Sharp weighted bounds for fractional integral operators in a space of homogeneous type

TL;DR

This paper extends the Hardy--Littlewood--Sobolev theorem to fractional integral operators on spaces of homogeneous type, establishing sharp weighted bounds and their dependence on weight constants, generalizing Euclidean results.

Contribution

It introduces sharp weighted bounds for fractional integral operators in spaces of homogeneous type, generalizing recent Euclidean space results.

Findings

Established sharp bounds for fractional integral operators in homogeneous spaces.

Demonstrated the dependence of operator norms on weight constants.

Proved the sharpness of bounds in spaces with infinitely many points.

Abstract

We consider a version of M. Riesz fractional integral operator on a space of homogeneous type and show an analogue of the well-known Hardy--Littlewood--Sobolev theorem in this context. In our main result, we investigate the dependence of the operator norm on weighted spaces on the weight constant, and find the relationship between these two quantities. It it shown that the estimate obtained is sharp in any given space of homogeneous type with infinitely many points. Our result generalizes the recent Euclidean result by Lacey, Moen, P{\'e}rez and Torres.