On the Laplace transform for tempered holomorphic functions

TL;DR

This paper extends the Fourier-Sato transform to non-conic sheaves by introducing an extra variable and a subanalytic sheaf of holomorphic functions with exponential growth, enabling Paley-Wiener type results.

Contribution

It introduces a novel approach to handle non-conic sheaves via an added variable and a new subanalytic sheaf, expanding the applicability of the Laplace transform in this context.

Findings

Established Paley-Wiener type theorems for tempered holomorphic functions.

Developed a new subanalytic sheaf of holomorphic functions with exponential growth.

Extended the Fourier-Sato transform to non-conic sheaves using the added variable.

Abstract

In order to discuss the Fourier-Sato transform of not necessarily conic sheaves, we compensate the lack of homogeneity by adding an extra variable. We can then obtain Paley-Wiener type results, using a theorem by Kashiwara and Schapira on the Laplace transform for tempered holomorphic functions. As a key tool in our approach, we introduce the subanalytic sheaf of holomorphic functions with exponential growth, which should be of independent interest.