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

TL;DR

This paper develops new ultradifferentiable function and ultradistribution spaces, analyzes their convolution properties, and characterizes the convolution of Roumieu ultradistributions using translation-invariant Banach spaces.

Contribution

It introduces new ultradifferentiable and ultradistribution spaces, studies their convolution, and characterizes Roumieu ultradistribution convolution through integrable ultradistributions.

Findings

Analysis of convolutors via duality with test function spaces

Full characterization of convolution of Roumieu ultradistributions

Existence criterion for convolution based on integrable ultradistributions

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 $\mathcal{O}'^{\ast}_{C}(\mathbb{R}^{d})$ for tempered ultradistributions are analyzed via the duality with respect to the test function spaces $\mathcal{O}^{\ast}_{C}(\mathbb{R}^{d})$, 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 $T,S\in \DD'^{\{M_p\}}\left(\RR^d\right)$ exists if and only if $\left(\varphi*\check{S}\right)T\in\DD'^{\{M_p\}}_{L^1}\left(\RR^d\right)$ for every $\varphi\in\DD^{\{M_p\}}\left(\RR^d\right)$.