Weighted bounds for multilinear operators with non-smooth kernels

Abstract

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 Hölder continuity of those in the class of multilinear Calderón-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 Schrödinger operators on ${\mathbb{R}}^n$. Our results are new even in the linear case.