The equivalence between pointwise Hardy inequalities and uniform fatness

Details The equivalence between pointwise Hardy inequalities and uniform fatness The equivalence between pointwise Hardy inequalities and uniform fatness

2009-06-11

arxiv.org/abs/0906.2086v1

Mathematics

Analysis of PDEs

18 pages

Scientific paper

We prove an equivalence result between the validity of a pointwise Hardy inequality in a domain and uniform capacity density of the complement. This result is new even in Euclidean spaces, but our methods apply in general metric spaces as well. We also present a new transparent proof for the fact that uniform capacity density implies the classical integral version of the Hardy inequality in the setting of metric spaces. In addition, we consider the relations between the above concepts and certain Hausdorff content conditions.

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