Weighted vector-valued bounds for a class of multilinear singular integral operators

TL;DR

This paper establishes weighted vector-valued bounds for multilinear singular integral operators and their commutators, extending the understanding of their behavior in weighted function spaces with multiple A_p weights.

Contribution

It introduces new weighted bounds for a class of multilinear operators and their commutators, including endpoint estimates, advancing the theoretical framework in harmonic analysis.

Findings

Proved weighted vector-valued bounds for multilinear singular integrals.

Established weighted weak type endpoint estimates for commutators.

Extended bounds to multiple A_p weight settings.

Abstract

In this paper, we investigate the weighted vector-valued bounds for a class of multilinear singular integral operators, and its commutators, from $L^{p_1}(l^{q_1};\,\mathbb{R}^n,w_1)\times\dots\times L^{p_m}(l^{q_m};\,\mathbb{R}^n,w_m)$ to $L^{p}(l^q;\,\mathbb{R}^n,\nu_{\vec{w}})$, with $p_1,\dots,p_m\in (1,\,\infty)$ and $1/p=1/p_1+\dots+1/p_m$ and $\vec{w}$ is a multiple $A_{\vec{P}}$ weights. Our argument also leads to the weighted weak type endpoint estimates for the commutators.