  In this paper we present a theorem that generalizes Sawyer's classic result
about mixed weighted inequalities to the multilinear context. Let
$\vec{w}=(w_1,...,w_m)$ and $\nu = w_1^\frac{1}{m}...w_m^\frac{1}{m}$, the main
result of the paper sentences that under different conditions on the weights we
can obtain $$\Bigg\| \frac{T(\vec f\,)(x)}{v}\Bigg\|_{L^{\frac{1}{m},
\infty}(\nu v^\frac{1}{m})} \leq C \ \prod_{i=1}^m{\|f_i\|_{L^1(w_i)}}, $$
where $T$ is a multilinear Calder\'on-Zygmund operator. To obtain this result
we first prove it for the $m$-fold product of the Hardy-Littlewood maximal
operator $M$, and also for $\mathcal{M}(\vec{f})(x)$: the multi(sub)linear
maximal function introduced in \cite{LOPTT}.
  As an application we also prove a vector-valued extension to the mixed
weighted weak-type inequalities of multilinear Calder\'on-Zygmund operators.
