A note on commutator in the multilinear setting

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 $$\underset{Q}{\sup }\frac{1}{|Q{|}^m}{\int }_{Q^m}\left|\sum _{i=1}^m\right(b_i(x_i)-(b_i{)}_Q)|d\vec{x}<\infty ,$$ 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 Calderón-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.

9 pages

References in corpus (2)

• Characterizations of BMO through commutators of bilinear singular integral operators
• Characterizations of weighted BMO space and its application