Boundedness of fractional integral operators on non-homogeneous metric measure spaces

TL;DR

This paper introduces a generalized fractional integral operator on non-homogeneous metric measure spaces and establishes its boundedness properties, extending classical results to more general settings including non-doubling measures.

Contribution

It defines a broad class of fractional integral operators on non-homogeneous spaces and proves their boundedness, including for commutators with RBMO functions, generalizing previous results.

Findings

Established $(L^{p}( u),L^{q}( u))$-boundedness for the operators.

Extended boundedness results to commutators with RBMO functions.

Unified results for both homogeneous and non-doubling measure spaces.

Abstract

In this paper, the fractional integral operator on non-homogeneous metric measure spaces is introduced, which contains the classic fractional integral operator, fractional integral with non-doubling measures and fractional integral with fractional kernel of order $\alpha$ and regularity $\epsilon$ introduced by Garc\'{i}a-Cuerva and Gatto as special cases. And the $(L^{p}(\mu),L^{q}(\mu))$-boundedness for fractional integral operators on non-homogeneous metric measure spaces is established. From this, the $(L^{p}(\mu),L^{q}(\mu))$-boundedness for commutators and multilinear commutators generated by fractional integral operators with $RBMO(\mu)$ function are further obtained. These results in this paper includes the corresponding results on both the homogeneous spaces and non-doubling measure spaces.