A note on commutator in the multilinear setting

Let $m\in \mathbb{N}$ and $\vec{b}=(b_1,\cdots ,b_m)$ be a collection of locally integrable functions. It is proved that $b_{1},b_{2},\cdots, b_{m}\in BMO$ifandonlyif$${\sup }_Q\frac{1}{\mathrm{\mid }Q{\mathrm{\mid }}^m}{\int }_{Q^m}\mid {\sum }_{i=1}^m(b_i(x_i)-(b_i{)}_Q)\mid d\vec{x}<\mathrm{\infty },$$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 $[\mathrm{\Sigma }\vec{b},T]$ 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.