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

TL;DR

This paper revisits Bloom's 1985 characterization of the boundedness of commutators in weighted $L^p$ spaces, providing a modern proof for the case $p=2$ and setting the stage for broader generalizations.

Contribution

It offers a new proof of Bloom's inequality for $p=2$, simplifying the understanding of commutator boundedness in weighted spaces and paving the way for future generalizations.

Findings

Modern proof of Bloom's inequality for $p=2$

Clarification of $BMO$ conditions in weighted spaces

Framework for extending results to Calderón-Zygmund operators

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 Calder\'on-Zygmund operators and dimensions.