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

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.