Weighted norm inequalities for multilinear operators and applications to multilinear Fourier multipliers

TL;DR

This paper establishes weighted norm inequalities for multilinear operators with mild kernel regularity conditions, extending boundedness results to weighted Lebesgue spaces and applying these to multilinear Fourier multipliers and their commutators.

Contribution

It introduces weaker regularity conditions for kernels of multilinear operators and proves their boundedness on weighted Lebesgue spaces, including applications to Fourier multipliers.

Findings

Boundedness of multilinear operators on weighted Lebesgue spaces.

Weighted norm inequalities for multilinear Fourier multipliers.

Boundedness of commutators with BMO functions on weighted spaces.

Abstract

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.