Multilinear commutators with vector symbol $\vec{b}=(b_1,\ldots,b_m)$ defined by\[T_{\vec{b}}(f)(x)=\int_{{\bb R}^n}\Bigg[\prod\limits^m_{j=1}(b_j(x)-b_j(y))\Bigg]K(x,y)f(y)dy\]are considered, where $K$ is a Calderón–Zygmund kernel. The following a priori estimates are proved for $w\in A_\infty$. For $0 < p < \infty$, there exists a constant $C$ such that\[\|\dot{T}_{{\vec{b}}}(f)\|_{L^P(w)}\le C\|\vec{b}\|\|M_{L(\log\,L)^{1/r}}(f)\|_{L^P(w)}\]and\[\sup_{t>0}\frac{1}{\Phi(\frac{1}{t})}w(\{y\in{\bb R}^n:|T_{\vec{b}}f(y)|>t\})\le C\sup_{t>0}\frac{1}{\Phi(\frac{1}{t})}w(\{y\in{\bb R}^n:M_{L(\log\,L)^{1/r}}(\|\vec{b}\|f)(y)>t\}),\]where\begin{eqnarray*}&\|\vec{b}\|=\prod\limits^m_{j=1}\|b_j\|_{osc_{\exp L}^r j},\\&\Phi(t)=t\log^{1/r}(e+t),\quad \frac{1}{r}=\frac{1}{r_1}+\cdots+\frac{1}{r_m},\end{eqnarray*}and $M_{L(\log L)^{\alpha}}$ is an Orlicz type maximal operator. This extends, with a different approach, classical results by Coifman.

As a corollary, it is deduced that the operators $T_{\vec{b}}$ are bounded on $L^p(w)$ when $w\in A_p$, and that they satisfy corresponding weighted $L(\log\,L)^{1/r}$-type estimates with $w\in A_1$.