Edge of the Wedge Theorem for Tempered Ultrahyperfunctions

TL;DR

This paper develops a local version of the edge of the wedge theorem for tempered ultrahyperfunctions, extending existing global results and addressing localization challenges in this function class.

Contribution

It introduces and proves a local formulation of the edge of the wedge theorem for tempered ultrahyperfunctions, which lacked such a version before.

Findings

Established a local edge of the wedge theorem for tempered ultrahyperfunctions

Extended the global theorem to a local context

Provided a cohomological approach for the proof

Abstract

Tempered ultra-hyperfunctions do not have the same type of localization properties as Schwartz distributions or Sato hyperfunctions; but the localization properties seem to play an important role in the proofs of the various versions of the edge of the wedge theorem. Thus, for tempered ultra hyper-functions, one finds a global form of this result in the literature, but no local version. In this paper we propose and prove a formulation of the edge of the wedge theorem for tempered ultra-hyperfunctions, both in global and local form. We explain our strategy first for the one variable case. We argue that in view of the cohomological definition of hyperfunctions and ultra-hyperfunctions, the global form of the edge of the wedge theorem is not surprising at all.