The equivalence between pointwise Hardy inequalities and uniform fatness

2009/06/11 by Riikka Korte, Korte, Riikka, Juha Lehrbäck +3 · 1 citation
Mathematics · #26D15 #31C45 #46E35 #Analysis of PDEs (math.AP) #Analytic and geometric function theory #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · doi:10.48550/arxiv.0906.2086

openalex publication_date 2009/06/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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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• The doubling property of the elliptic measure, for elliptic operators with drifts satisfying an average diverging condition

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