In this paper we present a theorem that generalizes Sawyer's classic result about mixed weighted inequalities to the multilinear context. Let w = (w1, ..., wm) and ν = w 1 m 1 ...w 1 m m , the main result of the paper sentences that under different conditions on the weights we can obtainwhere 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 M( f )(x): the multi(sub)linear maximal function introduced in [13].As an application we also prove a vector-valued extension to the mixed weighted weak-type inequalities of multilinear Calderón-Zygmund operators.where T is Hardy-Littlewood maximal function or any Calderón-Zygmund operator.Quantitative estimates of these mixed weighted results can be found in [16]. Moreover, it was conjectured in [7] 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 ∈ A 1 and v ∈ A ∞ . Recently, in [14] 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.
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