Paley-Wiener-Schwartz Theorem and Microlocal Analysis in Theory of Tempered Ultrahyperfunctions

TL;DR

This paper extends the Paley-Wiener-Schwartz theorem to tempered ultrahyperfunctions, linking support properties to holomorphicity of their Fourier-Laplace transforms, with applications in microlocal analysis and quantum field theories.

Contribution

It generalizes the Paley-Wiener-Schwartz theorem for tempered ultrahyperfunctions and explores their microlocal singularity structure, with implications for physical theories with fundamental length.

Findings

Generalized Paley-Wiener-Schwartz theorem for ultrahyperfunctions

Characterized singularities via analytic wave front set

Potential applications in quantum field theory with minimal length

Abstract

We give some precisions on the Fourier-Laplace transform theorem for tempered ultrahyperfunctions introduced by Sebasti\~ao e Silva and Hasumi, by considering the theorem in its simplest form: the equivalence between support properties of a distribution in a closed convex cone and the holomorphy of its Fourier-Laplace transform in a suitable tube with conical basis. We establish a generalization of Paley-Wiener-Schwartz theorem for this setting. This theorem is interesting in connection with the microlocal analysis, where a description of the singularity structure of tempered ultrahyperfunctions in terms of the concept of analytic wave front set is given. We also suggest a physical application of the results obtained in the construction and study of field theories with fundamental length.