Journal of Inequalities and Applications
Volume 1 (1997), Issue 2, Pages 125-137

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.


Let D be an open subset of n(n2) with finite Lebesgue n-measure, let d(x) be the distance from xn to the boundary D of D, and let 1<p<. We give a simple direct proof that if n\D satisfies the plumpness condition of Martio and Väisälä [10], then the inequality of Hardy type, D(|u(x)|/dα(x))pdxCD(|u(x)|/dβ(x))pdx,uC0(D), holds whenever βmax{0,α1}. 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.