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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 σ,ω∈Ap, the commutator [T,b] is compact from Lp(σ)→Lp(ω) if and only if b∈VMOν, where ν=(σ/ω)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 VMOν, but this need not hold in dimensions d≥2. Moreover, the commutator in the product setting with respect to little VMO space is also investigated.
- Two weight
- Riesz transform
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