---
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 Calderón-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.'
address:
- 'Department of Mathematics and Statistics, P.O. Box 68 (Pietari Kalmin katu 5), FI-00014 University of Helsinki, Finland'
- 'Center for Applied Mathematics, Tianjin University, Weijin Road 92, 300072 Tianjin, China'
- 'Department of Mathematics and Statistics, P.O. Box 68 (Pietari Kalmin katu 5), FI-00014 University of Helsinki, Finland'
author:
- Tuomas Hytönen
- Kangwei Li
- Tuomas Oikari
title: Iterated commutators under a joint condition on the tuple of multiplying functions
---

Introduction
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The study of commutators of Calderón-Zygmund operators with pointwise multiplication has been a long standing interest in the field of harmonic analysis; for example, in the fundamental paper of Coifman, Rochberg, Weiss [@CRW] a characterization of the space ${\operatorname{BMO}}({\mathbb{R}}^d)$ is given with respect to the commutator taken with the Riesz transforms: $$[b,R_i]:L^2({\mathbb{R}}^d)\to L^2({\mathbb{R}}^d)\quad\mbox{ boundedly for all} \quad i=1,\dots,d\quad$$ if and only if $b\in{\operatorname{BMO}}({\mathbb{R}}^d).$ Here 