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Pointwise Hardy inequalities

Authors:	Piotr Hajlasz

Institution:	Instytut Matematyki, Uniwersytet Warszawski, Banacha 2, 02--097 Warszawa, Poland

Abstract:	If $\Omega\subset{{\mathbb R}}^{n}$ is an open set with the sufficiently regular boundary, then the Hardy inequality $\int _{\Omega}\|u\|^{p}\varrho^{-p}\leq C\int _{\Omega}\|\nabla u\|^{p}$ holds for $u\in C_{0}^{\infty}(\Omega)$ and $1<p<\infty$ , where $\varrho(x)=\operatorname{dist}(x,\partial\Omega)$ . The main result of the paper is a pointwise inequality $\|u\|\leq\varrho M_{2\varrho}\|\nabla u\|$ , where on the right hand side there is a kind of maximal function. The pointwise inequality combined with the Hardy-Littlewood maximal theorem implies the Hardy inequality. This generalizes some recent results of Lewis and Wannebo.

Keywords:	Hardy inequalities Sobolev spaces capacity $p$-thick sets maximal function Wiener criterion

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