Compactness of commutator of Riesz transforms in the two weight setting

Abstract

We characterize the compactness of commutators in the Bloom setting. Namely, for a suitably non-degenerate Calderón--Zygmund operator $T$, and a pair of weights $\sigma ,\omega \in A_p$, the commutator $[T,b]$ is compact from $L^p(\sigma )\to L^p(\omega )$ if and only if $b\in VM{O}_{\nu }$, where $\nu =(\sigma /\omega {)}^{1/p}$. This extends the work of the first author, Holmes and Wick. The weighted $VMO$ spaces are different from the classical $VMO$ space. In dimension $d=1$, compactly supported and smooth functions are dense in $VM{O}_{\nu }$, but this need not hold in dimensions $d\ge 2$. Moreover, the commutator in the product setting with respect to little VMO space is also investigated.