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 $\vec{w}=(w_1,...,w_m)$ 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 $$\Vert \frac{T(\vec{f})(x)}{v}{\Vert }_{L^{\frac{1}{m},\infty }(\nu v^{\frac{1}{m}})}\le C\prod _{i=1}^m\Vert f_i{\Vert }_{L^1(w_i)},$$ 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 \cite{LOPTT}. As an application we also prove a vector-valued extension to the mixed weighted weak-type inequalities of multilinear Calderón-Zygmund operators.

10 pages

References in corpus (1)

• Mixed weak type estimates: Examples and counterexamples related to a problem of E. Sawyer

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• Sawyer-type inequalities for Lorentz spaces
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• Weak and strong type estimates for the multilinear Littlewood-Paley operators
• Weighted mixed weak-type inequalities for multilinear fractional operators