Schatten class properties of weighted commutators

Abstract

In this paper, we establish the Schatten class properties of the commutator $[M_f,(-\Delta {)}^{\epsilon /2}]$ of a power $0<\epsilon <1$ of the Laplacian with the multiplication operator on weighted spaces. For instance, we show that this commutator belongs to the weak Schatten class ${\mathcal{L}}_{d/(1-\epsilon ),\infty }$ associated with the weighted Hilbert space $L_2({\mathbb{R}}^d,w(x)dx)$ if and only if $f$ is from the homogeneous Sobolev space ${\dot{W}}_{d/(1-\epsilon )}^1({\mathbb{R}}^d)$. Our results generalize the recent result due to Frank, Sukochev and Zanin.