Microlocal hypoellipticity of linear partial differential operators with generalized functions as coefficients

TL;DR

This paper extends microlocal hypoellipticity results to linear PDEs with generalized function coefficients, introducing new methods to handle non-smooth lower order terms and providing simplified applicable conditions.

Contribution

It offers a novel extension of microlocal hypoellipticity theory to operators with generalized function coefficients, including methodological innovations and technical refinements.

Findings

Extended microlocal hypoellipticity results to generalized coefficients

Developed new techniques to manage non-smooth lower order terms

Provided simplified conditions for specific cases

Abstract

We investigate microlocal properties of partial differential operators with generalized functions as coefficients. The main result is an extension of a corresponding (microlocalized) distribution theoretic result on operators with smooth hypoelliptic symbols. Methodological novelties and technical refinements appear embedded into classical strategies of proof in order to cope with most delicate interferences by non-smooth lower order terms. We include simplified conditions which are applicable in special cases of interest.