  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.
