Schatten classes and commutators of Riesz transform on Heisenberg group and applications

TL;DR

This paper characterizes the Schatten norms of commutators with Riesz transforms on the Heisenberg group using Besov norms, extending classical Euclidean results and applying to various operators.

Contribution

It generalizes classical Euclidean results to the Heisenberg group and extends methods to analyze a broader class of operators including the Cauchy--Szego projection.

Findings

Schatten norm characterization via Besov norms

Extension of methods to second order Riesz transforms

Application to Cauchy--Szego projection

Abstract

We study commutators with the Riesz transforms on the Heisenberg group. The Schatten norm of these commutators is characterized in terms of Besov norms of the symbol. This generalizes the classical Euclidean results of Peller, Janson--Wolff and Rochberg--Semmes. The method of proof extends the earlier methods, allowing us to address not just the Riesz transforms, but also the Cauchy--Szego projection and second order Riesz transforms on the Heisenberg group, among other settings.