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

38 pages

References in corpus (2)

• New distribution spaces associated to translation-invariant Banach spaces
• Boundary values of holomorphic functions and heat kernel method in translation-invariant distribution spaces

Cited by in corpus (10)

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• On weighted inductive limits of spaces of ultradifferentiable functions and their duals
• Translation-modulation invariant Banach spaces of ultradistributions
• Factorization in Denjoy-Carleman classes associated to representations of $({\mathbb{R}}^d,+)$
• Modulation spaces associated to tensor products of amalgam spaces
• On the space of ultradistributions vanishing at infinity
• Convolutors of translation-modulation invariant Banach spaces of ultradistributions
• On the projective description of spaces of ultradifferentiable functions of Roumieu type
• A multidimensional Tauberian theorem for Laplace transforms of ultradistributions
• Extension of Localisation Operators to Ultradistributional Symbols With Super-Exponential Growth