New oscillation classes and two weight bump conditions for commutators

TL;DR

This paper introduces new oscillation classes and two weight bump conditions for higher order commutators of Calderón-Zygmund operators, expanding the understanding of their boundedness beyond traditional BMO functions.

Contribution

It generalizes previous results by replacing BMO with oscillation classes and establishes new sufficient and necessary conditions for commutator boundedness with two weight bump conditions.

Findings

Established new sufficient conditions for boundedness of higher order commutators.

Proved necessary conditions for the boundedness of iterated commutators.

Extended previous work by incorporating oscillation classes and two weight bump conditions.

Abstract

In this paper we consider two weight bump conditions for higher order commutators. Given $b$ and a Calder\'on-Zygmund operator $T$, define the commutator $T^1_bf=[T,b]f= bTf-T(bf)$, and for $m\geq 2$ define the iterated commutator $T^m_b f = [b,T_b^{m-1}]f$. Traditionally, commutators are defined for functions $b\in BMO$, but we show that if we replace $BMO$ by an oscillation class first introduced by P\'erez [31], we can give a range of sufficient conditions on a pair of weights $(u,v)$ for $T^m_b : L^p(v)\rightarrow L^p(u)$ to be bounded. Our results generalize work of the first two authors in [10], and more recent work by Lerner, et al. [28]. We also prove necessary conditions for the iterated commutators to be bounded, generalizing results of Isralowitz, et al.[20].