  Let $T$ be a multilinear operator which is bounded on certain products of
unweighted Lebesgue spaces of $\mathbb R^n$. We assume that the associated
kernel of $T$ satisfies some mild regularity condition which is weaker than the
usual H\"older continuity of those in the class of multilinear
Calder\'on-Zygmund singular integral operators. We then show the boundedness
for $T$ and the boundedness of the commutator of $T$ with BMO functions on
products of weighted Lebesgue spaces of $\mathbb R^n$. As an application, we
obtain the weighted norm inequalities of multilinear Fourier multipliers and of
their commutators with BMO functions on the products of weighted Lebesgue
spaces when the number of derivatives of the symbols is the same as the best
known result for the multilinear Fourier multipliers to be bounded on the
products of unweighted Lebesgue spaces.
