Geometric Microlocal Analysis in Denjoy-Carleman classes

TL;DR

This paper develops a geometric microlocal analysis framework for ultradifferentiable functions, extending classical wavefront set theory to quasianalytic and non-quasianalytic classes, with applications to elliptic regularity and Holmgren's theorem.

Contribution

It introduces a systematic geometric theory for ultradifferentiable wavefront sets, including invariance under diffeomorphisms and elliptic regularity results, extending classical microlocal analysis.

Findings

Established an analogue of Bony's theorem for ultradifferentiable wavefront sets

Proved invariance of the wavefront set under ultradifferentiable diffeomorphisms

Demonstrated microlocal elliptic regularity for operators on ultradifferentiable bundles

Abstract

A systematic geometric theory for the ultradifferentiable (non-quasianalytic and quasianalytic) wavefront set similar to the well-known theory in the classic smooth and analytic setting is developed. In particular an analogue of Bony's Theorem and the invariance of the ultradifferentiable wavefront set under diffeomorphisms of the same regularity is proven using a Theorem of Dyn'kin about the almost-analytic extension of ultradifferentiable functions. Furthermore we prove a microlocal elliptic regularity theorem for operators defined on ultradifferentiable vector bundles. As an application we show that Holmgren's theorem and several generalizations hold for operators with quasianalytic coefficients.