A note on commutator in the multilinear setting

TL;DR

This paper characterizes BMO functions via a multilinear commutator condition and proves that boundedness of certain multilinear Calderón-Zygmund commutators implies the functions are in BMO, extending previous results.

Contribution

It provides a new characterization of BMO functions in the multilinear setting and extends existing boundedness results of multilinear commutators to more general cases.

Findings

Characterization of BMO functions via multilinear commutator condition.

Boundedness of multilinear Calderón-Zygmund commutators implies BMO membership.

Extension of previous results to the general multilinear case.

Abstract

Let $m\in \mathbb{N}$ and $\vec{b}=(b_{1},\cdots,b_{m})$ be a collection of locally integrable functions. It is proved that $b_{1},b_{2},\cdots, b_{m}\in BMO$ if and only if $$\sup_{Q}\frac{1}{|Q|^{m}}\int_{Q^{m}}\Big|\sum_{i=1}^{m}\big(b_{i}(x_{i})-(b_{i})_{Q}\big)\Big|d\vec{x}<\infty,$$ where $(b_{i})_{Q}=\frac{1}{|Q|}\int_{Q}b_{i}(x)dx$. As an application, we show that if the linear commutator of certain multilinear Calder\'{o}n-Zygmund operator $[\Sigma \vec{b},T]$ is bounded from $L^{p_{1}}\times\cdots\times L^{p_{m}}$ to $L^{p}$ with $\sum_{i=1}^{m}1/p_{i}=1/p$ and $1<p,p_{1},\cdots,p_{m}<\infty$, then $b_{1},\cdots,b_{m}\in BMO$. Therefore, the result of Chaffee \cite{C} (or Li and Wick \cite{LW}) is extended to the general case.