Connes' trace theorem and the log-polyhomogeneous calculus for 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. There are also ``higher" residues, defined on the set of operators whose symbols that can be expressed as polynomials of a logarithm, introduced by Lesch, and these are related to other singular traces. 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 $M.$ Associated to a Carnot manifold is a pseudodifferential calculus $\Psi_H(M),$ which measures sections of $H_k$ as having order $k.$ 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. We also introduce an analogy of Lesch's log-polyhomogeneous calculus for Carnot manifolds, define the corresponding higher residues, and give their spectral description in terms of singular traces.