Micro-support and Cauchy problem for temperate solutions of regular $\cal D$-Modules

TL;DR

This paper investigates the properties of temperate microfunction solutions for regular microdifferential systems on complex manifolds, providing bounds on their micro-support and solving the Cauchy problem under hyperbolicity conditions.

Contribution

It establishes bounds on the micro-support of temperate microfunction solutions and addresses the Cauchy problem for regular $ ext{D}$-modules in the temperate setting.

Findings

Bound on the micro-support of temperate microfunction solutions

Solution to the Cauchy problem under hyperbolicity conditions

Enhanced understanding of temperate solutions in microdifferential systems

Abstract

Let $X$ be a complex manifold, $V$ a smooth involutive submanifold of $T^*X$, $\cal M$ a microdifferential system regular along $V$, and $F$ an $\mathbb{R}$-constructible sheaf on $X$. The complex of temperate microfunction solutions of $\cal M$ associated with $F$ is now well understood. In this paper we study the temperate case and give a bound to the micro-support of the temperate microfunction solution sheaves and solve the Cauchy problem under suitable hyperbolicity conditions.