# Weighted mixed weak-type inequalities for multilinear operators

## TL;DR

This paper extends classical weighted inequalities to multilinear operators, providing new bounds for Calderón-Zygmund operators and their vector-valued versions under mixed weight conditions.

## Contribution

It generalizes Sawyer's result to the multilinear setting and establishes new weighted weak-type inequalities for multilinear Calderón-Zygmund operators.

## Key findings

- Established a generalized weak-type inequality for multilinear Calderón-Zygmund operators.
- Proved the inequality for the multilinear maximal function and the Hardy-Littlewood maximal operator.
- Extended results to vector-valued inequalities for these 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 $$\Bigg\| \frac{T(\vec f\,)(x)}{v}\Bigg\|_{L^{\frac{1}{m}, \infty}(\nu v^\frac{1}{m})} \leq C \ \prod_{i=1}^m{\|f_i\|_{L^1(w_i)}}, $$ where $T$ is a multilinear Calder\'on-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 Calder\'on-Zygmund operators.