Weighted inequalities and pointwise estimates for the multilinear fractional integral and maximal operators

TL;DR

This paper establishes weighted inequalities and pointwise estimates for multilinear fractional integral and maximal operators, extending previous results and providing new boundedness conditions in various function spaces.

Contribution

It extends weighted inequalities and pointwise estimates to multilinear fractional operators, generalizing prior results and introducing new boundedness conditions.

Findings

Weighted norm inequalities for multilinear fractional integral operators.

Pointwise estimates between multilinear fractional maximal and integral operators.

Boundedness results in Banach function spaces for generalized multilinear fractional maximal operators.

Abstract

In this article we prove weighted norm inequalities and pointwise estimates between the multilinear fractional integral operator and the multilinear fractional maximal. As a consequence of these estimations we obtain weighted weak and strong inequalities for the multilinear fractional integral operator. In particular, we extend some results given in \cite{CPSS} to the multilinear context. On the other hand we prove weighted pointwise estimates between the multilinear fractional maximal operator ${\cal M}_{\alpha,B}$ associated to a Young function $B$ and the multilinear maximal operators ${\cal M}_{\psi}={\cal M}_{0,\psi}$, $\psi(t)=B(t^{1-\alpha/(nm)})^{{nm}/{(nm-\alpha)}}$. As an application of these estimate we obtain a direct proof of the $L^p-L^q$ boundedness results of ${\cal M}_{\alpha,B}$ for the case $B(t)=t$ and $B_k(t)=t(1+\log^+t)^k$ when $1/q=1/p-\alpha/n$. We also give sufficient conditions on the weights involved in the boundedness results of ${\cal M}_{\alpha,B}$ that generalizes those given in \cite{M} for $B(t)=t$. Finally, we prove some boundedness results in Banach function spaces for a generalized version of the multilinear fractional maximal operator.