Generalized Fractional Integrals and Their Commutators over Non-homogeneous Metric Measure Spaces

TL;DR

This paper characterizes the boundedness of fractional integrals and their multilinear commutators on non-homogeneous metric measure spaces, extending classical results to more general settings with applications to Orlicz spaces.

Contribution

It provides new equivalent conditions for fractional integral boundedness and establishes boundedness of multilinear commutators on Orlicz spaces in non-homogeneous spaces.

Findings

Equivalent characterizations of fractional integral boundedness.

Boundedness of multilinear commutators on Orlicz spaces.

Weak type endpoint estimates for commutators.

Abstract

Let $({\mathcal X},d,\mu)$ be a metric measure space satisfying both the upper doubling and the geometrically doubling conditions. In this paper, the authors establish some equivalent characterizations for the boundedness of fractional integrals over $({\mathcal X},d,\mu)$. The authors also prove that multilinear commutators of fractional integrals with ${\mathop\mathrm{\,RBMO(\mu)}}$ functions are bounded on Orlicz spaces over $({\mathcal X},d,\mu)$, which include Lebesgue spaces as special cases. The weak type endpoint estimates for multilinear commutators of fractional integrals with functions in the Orlicz-type space ${\mathrm{Osc}_{\exp L^r}(\mu)}$, where $r\in [1,\infty)$, are also presented. Finally, all these results are applied to a specific example of fractional integrals over non-homogeneous metric measure spaces.