Strong Regularity and Microsupport Estimates for Multi-Microlocalizations of Subanalytic Sheaves

TL;DR

This paper develops strong regularity concepts for subanalytic sheaves and provides estimates for their multi-microlocalizations, with applications to solution sheaves of D-modules and a multi-microlocal Bochner's tube theorem.

Contribution

It introduces strong regularity for subanalytic sheaves and establishes support and microsupport estimates for their multi-microlocalizations, advancing microlocal analysis techniques.

Findings

Established support and microsupport estimates for multi-microlocalizations.

Proved initial value theorems for multi-microlocal objects with growth conditions.

Derived a multi-microlocal version of Bochner's tube theorem.

Abstract

We introduce the notion of strong regularity for subanalytic sheaves and establish estimates for the supports and microsupports of their multi-microlocalizations. As applications, we study subanalytic sheaves of Whit- ney and temperate holomorphic solutions of regular D-modules along an involutive subbundle. In this setting we prove initial value theorems for multi-microlocal objects with growth conditions and division theorems for temperate and Whitney multi-microfunctions. As a consequence, we obtain a multi-microlocal version of Bochner's tube theorem for solution sheaves of strongly asymptotically developable functions.