Commutators in the Two-Weight Setting

TL;DR

This paper characterizes the boundedness of commutators of Riesz transforms between weighted Lebesgue spaces using a BMO space adapted to the weights, extending classical results to a two-weight setting.

Contribution

It extends the characterization of commutator boundedness to the two-weight setting with general weights, generalizing classical one-weight and one-dimensional results.

Findings

Two-weight norm inequality for commutators is equivalent to BMO condition.

Generalizes classical results of Coifman-Rochberg-Weiss and Bloom.

Provides a unified framework for Riesz transform commutators with weights.

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.