Expansion of traces and Dixmier traceability for global pseudo-differential operators on manifolds with boundary

TL;DR

This paper extends the understanding of trace regularisation and Dixmier traceability for global pseudo-differential operators on manifolds with boundary, using non-harmonic analysis techniques to generalize classical microlocal results.

Contribution

It introduces non-harmonic analysis methods to study trace regularisation and Dixmier traceability for global pseudo-differential operators on manifolds with boundary, expanding classical microlocal analysis results.

Findings

Determined singularity orders in trace regularisation.

Identified sharp regularity orders for Dixmier traceability.

Developed coordinate-free analysis methods.

Abstract

Given a smooth manifold $M$ (with or without boundary), in this paper we study the regularisation of traces for the global pseudo-differential calculus in the context of non-harmonic analysis. Indeed, using the global pseudo-differential calculus on manifolds (with or without boundary) developed in [30], the Calder\'on-Vaillancourt Theorem and the global functional calculus in [6], we determine the singularity orders in the regularisation of traces and the sharp regularity orders for the Dixmier traceability of the global H\"ormander classes. Our analysis (free of coordinate systems) allows us to obtain non-harmonic analogues of several classical results arising from the microlocal analysis of regularised traces for pseudo-differential operators with symbols defined by localisations.