Convolutors on $\mathcal{S}_\omega(\mathbb{R}^N)$

TL;DR

This paper investigates the structure and representations of certain ultradifferentiable function spaces and their duals, establishing their convolution properties and Fourier transform isomorphisms within the framework of Beurling type ultradifferentiability.

Contribution

It provides new representations and structure theorems for ultradifferentiable function spaces and characterizes their duals as convolutors, including the Fourier transform isomorphism results.

Findings

$ ext{O}'_{C, ext{} ext{omega}}( ext{R}^N)$ is the space of convolutors of $ ext{S}_ ext{omega}( ext{R}^N)$

Fourier transform is an isomorphism between $ ext{O}'_{C, ext{} ext{omega}}( ext{R}^N)$ and $ ext{O}_{M, ext{} ext{omega}}( ext{R}^N)$

The Fourier transform is a topological isomorphism under specified lc-topologies

Abstract

In this paper we continue the study of the spaces $\mathcal{O}_{M,\omega}(\mathbb{R}^N)$ and $\mathcal{O}_{C,\omega}(\mathbb{R}^N)$ undertaken in [1]. We determine new representations of such spaces and we give some structure theorems for their dual spaces. Furthermore, we show that $\mathcal{O}'_{C,\omega}(\mathbb{R}^N)$ is the space of convolutors of the space $\mathcal{S}_\omega(\mathbb{R}^N)$ of the $\omega$-ultradifferentiable rapidly decreasing functions of Beurling type (in the sense of Braun, Meise and Taylor) and of its dual space $\mathcal{S}'_\omega(\mathbb{R}^N)$. We also establish that the Fourier transform is an isomorphism from $\mathcal{O}'_{C,\omega}(\mathbb{R}^N)$ onto $\mathcal{O}_{M,\omega}(\mathbb{R}^N)$. In particular, we prove that this isomorphism is topological when the former space is endowed with the strong operator lc-topology induced by $\mathcal{L}_b(\mathcal{S}_\omega(\mathbb{R}^N))$ and the last space is endowed with its natural lc-topology.