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
$w\in A_1$
.