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Weighted mixed weak-type inequalities for multilinear operators

Volume 244 / 2019

Kangwei Li, Sheldy J. Ombrosi, M. Belén Picardi Studia Mathematica 244 (2019), 203-215 MSC: Primary 42B20; Secondary 42B25. DOI: 10.4064/sm170529-31-8 Published online: 21 May 2018

Abstract

We generalize Sawyer’s classical result about mixed weighted inequalities to the multilinear context. Let $\vec{w}=(w_1,\dots,w_m)$ and $\nu = w_1^{1/m}\ldots w_m^{1/m}$. The main result states that under different conditions on the weights we can obtain $$\biggl\| \frac{T(\vec f\,)}{v}\bigg\|_{L^{{1/m}, \infty}(\nu v^{1/m})} \leq C \prod_{i=1}^m{\|f_i\|_{L^1(w_i)}}, $$ 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 $\mathcal{M}(\vec{f}\,)$, the multi(sub)linear maximal function introduced by Lerner et al. (2009).

As an application we also prove a vector-valued extension of mixed weighted weak-type inequalities for multilinear Calderón–Zygmund operators.

Authors

  • Kangwei LiBCAM, Basque Center for Applied Mathematics
    Alameda de Mazarredo 14
    48009 Bilbao, Spain
    e-mail
  • Sheldy J. OmbrosiDepartamento de Matemática
    Universidad Nacional del Sur
    Bahía Blanca, 8000, Argentina
    e-mail
  • M. Belén PicardiDepartamento de Matemática
    Universidad Nacional del Sur
    Bahía Blanca, 8000, Argentina
    e-mail

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