AbstractIn this paper we continue the study of the spaces $${\mathcal O}_{M,\omega }({\mathbb R}^N)$$
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ω
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and $${\mathcal O}_{C,\omega }({\mathbb R}^N)$$
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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 $${\mathcal O}'_{C,\omega }({\mathbb R}^N)$$
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′
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is the space of convolutors of the space $${\mathcal S}_\omega ({\mathbb R}^N)$$
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of the $$\omega $$
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-ultradifferentiable rapidly decreasing functions of Beurling type (in the sense of Braun, Meise and Taylor) and of its dual space $${\mathcal S}'_\omega ({\mathbb R}^N)$$
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. We also establish that the Fourier transform is an isomorphism from $${\mathcal O}'_{C,\omega }({\mathbb R}^N)$$
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onto $${\mathcal O}_{M,\omega }({\mathbb R}^N)$$
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. In particular, we prove that this isomorphism is topological when the former space is endowed with the strong operator lc-topology induced by $${\mathcal L}_b({\mathcal S}_\omega ({\mathbb R}^N))$$
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b
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and the last space is endowed with its natural lc-topology.
Examples of measures with slow decay of the spherical means of the Fourier transform