Weighted mixed weak-type inequalities for multilinear operators
Volume 244 / 2019
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.
