Iterated Riesz Commutators: A Simple Proof of Boundedness

TL;DR

This paper presents a straightforward proof demonstrating the boundedness of iterated Riesz transform commutators on L^p spaces, utilizing dyadic shift representations to simplify the analysis.

Contribution

It introduces a simplified proof technique for boundedness of iterated Riesz commutators using dyadic operators and paraproduct estimates, streamlining previous approaches.

Findings

Proves L^p boundedness of iterated Riesz commutators

Uses dyadic shift representations to simplify proofs

Reduces complex estimates to paraproduct bounds

Abstract

We give a simple proof of L^p boundedness of iterated commutators of Riesz transforms and a product BMO function. We use a representation of the Riesz transforms by means of simple dyadic operators - dyadic shifts - which in turn reduces the estimate quickly to paraproduct estimates.