On a class of translation-invariant spaces of quasianalytic ultradistributions

TL;DR

This paper introduces and studies a new class of translation-invariant Banach spaces of quasianalytic ultradistributions, extending previous frameworks to include Fourier hyperfunctions and ultrahyperfunctions, with applications to convolution and multiplication operations.

Contribution

It defines novel translation-invariant Banach spaces of quasianalytic ultradistributions and their duals, generalizing prior work and enabling analysis of new weighted ultradistribution spaces.

Findings

Established properties of the new Banach spaces of quasianalytic ultradistributions.

Extended the framework to include Fourier hyperfunctions and ultrahyperfunctions.

Analyzed convolution and multiplication in the new spaces.

Abstract

A class of translation-invariant Banach spaces of quasianalytic ultradistributions is introduced and studied. They are Banach modules over a Beurling algebra. Based on this class of Banach spaces, we define corresponding test function spaces $\mathcal{D}^*_E$ and their strong duals $\mathcal{D}'^*_{E'_{\ast}}$ of quasianalytic type, and study convolution and multiplicative products on $\mathcal{D}'^*_{E'_{\ast}}$. These new spaces generalize previous works about translation-invariant spaces of tempered (non-quasianalytic ultra-) distributions; in particular, our new considerations apply to the settings of Fourier hyperfunctions and ultrahyperfunctions. New weighted $\mathcal{D}'^{\ast}_{L^{p}_{\eta}}$ spaces of quasianalytic ultradistributions are analyzed.