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 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.