Two-weighted estimates of the multilinear fractional integral operator between weighted Lebesgue and Lipschitz spaces with optimal parameters

TL;DR

This paper characterizes the boundedness of multilinear fractional integral operators between weighted Lebesgue and Lipschitz spaces, providing optimal parameter ranges and examples of weight pairs, generalizing previous linear and unweighted results.

Contribution

It introduces new characterizations of weight classes for the boundedness of multilinear fractional integrals, extending known results to broader weighted and multilinear settings.

Findings

Established optimal parameter ranges for boundedness

Provided examples of weight pairs satisfying the conditions

Generalized previous linear and unweighted estimates

Abstract

Given an $m$-tuple of weights $\vec{v}=(v_1,\dots,v_m)$, we characterize the classes of pairs $(w,\vec{v})$ involved with the boundedness properties of the multilinear fractional integral operator from $\prod_{i=1}^mL^{p_i}\left(v_i^{p_i}\right)$ into suitable Lipschitz spaces associated to a parameter $\delta$, $\mathcal{L}_w(\delta)$. Our results generalize some previous estimates not only for the linear case but also for the unweighted problem in the multilinear context. We emphasize the study related to the range of the parameters involved with the problem described above, which is optimal in the sense that they become trivial outside of the region obtained. We also exhibit nontrivial examples of pairs of weights in this region.