Equivalence and self-improvement of p-fatness and Hardy's inequality, and association with uniform perfectness

TL;DR

This paper proves that in metric spaces, p-Hardy inequalities imply uniform p-fatness and explores their equivalence with uniform perfectness, revealing self-improving properties and establishing key relationships among these geometric and analytic conditions.

Contribution

The paper provides a simplified proof of the implication from p-Hardy inequality to uniform p-fatness at p=n, extends the results to metric spaces, and demonstrates the equivalence of these properties with uniform perfectness in Ahlfors Q-regular spaces.

Findings

p-Hardy inequality implies uniform p-fatness when p=n.

The proof extends to metric space settings.

In Ahlfors Q-regular spaces, p-fatness, p-Hardy inequality, and uniform perfectness are equivalent.

Abstract

We present an easy proof that $p$--Hardy's inequality implies uniform $p$--fatness of the boundary when $p=n$. The proof works also in metric space setting and demonstrates the self--improving phenomenon of the $p$--fatness. We also explore the relationship between $p$-fatness, $p$-Hardy inequality, and the uniform perfectness for all $p\ge 1$, and demonstrate that in the Ahlfors $Q$-regular metric measure space setting with $p=Q$, these three properties are equivalent.