Compactness of commutator of Riesz transforms in the two weight setting

We characterize the compactness of commutators in the Bloom setting. Namely, for a suitably non-degenerate Calder\'on--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 $b \in VMO _{\nu }$, 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 $VMO _{\nu }$, but this need not hold in dimensions $d \geq 2$. Moreover, the commutator in the product setting with respect to little VMO space is also investigated.