  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.
