Journal of Inequalities and Applications
Volume 1 (1997), Issue 2, Pages 125-137
doi:10.1155/S1025583497000088

Remarks on the hardy inequality

D. E. Edmunds1 and R. Hurri–Syrjänen2

1Centre for Mathematical Analysis and its Applications, University of Sussex, Falmer, East Sussex, Brighton BN1 9QH, UK
2Department of Mathematics, University of Helsinki, Hallituskatu 15, Helsinki SF-00100, Finland

Received 3 July 1996

Copyright © 1997 D. E. Edmunds and R. Hurri–Syrjänen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let D be an open subset of ℝn(n≥2) with finite Lebesgue n-measure, let d(x) be the distance from x∈ℝn to the boundary ∂D of D, and let 1<p<∞. We give a simple direct proof that if ℝn\D satisfies the plumpness condition of Martio and Väisälä [10], then the inequality of Hardy type, ∫D(|u(x)|/dα(x))pdx≤C∫D(|∇u(x)|/dβ(x))pdx,  u∈C0∞(D), holds whenever β≥max{0,α−1}. We also show that the plumpness condition may be replaced by ones. which enable domains with lower-dimensional portions of their boundaries to be handled.