Let R be the vector of Riesz transforms on Rn, and let
μ,λ∈Ap be two weights on Rn, 1<p<∞. The
two-weight norm inequality for the commutator [b,R]:Lp(Rn;μ)→Lp(Rn;λ) is shown to be equivalent to the function b
being in a BMO space adapted to μ and λ. This is a common extension
of a result of Coifman-Rochberg-Weiss in the case of both λ and μ
being Lebesgue measure, and Bloom in the case of dimension one.Comment: v3: suggestions from two referees incorporate