Characterizations of weighted compactness of commutators via $\textrm {CMO}(\mathbb R^n)$

Abstract:

In this paper, the authors show that a function $b\in \textrm {BMO}(\mathbb R^n)$ is in $\textrm {CMO}(\mathbb R^n)$ if and only if the Riesz transform commutator $[b, R_i]$ is compact on $L^p_w(\mathbb R^n)$ for $i\in \{1, 2,\cdots ,n\}$, $p\in (1, \infty )$, and $w\in A_p(\mathbb R^n)$, and if and only if the fractional integral commutator $[b, I_\alpha ]$ is compact from $L^p_{w^p}(\mathbb R^n)$ to $L^q_{w^q}(\mathbb R^n)$, where $\alpha \in (0, n)$, $p,q\in (1, \infty )$ with $\frac 1p=\frac 1q+\frac \alpha n$ and $w\in A_{p, q}(\mathbb R^n)$.