Remarks on the hardy inequality - R. Hurri–Syrjänen (Journal of Inequalities and Applications, 1997)
Let D be an open subset of Ã¢Â„Ân(nÃ¢Â‰Â¥2) with finite Lebesgue n-measure, let d(x) be the distance from xÃ¢ÂˆÂˆÃ¢Â„Ân to the boundary Ã¢ÂˆÂ‚D of D, and let 1<p<Ã¢ÂˆÂž. We give a simple direct proof that if Ã¢Â„ÂnD satisfies the plumpness condition of Martio and VÃƒÂ¤isÃƒÂ¤lÃƒÂ¤ [10], then the inequality of Hardy type, Ã¢ÂˆÂ«D(|u(x)|/dÃŽÂ±(x))pdxÃ¢Â‰Â¤CÃ¢ÂˆÂ«D(|Ã¢ÂˆÂ‡u(x)|/dÃŽÂ²(x))pdx,Ã¢Â€Âƒ uÃ¢ÂˆÂˆC0Ã¢ÂˆÂž(D), holds whenever ÃŽÂ²Ã¢Â‰Â¥max{0,ÃŽÂ±Ã¢ÂˆÂ’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.