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

Abstract: In this paper we continue the study of the spaces $\mathcal{O}<em>{M,\omega}(\mathbb{R}<sup>N)$ and $\mathcal{O}</em>{C,\omega}(\mathbb{R}<sup>N)$ undertaken in [1]. We determine new representations of such spaces and we give some structure theorems for their dual spaces. Furthermore, we show that $\mathcal{O}&#39;<em>{C,\omega}(\mathbb{R}<sup>N)$ is the space of convolutors of the space $\mathcal{S}</em>\omega(\mathbb{R}<sup>N)$ of the $\omega$-ultradifferentiable rapidly decreasing functions of Beurling type (in the sense of Braun, Meise and Taylor) and of its dual space $\mathcal{S}&#39;<em>\omega(\mathbb{R}<sup>N)$. We also establish that the Fourier transform is an isomorphism from $\mathcal{O}&#39;</em>{C,\omega}(\mathbb{R}<sup>N)$ onto $\mathcal{O}<em>{M,\omega}(\mathbb{R}<sup>N)$. In particular, we prove that this isomorphism is topological when the former space is endowed with the strong operator lc-topology induced by $$\mathcal{L}_b(\mathcal{S}</em>\omega(\mathbb{R}<sup>N))$$ and the last space is endowed with its natural lc-topology.