Hypoellipticity of analytic differential operators in general ultradifferentiable classes

TL;DR

This paper extends the understanding of hypoellipticity for analytic differential operators to ultradifferentiable classes, showing that pseudodifferential and Fourier integral operators maintain good behavior in these settings.

Contribution

It generalizes Treves' hypoellipticity characterization from analytic to ultradifferentiable operators, broadening the scope of regularity analysis.

Findings

Ultradifferentiable classes satisfy minimal regularity properties.

Analytic pseudodifferential and Fourier integral operators behave well in these classes.

Extension of Treves' hypoellipticity characterization to ultradifferentiable operators.

Abstract

We show that analytic pseudodifferential and Fourier integral operators behave well for ultradifferentiable classes satisfying minimal regularity properties. As an application we investigate the ultradifferentiable regularity properties of several examples of analytic differential operators. In particular we extend Treves' characterization of the hypoellipticity of analytic operators of principal type to the ultradifferentiable category.