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

Original · EN

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≥ 1, and demonstrate that in the Ahlfors Q-regular metric measure space setting with p=Q, these three properties are equivalent.