Endpoint weak Schatten class estimates and trace formula for commutators of Riesz transforms with multipliers on Heisenberg groups

TL;DR

This paper establishes endpoint weak Schatten class estimates and a trace formula for commutators of Riesz transforms with multipliers on Heisenberg groups, advancing understanding of their spectral properties without relying on Fourier analysis.

Contribution

It introduces a new singular trace formula on Heisenberg groups and applies double operator integrals to analyze commutators, bypassing traditional Fourier analysis methods.

Findings

Established endpoint weak Schatten class estimates for commutators.

Constructed a singular trace formula on Heisenberg groups.

Provided a framework applicable to general stratified Lie groups.

Abstract

Along the line of singular value estimates for commutators by Rochberg-Semmes, Lord-McDonald-Sukochev-Zanin and Fan-Lacey-Li, we establish the endpoint weak Schatten class estimate for commutators of Riesz transforms with multiplication operator $M_f$ on Heisenberg groups via homogeneous Sobolev norm of the symbol $f$. The new tool we exploit is the construction of a singular trace formula on Heisenberg groups, which, together with the use of double operator integrals, allows us to bypass the use of Fourier analysis and provides a solid foundation to investigate the singular values estimates for similar commutators in general stratified Lie groups.