Compactness of commutator of Riesz transforms in the two weight setting

We characterize the compactness of commutators in Bloom setting. Namely, for a suitably non-degenerate Calderón–Zygmund operator T, and pair weights σ , ω ∈ A p commutator [ T b ] is compact from L ( ) → if only V M O ν where = / 1 . This extends work first author, Holmes Wick. The weighted VMO spaces are different classical space. In dimension d compactly supported smooth functions dense but this need not hold dimensions ≥ 2 Moreover, product setting with respect to little space also investigated.