  Via the new weight function $A_{\vec p}^{\theta }(\varphi )$, the authors
introduce a new class of multilinear Littlewood--Paley $g_{\lambda}^{*}$
functions and establish the boundedness on weighted Lebesgue spaces. In
addition, the authors obtain the boundedness of the multilinear commutator and
multilinear iterated commutator generated by the multilinear Littlewood--Paley
$g_{\lambda}^{*}$ function and the new $BMO$ function on weighted Lebesgue
spaces. The results in this article include the known results in
\cite{XY2015,SXY2014}. When $m=1$, that is, in the case of one-linear, our
conclusions are also new, further extending the results in \cite{S1961}.
