Some extensions of classes involving pair of weights related to the boundedness of multilinear commutators associated to generalized fractional integral operators

TL;DR

This paper investigates the boundedness of higher order multilinear commutators related to generalized fractional integral operators, establishing new weighted inequalities and optimal parameter ranges.

Contribution

It extends previous linear results to multilinear settings, providing new conditions on weights for boundedness of generalized fractional commutators.

Findings

Established boundedness of multilinear commutators on weighted spaces.

Identified optimal parameter ranges for the weights and spaces involved.

Provided examples of weights covering the optimal parameter regions.

Abstract

We deal with the boundedness properties of higher order commutators related to some generalizations of the multilinear fractional integral operator of order $m$, $I_\alpha ^m$, from a product of weighted Lebesgue spaces into adequate weighted Lipschitz spaces, extending some previous estimates for the linear case. Our study includes two different types of commutators and sufficient conditions on the weights in order to guarantee the continuity properties described above. We also exhibit the optimal range of the parameters involved. The optimality is understood in the sense that the parameters defining the corresponding spaces belong to a certain region, being the weights trivial outside of it. We further show examples of weights for the class which cover the mentioned area.