Weighted bounds for multilinear operators with non-smooth kernels

TL;DR

This paper establishes weighted bounds for multilinear operators with less regular kernels, extending results to Fourier multipliers and Riesz transforms, with new findings even in the linear case.

Contribution

It introduces weighted bounds for multilinear operators with non-smooth kernels, broadening the class of operators for which weighted estimates are known.

Findings

Weighted bounds for multilinear operators with mild kernel regularity.

Extension of weighted estimates to Fourier multipliers and Schr"odinger Riesz transforms.

Results are new even for linear operators.

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 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.