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--- abstract: | 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 Calderón-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 [@LOPTT]. As an application we also prove a vector-valued extension to the mixed weighted weak-type inequalities of multilinear Calderón-Zygmund operators. address: - 'BCAM, Basque Center for Applied Mathematics, Mazarredo, 14. 48009 Bilbao Basque Country, Spain' - | Departamento de Matemática\ Universidad Nacional del Sur\ Bahía Blanca, 8000, Argentina - | Departamento de Matemática\ Universidad Nacional del Sur\ Bahía Blanca, 8000, Argentina author: - Kangwei Li - 'Sheldy J. Ombrosi' - 'M. Belén Picardi' title: 'Weighted mixed weak-type inequalities for multilinear operators' --- [^1] Introduction ============ In 1985, E. Sawyer [@S] proved the following mixed weak-type inequality: \[sawyer\] If $u,v \in A_1$, then there is a constant $C$ such that for all $t>0$, $$uv\Big(\Big\{ x \in \mathbb{R} :\; \frac{M(fv)(x)}{v(x)} \; > t \Big\}\Big) \; \leq \; \frac{C}{t} \int_{\mathbb{R}} |f(x)|u(x)v(x)\,dx.$$ In the same work E. Sawyer conjectured that the previous theorem is valid if the maximal operator is replaced by the Hilbert transform. In 2005, D. Cruz-Uribe, J. M. Martell and C. Pérez [@CMP] extended this result to $\mathbb{R}^n$. Furthermore, they proved it for Calderón-Zygmund operators, solving this Sawyer’s conjecture. \[cmpteo\] If $u,v \in A_1$, or $u\in A_{1}$ and $uv\in A_{\infty}$, then there is a constant $C$ such that for all $t>0$, $$uv\Bigg(\Big\{ x \in \mathbb{R}^n :\; \frac{|T(fv)(x)|}{v(x)} \; > t \Big\}\Bigg) \; \leq \; \frac{C}{t} \int_{\mathbb{R}^n} |f(x)|u(x)v(x)\,dx,$$ where $T$ is Hardy-Littlewood maximal function or any Calderón-Zygmund operator. Quantitative estimates of these mixed weighted results can be found in [@OPR]. Moreover, it was conjectured in [@CMP] that the conclusion of the previous theorem still holds if a weaker and more general hypothesis is satisfied. That is, if we have the following conditions on the weights, $u\in A_{1}$ and $v\in A_{\infty}$. Recently, in [@LOP] the first two authors and C. Pérez solved such conjecture. Namely, the following theorem was proved, which constitutes the more difficult case of this class of mixed weighted inequalities. \[LOPinf\] Let $T$ be a Calderón-Zygmund operator or the Hardy-Littlewood maximal operator and let $u\in A_1$ and $v\in A_{\infty}$. Then there is a finite constant $C$ depending on the $A_1$ constant of $u$ and the $A_{\infty}$ constant of $v$ such that $$\label{a1ainf} \ \Big\| \frac{T(fv)}{v}\Big\|_{L^{1, \infty}(uv)} \leq C \|f\|_{L^1(uv)}.$$ On the other hand, the study of multilinear Calderón-Zygmund theory started in the seventies with the works of R. Coifman and Y. Meyer ([@CoMe1] and [@CoMe2]). However, a systematic treatment of this topic appears later with works of L. Grafakos and R. Torres [@GraTor1; @GraTor2]. We recall the definition of a