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Commutators in the Two-Weight Setting

Abstract

Let RR be the vector of Riesz transforms on Rn\mathbb{R}^n, and let μ,λAp\mu,\lambda \in A_p be two weights on Rn\mathbb{R}^n, 1<p<1 < p < \infty. The two-weight norm inequality for the commutator [b,R]:Lp(Rn;μ)Lp(Rn;λ)[b, R] : L^p(\mathbb{R}^n;\mu) \to L^p(\mathbb{R}^n;\lambda) is shown to be equivalent to the function bb being in a BMO space adapted to μ\mu and λ\lambda. This is a common extension of a result of Coifman-Rochberg-Weiss in the case of both λ\lambda and μ\mu being Lebesgue measure, and Bloom in the case of dimension one.Comment: v3: suggestions from two referees incorporate

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