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

In this paper we prove some sharp weighted norm inequalities for the multi(sub)linear maximal function $\text{Mm}$ introduced in \cite{LOPTT} and for multilinear Calder'on-Zygmund operators. In particular we obtain a sharp mixed “$A_p-A_{\mathrm{\infty }}$” bound for $\text{Mm}$, some partial results related to a Buckley-type estimate for $\text{Mm}$, and a sufficient condition for the boundedness of $\text{Mm}$ 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.