Sharp weighted bounds for multilinear maximal functions and Calder\'on-Zygmund operators

TL;DR

This paper establishes sharp weighted bounds for multilinear maximal functions and Calderón-Zygmund operators, advancing understanding of their behavior under weighted norms and introducing new bounds and conditions.

Contribution

It provides the first sharp mixed $A_p-A_{}$ bounds for multilinear maximal functions and extends the $A_2$ conjecture to multilinear operators.

Findings

Sharp mixed $A_p-A_{}$ bounds for multilinear maximal functions

Bound for multilinear Calderón-Zygmund operators via dyadic positive operators

Multilinear $A_2$ conjecture is addressed

Abstract

In this paper we prove some sharp weighted norm inequalities for the multi(sub)linear maximal function $\Mm$ introduced in \cite{LOPTT} and for multilinear Calder\'on-Zygmund operators. In particular we obtain a sharp mixed "$A_p-A_{\infty}$" bound for $\Mm$, some partial results related to a Buckley-type estimate for $\Mm$, and a sufficient condition for the boundedness of $\Mm$ between weighted $L^p$ spaces with different weights taking into account the precise bounds. Next we get a bound for multilinear Calder\'on-Zygmund operators in terms of dyadic positive multilinear operators in the spirit of the recent work. Then we obtain a multilinear version of the "$A_2$ conjecture". Several open problems are posed.