  In this paper, we first introduce the new class of multiple weights
$A^\vc_{\vec{p}}$ which is larger than the class of multiple weights in
\cite{LOPTG}. Then, using this class of weights, we study the weighted norm
inequalities for certain classes of multilinear operators and their commutators
with new BMO functions introduced by \cite{BHS1}. Finally, we show that some
multilinear pseudodifferential operators fall within the scope of the theory
obtained in this paper.
