A Note on Fourier-Laplace Transform and Analytic Wave front Set in Theory of Tempered Ultrahyperfunctions

TL;DR

This paper explores the Fourier-Laplace transform of tempered ultrahyperfunctions, generalizes a classical theorem, and analyzes their singularity structure via the analytic wave front set, advancing microlocal analysis.

Contribution

It extends the Paley-Wiener-Schwartz theorem to tempered ultrahyperfunctions and describes their singularities using the analytic wave front set.

Findings

Generalized Paley-Wiener-Schwartz theorem for ultrahyperfunctions

Characterized singularity structures via analytic wave front set

Connected microlocal analysis with ultrahyperfunction theory

Abstract

In this paper we study the Fourier-Laplace transform of tempered ultrahyperfunctions introduced by Sebasti\~ao e Silva and Hasumi. We establish a generalization of Paley-Wiener-Schwartz theorem for this setting. This theorem is interesting in connection with the microlocal analysis. For this reason, the paper also contains a description of the singularity structure of tempered ultrahyperfunctions in terms of the concept of analytic wave front set.