  Let $T$ be a multilinear integral operator which is bounded on certain
products of Lebesgue spaces on $\mathbb R^n$. We assume that its associated
kernel 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. In this paper, given a suitable multiple weight
$\vec{w}$, we obtain the bound for the weighted norm of multilinear operators
$T$ in terms of $\vec{w}$. As applications, we exploit this result to obtain
the weighted bounds for certain singular integral operators such as linear and
multilinear Fourier multipliers and the Riesz transforms associated to
Schr\"odinger operators on $\mathbb{R}^n$. Our results are new even in the
linear case.
