A new microlocal analysis of hyperfunctions

TL;DR

This paper develops a microlocal analysis framework for hyperfunctions using generalized FBI transforms, enabling a detailed characterization of their wave-front sets and extending classical regularity results to hyperfunctions.

Contribution

It introduces a microlocal decomposition and generalized FBI transforms for hyperfunctions, generalizing classical results and linking elliptic regularity from distributions to hyperfunctions.

Findings

Characterization of hyperfunction wave-front sets using microlocal decomposition

Extension of elliptic regularity theorems from distributions to hyperfunctions

Recovery and generalization of classical microlocal regularity results

Abstract

In this work we study microlocal regularity of hyperfunctions defining in this context a class of generalized FBI transforms first introduced for distributions by Berhanu and Hounie. Using a microlocal decomposition of a hyperfunction and the generalized FBI transforms we were able to characterize the wave-front set of hyperfunctions according several types of regularity. The microlocal decomposition allowed us to recover and generalize both classical and recent results and, in particular, we proved for differential operators with real-analytic coefficients that if the elliptic regularity theorem regarding any reasonable regularity holds for distributions, then it is automatically true for hyperfunctions.