Self-improvement of uniform fatness revisited

TL;DR

This paper presents a new, simplified proof for the self-improvement property of uniform p-fatness in metric spaces, avoiding complex potential theory and leveraging local Hardy inequalities.

Contribution

It introduces a novel proof technique for self-improvement of uniform p-fatness that relies on standard geometric analysis methods.

Findings

Proof avoids deep potential theory results

Establishes self-improvement via local Hardy inequalities

Simplifies understanding of uniform p-fatness in metric spaces

Abstract

We give a new proof for the self-improvement of uniform p-fatness in the setting of general metric spaces. Our proof is based on rather standard methods of geometric analysis, and in particular the proof avoids the use of deep results from potential theory and analysis on metric spaces that have been indispensable in the previous proofs of the self-improvement. A key ingredient in the proof is a self-improvement property for local Hardy inequalities.