The Holmes--Wick theorem on two-weight bounds for higher order commutators revisited

TL;DR

This paper revisits the Holmes--Wick theorem, providing an alternative proof for two-weight bounds of higher order commutators using classical Cauchy integral techniques, building on prior first-order results.

Contribution

It offers a new proof method for the higher order case, simplifying the understanding of two-weight bounds for commutators in harmonic analysis.

Findings

Alternative proof of higher order commutator bounds

Reduction of higher order case to first order results

Utilization of classical Cauchy integral argument

Abstract

A sufficient condition for the two-weight boundedness of higher order commutators was recently obtained by Holmes and Wick in terms of an intersection of two BMO spaces. We provide an alternative proof, showing that the higher order case can be deduced by a classical Cauchy integral argument from the corresponding first order result of Holmes, Lacey and Wick.