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

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

2007-09-13

arxiv.org/abs/0709.2013v2

Math. Z. 246(1), 99-110, 2010.

Mathematics

Functional Analysis

12 pages

Scientific paper

10.1007/s00209-008-0454-y

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.

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