A note on weighted bounds for singular operators with nonsmooth kernels

Let T be a multilinear integral operator which is bounded on certain products of Lebesgue spaces on ℝⁿ. We assume that its associated kernel satisfies some mild regularity condition which is weaker than the usual Hölder continuity of kernels of multilinear Calderón-Zygmund singular integral operators. In this paper, given a suitable multiple weight w→, we obtain a bound for the weighted norm of T in terms of w→. As applications, we obtain new weighted bounds for certain singular integral operators such as linear and multilinear Fourier multipliers and the Riesz transforms associated to Schrödinger operators on ℝⁿ.