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.
Categories: math.CA