Commutators in the Two-Weight Setting

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