Weighted mixed weak-type inequalities for multilinear operators

Abstract

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 mm , the main result of the paper sentences that under different conditions on the weights we can obtain T ( ~f )(x) v L 1m ,∞(νv 1m ) ≤ C Ym i=1 kfikL1(wi), 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 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 Calder´on-Zygmund operators.