Two Weight Inequalities for Iterated Commutators with Calder\'on-Zygmund Operators

TL;DR

This paper extends weighted inequalities for iterated commutators of Calderón-Zygmund operators, connecting their boundedness to weighted BMO norms and generalizing previous one- and multi-weight results.

Contribution

It introduces new bounds for higher iterates of commutators in weighted settings, broadening the understanding of their behavior in harmonic analysis.

Findings

Extended two-weight inequalities to iterated commutators

Reproduced a one-weight result by Chung-Pereyra-Perez

Established bounds relating commutator norms to weighted BMO spaces

Abstract

Given a Calder\'on-Zygmund operator $T$, a classic result of Coifman-Rochberg-Weiss relates the norm of the commutator $[b, T]$ with the BMO norm of $b$. We focus on a weighted version of this result, obtained by Bloom and later generalized by Lacey and the authors, which relates $\| [b, T] : L^p(\mathbb{R}^n; \mu) \to L^p(\mathbb{R}^n; \lambda) \|$ to the norm of $b$ in a certain weighted BMO space determined by $A_p$ weights $\mu$ and $\lambda$. We extend this result to higher iterates of the commutator and recover a one-weight result of Chung-Pereyra-Perez in the process.