Compactness of commutator of Riesz transforms in the two weight setting

TL;DR

This paper characterizes when commutators of Calderón–Zygmund operators are compact in a two-weight setting, linking compactness to membership in a weighted VMO space, and extends previous work to higher dimensions and product spaces.

Contribution

It provides a characterization of compactness of commutators in the Bloom setting using weighted VMO spaces, extending prior results to higher dimensions and product spaces.

Findings

Commutator compactness characterized by weighted VMO membership.

Extension of results to higher dimensions and product spaces.

Weighted VMO spaces differ from classical VMO, especially in higher dimensions.

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.