Connes trace theorem for Carnot manifolds

TL;DR

This paper extends Connes' trace theorem to Carnot manifolds, showing that the Wodzicki residue coincides with the Dixmier trace in this more general geometric setting.

Contribution

It proves that Connes' trace theorem holds for pseudodifferential operators on Carnot manifolds, generalizing previous results to a broader class of geometric structures.

Findings

Connes' trace theorem is valid for Carnot manifolds.

The Wodzicki residue coincides with the Dixmier trace on Carnot manifolds.

The residue functional adapts to the filtration structure of Carnot manifolds.

Abstract

The Wodzicki residue is the unique trace on the algebra of classical pseudodifferential operators on a closed manifold, and Connes in 1988 proved that it coincides with the Dixmier trace. A Carnot manifold is a manifold $M$ whose tangent bundle $TM$ is equipped with a nested family $H$ of sub-bundles $H_0\leq H_1 \leq \cdots \leq TM$ which defines a filtration of the Lie algebra of vector fields on $M.$ Differential operators on Carnot manifolds have their order measured in terms of the filtration defined by $H,$ and the algebra of differential operators can be extended to an algebra of pseudodifferential operators. Recently, Dave-Haller and Couchet-Yuncken proposed definitions of a residue functional on the algebra of pseudodifferential operators adapted to a Carnot manifold. We prove that Connes' trace theorem holds in this setting.