New distribution spaces associated to translation-invariant Banach spaces

Abstract

We introduce and study new distribution spaces, the test function space ${\mathcal{D}}_E$ and its strong dual ${\mathcal{D}}_{E_{\ast }^{\prime }}^{\prime }$. These spaces generalize the Schwartz spaces ${\mathcal{D}}_{L^q}$, ${\mathcal{D}}_{L^p}^{\prime }$, ${\mathcal{B}}^{\prime }$ and their weighted versions. The construction of our new distribution spaces is based on the analysis of a suitable translation-invariant Banach space of distributions $E$ with continuous translation group, which turns out to be a convolution module over a Beurling algebra $L_{\omega }^1$. The Banach space $E_{\ast }^{\prime }$ stands for $L_{\check{\omega }}^1\ast E^{\prime }$. We also study convolution and multiplicative products on ${\mathcal{D}}_{E_{\ast }^{\prime }}^{\prime }$.

19 pages

References in corpus (2)

• Boundary values of holomorphic functions and heat kernel method in translation-invariant distribution spaces
• Multidimensional Tauberian theorems for wavelet and non-wavelet transforms

Cited by in corpus (16)

• On quasianalytic classes of Gelfand-Shilov type. Parametrix and convolution
• Convolution of ultradistributions and ultradistribution spaces associated to translation-invariant Banach spaces
• On PNT equivalences for Beurling numbers
• Translation-modulation invariant Banach spaces of ultradistributions
• Homogeneous Banach spaces as Banach convolution modules over $M(G)$
• Complex Tauberian theorems for Laplace transforms with local pseudofunction boundary behavior
• Tauberian class estimates for vector-valued distributions
• Modulation spaces associated to tensor products of amalgam spaces
• Vector valued Hardy spaces related to analytic functions having distributional boundary values
• Sequential conditions of integrability of Roumieu ultradistributions
• Convolutors of translation-modulation invariant Banach spaces of ultradistributions
• Convolution of Roumieu ultradistributions in sequential approach
• Besov regularity in non-linear generalized functions
• Sequence space representations for translation-modulation invariant function and distribution spaces
• Quasinormable $C_0$-groups and translation-invariant Fréchet spaces of type ${\mathcal{D}}_E$
• Generalized exponentially bounded integrated semigroups