Weighted mixed weak-type inequalities for multilinear fractional operators

TL;DR

This paper establishes new mixed weak-type inequalities for multilinear fractional operators, extending previous results and providing applications to vector-valued inequalities under certain weight conditions.

Contribution

It introduces novel mixed weak-type bounds for multilinear fractional operators, including maximal functions and integrals, under specific weight assumptions.

Findings

Proves inequalities for multilinear fractional operators with weights.

Extends previous results by Berra, Carena, and Pradolini.

Provides applications to vector-valued inequalities.

Abstract

The aim of this paper is to obtain mixed weak-type inequalities for multilinear fractional operators, extending results by F. Berra, M. Carena and G. Pradolini \cite{BCP}. We prove that, under certain conditions on the weights, there exists a constant $C$ such that $$\Bigg\| \frac{\mathcal G_{\alpha}(\vec f \,)}{v}\Bigg\|_{L^{q, \infty}(\nu v^q)} \leq C \ \prod_{i=1}^m{\|f_i\|_{L^1(u_i)}},$$ where $\mathcal G_{\alpha}(\vec f \,)$ is the multilinear maximal function $\mathcal M_{\alpha}(\vec f\,)$ that was introduced by K. Moen in \cite{M} or the multilineal fractional integral $\mathcal I_{\alpha}(\vec f \,)$. As an application a vector-valued weighted mixed inequality for $\mathcal I_{\alpha}(\vec f \,)$ will be provided as well.