Laplace hyperfunctions via Čech-Dolbeault cohomology
Naofumi Honda, Kohei Umeta
TL;DR
This work develops a comprehensive Čech-Dolbeault framework for Laplace hyperfunctions, extending Komatsu's one-dimensional theory to several variables through the radial compactification and exponential-type holomorphic data. It provides multiple, equivalent representations of Laplace hyperfunctions via relative Čech Dolbeault and relative Čech cohomologies, and introduces an intuitive wedge-based viewpoint that unifies these descriptions. The paper then defines and analyzes the Laplace transform and its inverse within this cohomological setting, establishing their independence from choices and showing they are mutual inverses under natural regularity conditions. Boundary value maps are constructed in both functorial and Čech-Dolbeault fashions and shown to be consistent, enabling explicit formulas for transforming between holomorphic data on wedges and hyperfunctions. Applications to constant-coefficient PDEs demonstrate the utility of the machinery for operational calculus in several variables.
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 Čech-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.