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arXiv · 1506.05747

Commutators in the Two-Weight Setting

Irina Holmes·Michael T. Lacey·Brett D. Wick
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Abstract

Let $R$ be the vector of Riesz transforms on $\mathbb{R}^n$, and let $μ,λ\in A_p$ be two weights on $\mathbb{R}^n$, $1 < p < \infty$. The two-weight norm inequality for the commutator $[b, R] : L^p(\mathbb{R}^n;μ) \to L^p(\mathbb{R}^n;λ)$ 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.

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Irina Holmes, Michael T. Lacey, Brett D. Wick. 2016-01-04. Commutators in the Two-Weight Setting. https://doi.org/10.1007/s00208-016-1378-1

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