Multilinear Fractional Integral Operators with Generalized Kernels

TL;DR

This paper introduces a new class of multilinear fractional integral operators with generalized kernels weaker than Dini kernels, establishing their boundedness on various function spaces and analyzing associated commutators.

Contribution

It extends the theory of multilinear fractional integrals by weakening kernel conditions and studying boundedness and commutators in broader function spaces.

Findings

Boundedness on weighted Lebesgue spaces

Boundedness on variable exponent Lebesgue spaces

New results on multilinear commutators with BMO functions

Abstract

In this article, we introduce a class of multilinear fractional integral operators with generalized kernels that are weaker than the Dini kernel condition. We establish the boundedness of multilinear fractional integral operators with generalized kernels on weighted Lebesgue spaces and variable exponent Lebesgue spaces, as well as the boundedness of multilinear commutators generated by multilinear fractional integral operators with generalized kernels and $BMO$ functions. Even when the generalized kernels condition goes back to the Dini kernel condition, the conclusions on the commutators remain new.