A holomorphic mapping property of analytic pseudo-differential operators

TL;DR

This paper investigates the conditions under which analytic pseudo-differential operators preserve holomorphic extendibility, providing explicit domain estimates based on the properties of the symbol and the function.

Contribution

It introduces a contour deformation method to precisely estimate the domain of holomorphy for pseudo-differential operators with analytic symbols.

Findings

Explicit domain estimates for holomorphic extendibility of $ ext{Op}(p)u$

A contour deformation technique for analyzing holomorphy

Conditions linking symbol and function holomorphy domains

Abstract

We study the holomorphic extendibility of $\text{Op}(p)u$, when $p$ is an analytic symbol, and explicit information is available on the domains of holomorphic extendibility of both $p$ and $u$. By a contour deformation argument, we obtain a precise local estimate of the domain of holomorphy of $\text{Op}(p)u$ in terms of the information on $p$ and $u$.