Iterated commutators under a joint condition on the tuple of multiplying functions

TL;DR

This paper introduces new joint conditions on functions weaker than BMO that nearly characterize the L^2 boundedness of iterated commutators with Calderón-Zygmund operators, advancing understanding of their boundedness criteria.

Contribution

It proposes novel joint conditions on functions that are weaker than BMO, providing a near characterization of the boundedness of iterated commutators with Calderón-Zygmund operators.

Findings

New joint conditions weaker than BMO for boundedness

Characterization of iterated commutator boundedness

Sandwiching boundedness between bisublinear mean oscillation conditions

Abstract

We present a pair of joint conditions on the two functions $b_1,b_2$ strictly weaker than $b_1,b_2\in \operatorname{BMO}$ that almost characterize the $L^2$ boundedness of the iterated commutator $[b_2,[b_1,T]]$ of these functions and a Calder\'on-Zygmund operator $T.$ Namely, we sandwich this boundedness between two bisublinear mean oscillation conditions of which one is a slightly bumped up version of the other.