Iterated commutators under a joint condition on the tuple of multiplying functions
Tuomas Hytönen, Kangwei Li, Tuomas Oikari
TL;DR
The paper investigates joint, weaker-than-BMO conditions on a pair of functions $(b_1,b_2)$ that ensure $L^2$ boundedness of the iterated commutator $[b_2,[b_1,T]]$ for Calderón-Zygmund operators. It establishes a two-sided sandwich: a lower bound in terms of $S_2(b_1,b_2)+T_2(b_1,b_2)$ and an upper bound via $S_{2+\\varepsilon}(b_1,b_2)+T_{2+\\varepsilon}(b_1,b_2)$, and then introduces genuinely weaker joint conditions $S_{A,B}$ and $T_C$ built from Young functions, shown to yield boundedness through sparse domination. The work also demonstrates the insufficiency of the $S_2+T_2$ (and even $S_{2+\\varepsilon}+T_{2+\\varepsilon}$) framework and provides counterexamples that motivate the shift to the Young-function setting. A conjecture is proposed that the $L^2$-boundedness is equivalent to the existence of suitable Young functions with $S_{A,B}+T_C<\\infty$, supported by intricate constructions using log-bumps and related examples. This advances the understanding of joint conditions in the bilinear/iterated commutator context and informs the broader theory of sparse domination and weighted inequalities.
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.