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

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.