Spectral Invariance of Non-Smooth Pseudodifferential Operators

TL;DR

This paper investigates spectral invariance of non-smooth pseudodifferential operators with H"older continuous coefficients, extending classical results to less regular settings using advanced operator characterization techniques.

Contribution

It extends spectral invariance results to non-smooth pseudodifferential operators by improving their characterization and addressing composition challenges.

Findings

Spectral invariance holds for certain non-smooth pseudodifferential operators.

Improved characterization enables better understanding of operator composition.

Results extend classical smooth case to non-smooth settings.

Abstract

In this paper we discuss some spectral invariance results for non-smooth pseudodifferential operators with coefficients in H\"older spaces. In analogy to the proof in the smooth case of Beals and Ueberberg, we use the characterization of non-smooth pseudodifferential operators to get such a result. The main new difficulties are the limited mapping properties of pseudodifferential operators with non-smooth symbols and the fact, that in general the composition of two non-smooth pseudodifferential operators is not a pseudodifferential operator. In order to improve these spectral invariance results for certain subsets of non-smooth pseudodifferential operators with coefficients in H\"older spaces, we improve the characterization of non-smooth pseudodifferential operators in a previous work by the authors.