The sharp weighted bound for multilinear maximal functions and Calder\'{o}n-Zygmund operators

TL;DR

This paper establishes the optimal weighted bounds for multilinear maximal functions and Calderón-Zygmund operators within a specific weighted Lebesgue space framework, advancing the understanding of their behavior under multiple weights.

Contribution

It provides the first sharp bounds for multilinear maximal functions across all relevant exponents and for Calderón-Zygmund operators when p ≥ 1, in the context of multiple A_p weights.

Findings

Sharp bounds for multilinear maximal functions for all p_i

Sharp bounds for m-linear Calderón-Zygmund operators when p ≥ 1

Extension of weighted inequalities to multiple A_p weights

Abstract

We investigate the weighted bounds for multilinear maximal functions and Calder\'on-Zygmund operators from $L^{p_1}(w_1)\times...\times L^{p_m}(w_m)$ to $L^{p}(v_{\vec{w}})$, where $1<p_1,...,p_m<\infty$ with $1/{p_1}+...+1/{p_m}=1/p$ and $\vec{w}$ is a multiple $A_{\vec{P}}$ weight. We prove the sharp bound for the multilinear maximal function for all such $p_1,..., p_m$ and prove the sharp bound for $m$-linear Calder\'on-Zymund operators when $p\geq 1$.