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

In this paper we continue the study of the spaces $${\mathcal O}_{M,\omega }({\mathbb R}^N)$$ O M , ω ( R N ) and $${\mathcal O}_{C,\omega }({\mathbb R}^N)$$ O C , ω ( R N ) undertaken in Albanese and Mele (J Pseudo-Differ Oper Appl, 2021). We determine new representations of such spaces and we give some structure theorems for their dual spaces. Furthermore, we show that $${\mathcal O}'_{C,\omega }({\mathbb R}^N)$$ O C , ω ′ ( R N ) is the space of convolutors of the space $${\mathcal S}_\omega ({\mathbb R}^N)$$ S ω ( R N ) of the $$\omega $$ ω -ultradifferentiable rapidly decreasing functions of Beurling type (in the sense of Braun, Meise and Taylor) and of its dual space $${\mathcal S}'_\omega ({\mathbb R}^N)$$ S ω ′ ( R N ) . We also establish that the Fourier transform is an isomorphism from $${\mathcal O}'_{C,\omega }({\mathbb R}^N)$$ O C , ω ′ ( R N ) onto $${\mathcal O}_{M,\omega }({\mathbb R}^N)$$ O M , ω ( 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 $${\mathcal L}_b({\mathcal S}_\omega ({\mathbb R}^N))$$ L b ( S ω ( R N ) ) and the last space is endowed with its natural lc-topology.