Distribution of points and Hardy type inequalities in spaces of homogeneous type

TL;DR

This paper investigates Hardy type inequalities in spaces of homogeneous type, establishing conditions based on geometry and point distribution that ensure their validity, with examples and counterexamples illustrating these conditions.

Contribution

It introduces new sufficient conditions for Hardy inequalities in homogeneous spaces, linking geometric properties to inequality validity, supported by examples and counterexamples.

Findings

Identified geometric conditions ensuring Hardy inequalities

Established relationships between point distribution and inequality validity

Provided examples and counterexamples in the complex plane

Abstract

In the setting of spaces of homogeneous type, we study some Hardy type inequalities, which notably appeared in the proofs of local T(b) theorems as in [AR]. We give some suffi cient conditions ensuring their validity, related to the geometry and distribution of points in the homogeneous space. We study the relationships between these conditions and give some examples and counterexamples in the complex plane.