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.

v3: suggestions from two referees incorporated

References in corpus (4)

• Sharp weighted estimates for dyadic shifts and the $A_2$ conjecture
• Two Weight Inequalities for Riesz Transforms: Uniformly Full Dimension Weights
• Two Weight Inequalities for the Cauchy Transform from $\mathbb{R}$ to ${\mathbb{C}}_{+}$
• Sharp $A_2$ estimates of Haar shifts via Bellman function

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• Two-weight inequalities for multilinear commutators
• Three observations on commutators of Singular Integral Operators with BMO functions
• Off-diagonal estimates for bi-commutators
• Fractional Bloom boundedness and compactness of commutators
• The unified theory for the necessity of bounded commutators and applications
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• Two weight commutators on spaces of homogeneous type and applications
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• Two-Weight Inequalities for Commutators with Fractional Integral Operators
• Some new weighted estimates on product spaces
• Modern singular integral theory with mild kernel regularity
• Genuinely multilinear weighted estimates for singular integrals in product spaces
• Weighted Estimates of Singular Integrals and Commutators in the Zygmund Dilation Setting
• A note on two weight commutators of maximal functions on spaces of homogeneous type
• Variational characterizations of weighted Hardy spaces and weighted $BMO$ spaces
• Two weight commutators for Beurling--Ahlfors operator
• Boundedness of commutators and H${}^1$-BMO duality in the two matrix weighted setting
• A note on commutators on weighted Morrey spaces on spaces of homogeneous type
• Riesz transform and commutators in the Dunkl setting
• Compactness of commutator of Riesz transforms in the two weight setting
• Sparse dominations and weighted variation inequalities for singular integrals and commutators
• A note on extrapolation of compactness
• Weighted $L^p\to L^q$-boundedness of commutators and paraproducts in the Bloom setting
• Bloom type upper bounds in the product BMO setting
• Quasi-Banach estimates of commutators of bilinear bi-parameter singular integrals: paraproducts
• Bloom type inequality for bi-parameter singular integrals: efficient proof and iterated commutators