Weak and Strong Type Weighted Estimates for Multilinear Calder\'{o}n-Zygmund Operators

TL;DR

This paper establishes weak and strong weighted estimates for multilinear Calderón-Zygmund operators, providing sharp bounds under certain conditions and advancing understanding of weighted inequalities in harmonic analysis.

Contribution

It introduces new weak and strong type weighted estimates for multilinear Calderón-Zygmund operators with multiple weights, including sharp bounds in specific parameter regimes.

Findings

Derived weak and strong weighted estimates for multilinear operators.

Established sharp bounds for certain parameter ranges.

Extended the theory of weighted inequalities in multilinear harmonic analysis.

Abstract

In this paper, we study the weighted estimates for multilinear Calder\'{o}n-Zygmund operators %with multiple $A_{\vec{P}}$ weights from $L^{p_1}(w_1)\times...\times L^{p_m}(w_m)$ to $L^{p}(v_{\vec{w}})$, where $1<p, p_1,...,p_m<\infty$ with $1/{p_1}+...+1/{p_m}=1/p$ and $\vec{w}=(w_1,...,w_m)$ is a multiple $A_{\vec{P}}$ weight. We give weak and strong type weighted estimates of mixed $A_p$-$A_\infty$ type. Moreover, the strong type weighted estimate is sharp whenever $\max_i p_i \le p'/(mp-1)$.