Laplace hyperfunctions via \v{C}ech-Dolbeault cohomology

TL;DR

This paper explores Laplace hyperfunctions using cech-Dolbeault cohomology, providing new methods to construct Laplace transforms and their inverses, with applications to PDE systems with constant coefficients.

Contribution

It introduces a cohomological framework for Laplace hyperfunctions in higher dimensions, simplifying the construction of Laplace transforms and their inverses.

Findings

Constructed Laplace transformation and inverse via cohomology

Extended hyperfunction theory to higher dimensions

Applied results to PDE systems with constant coefficients

Abstract

The paper studies several properties of Laplace hyperfunctions introduced by H.~Komatsu in the one dimensional case and by the authors in the higher dimensional cases from the viewpoint of \v{C}ech-Dolbeault cohomology theory, which enables us, for example, to construct the Laplace transformation and its inverse in a simple way. We also give some applications to a system of PDEs with constant coefficients.