A note on the barrelledness of weighted $(PLB)$-spaces of ultradifferentiable functions

TL;DR

This paper characterizes when weighted $(PLB)$-spaces of ultradifferentiable functions are ultrabornological or barrelled, improving previous results and applying to Gelfand-Shilov spaces' multiplier spaces.

Contribution

It provides a complete characterization of barrelledness and ultrabornologicity for these spaces based on their weight systems, refining earlier theorems.

Findings

Multiplier space of $oldsymbol{ ext{Gelfand-Shilov}}$ $oldsymbol{ ext{space }oldsymbol{ extstyleoldsymbol{ ext{Sigma}}^{r}_{s}}(oldsymbol{ extbf{R}^{d}})}$ is ultrabornological.

Multiplier space of $oldsymbol{ ext{Gelfand-Shilov}}$ $oldsymbol{ ext{space }oldsymbol{ extstyleoldsymbol{ ext{S}}^{r}_{s}}(oldsymbol{ extbf{R}^{d}})}$ is not barrelled.

Characterization depends on the properties of the defining weight system.

Abstract

In this note we consider weighted $(PLB)$-spaces of ultradifferentiable functions defined via a weight function and a weight system, as introduced in our previous work [4]. We provide a complete characterization of when these spaces are ultrabornological and barrelled in terms of the defining weight system, thereby improving the main Theorem 5.1 of [4]. In particular, we obtain that the multiplier space of the Gelfand-Shilov space $\Sigma^{r}_{s}(\mathbb{R}^{d})$ of Beurling type is ultrabornological, whereas the one of the Gelfand-Shilov space $\mathcal{S}^{r}_{s}(\mathbb{R}^{d})$ of Roumieu type is not barrelled.