Besov Spaces, Schatten Classes and Weighted Versions of the Quantised Derivative

TL;DR

This paper investigates the properties of commutators of Riesz transforms on weighted spaces, establishing Schatten class estimates and extending the concept of quantised derivatives in the context of weighted $L^2$ spaces.

Contribution

It introduces weighted Schatten class estimates for Riesz transform commutators and extends the quantised derivative framework to weighted settings.

Findings

Established Schatten class and weak Schatten class estimates for Riesz transform commutators.

Provided a weighted extension of the quantised derivative concept.

Enhanced understanding of operator behavior in weighted harmonic analysis.

Abstract

In this paper, we establish the Schatten class and endpoint weak Schatten class estimates for the commutator of Riesz transforms on weighted $L^2$ spaces. It provides a weighted version for the estimate of the quantised derivative introduced by Alain Connes and studied recently by Lord--McDonald--Sukochev--Zanin.