi^{(p)}(0)=0$ for all $p\in\mathbb{N}$. Since $f$ is bounded, $\varphi$ is not identically zero. Hence, the Denjoy-Carleman theorem implies that $(M_{p})_{p\in\mathbb{N}}$ satisfies $(\gamma)$. Consequently, we have that $\lim_{p\to\infty}(p!/M_{p})^{1/p}=0$ [12, Lemma 4.1]. We now turn to the actual proof. It is based on the following observation [5, Lemma 3.6]: Let $(a_{p})_{p\in\mathbb{N}}\in\mathbb{C}^{\mathbb{N}}$ and let $G\in C^{\infty}((-\delta,\delta))$, $\delta>0$, with $G(0)\neq 0$. Set

[TABLE]

Then,

[TABLE]

Set $V=\{z\in\mathbb{C}\,|\,\Im mz>-1\}$. By [5, Lemma 3.1] and [12, Lemma 4.3], there is $G\in\mathcal{O}(V)$ satisfying the following properties:

• $(i)$

$G$ does not vanish on $V$.

• $(ii)$

$\displaystyle\sup_{z\in V}|G(z)|e^{{A}(h|z|)}<\infty$ for all $h>0$.

• $(iii)$

$\displaystyle\sup_{p\in\mathbb{N}}\sup_{x\in\mathbb{R}}\frac{|G^{(p)}(x)|e^{A(h|x|)}}{2^{p}p!}<\infty$ for all $h>0$.

Lemma 7.1 implies that the mapping $T:\mathcal{A}^{(M_{p})}(\mathbb{H})\rightarrow\mathcal{S}^{(A_{p})}_{(M_{p})}(0,\infty):f\rightarrow\mathcal{F}^{-1}((fG)_{|\mathbb{R}})$ is well-defined. Since $\mathcal{F}:\mathcal{S}^{(A_{p})}_{(M_{p})}(\mathbb{R})\rightarrow\mathcal{S}_{(A_{p})}^{(M_{p})}(\mathbb{R})$ is a topological isomorphism and (7.1) holds topologically, $T$ is also continuous. Next, the Cauchy estimates yield that

[TABLE]

Hence, the mapping

[TABLE]

is well-defined and continuous. Let $R:\Lambda_{(M_{p})}\rightarrow\mathcal{A}^{(M_{p})}(\mathbb{H})$ be a continuous linear right inverse of $\mathcal{B}:\mathcal{A}^{(M_{p})}(\mathbb{H})\rightarrow\Lambda_{(M_{p})}$. Then, $T\circ R\circ S:\Lambda_{(M_{p})}\rightarrow\mathcal{S}^{(A_{p})}_{(M_{p})}(0,\infty)$ is a continuous linear right inverse of $\mathcal{M}:\mathcal{S}^{(A_{p})}_{(M_{p})}(0,\infty)\rightarrow\Lambda_{(M_{p})}$. ∎