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

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 ${\mathcal{O}}_{C,\omega }^{\prime }({\mathbb{R}}^N)$ 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 ${\mathcal{S}}_{\omega }^{\prime }({\mathbb{R}}^N)$. We also establish that the Fourier transform is an isomorphism from ${\mathcal{O}}_{C,\omega }^{\prime }({\mathbb{R}}^N)$ 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.

arXiv admin note: text overlap with arXiv:2011.03961