Bloom's Inequality: Commutators in a Two-Weight Setting

Abstract

In 1985, Bloom characterized the boundedness of the commutator $[b,H]$ as a map between a pair of weighted $L^p$ spaces, where both weights are in $A_p$. The characterization is in terms of a novel $BMO$ condition. We give a 'modern' proof of this result, in the case of $p=2$. In a subsequent paper, this argument will be used to generalize Bloom's result to all Calderón-Zygmund operators and dimensions.

v1: 9 pages. v2: 9 pages, typos corrected

Cited by in corpus (8)

• Matrix weighted norm inequalities for commutators and paraproducts with matrix symbols
• Dyadic harmonic analysis and weighted inequalities: the sparse revolution
• Bilinear Calderón-Zygmund theory on product spaces
• Fractional Bloom boundedness and compactness of commutators
• Off-diagonal estimates for bi-commutators
• Iterated commutators under a joint condition on the tuple of multiplying functions
• Boundedness of commutators and H${}^1$-BMO duality in the two matrix weighted setting
• Weighted $L^p\to L^q$-boundedness of commutators and paraproducts in the Bloom setting