Convolution of ultradistributions and ultradistribution spaces associated to translation-invariant Banach spaces

Abstract

We introduce and study a number of new spaces of ultradifferentiable functions and ultradistributions and we apply our results to the study of the convolution of ultradistributions. The spaces of convolutors $\mathcal{O}{\text{'}}_C^{*}\left({\mathbb{R}}^d\right)$ for tempered ultradistributions are analyzed via the duality with respect to the test function spaces ${\mathcal{O}}_C^{*}\left({\mathbb{R}}^d\right)$ introduced in this article. We also study ultradistribution spaces associated to translation-invariant Banach spaces of tempered ultradistributions and use their properties to provide a full characterization of the general convolution of Roumieu ultradistributions via the space of integrable ultradistributions. We show that the convolution of two Roumieu ultradistributions $T,S\in \mathcal{D}{\text{'}}^{\left\{M_p\right\}}\left({\mathbb{R}}^d\right)$ exists if and only if $\left(\phi \check{S}\right)T\in \mathcal{D}{\text{'}}_{L^1}^{\left\{M_p\right\}}\left({\mathbb{R}}^d\right)$ for every $\phi \in {\mathcal{D}}^{\left\{M_p\right\}}\left({\mathbb{R}}^d\right)$.