Boundary values in ultradistribution spaces related to extended Gevrey regularity

TL;DR

This paper introduces new ultradistribution spaces linked to extended Gevrey regularity, demonstrating boundary value properties of analytic functions with logarithmic growth and analyzing their wave front sets.

Contribution

It develops novel ultradistribution spaces using a new condition, $ ilde{(M.2)}$, expanding the classical theory and providing new techniques for boundary value analysis.

Findings

Boundary values of certain analytic functions are ultradistributions.

New ultradistribution spaces are constructed with a modified growth condition.

Wave front sets are characterized within the new framework.

Abstract

Following the well-known theory of Beurling and Roumieu ultradistributions, we investigate new spaces of ultradistributions as dual spaces of test functions which correspond to associated functions of logarithmic-type growth at infinity. In the given framework we prove that boundary values of analytic functions with the corresponding logarithmic growth rate towards the real domain are ultradistributions. The essential condition for that purpose, condition $(M.2)$ in the classical ultradistribution theory, is replaced by the new one, $\widetilde{(M.2)}$. For that reason, new techniques were performed in the proofs. As an application, we discuss the corresponding wave front sets.