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Pointwise Hardy inequalities

Author: Piotr Hajlasz
Journal: Proc. Amer. Math. Soc. 127 (1999), 417-423
MSC (1991): Primary 31C15, 46E35; Secondary 42B25
MathSciNet review: 1458875
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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.

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Additional Information

Piotr Hajlasz
Affiliation: Instytut Matematyki, Uniwersytet Warszawski, Banacha 2, 02–097 Warszawa, Poland

Keywords: Hardy inequalities, Sobolev spaces, capacity, $p$-thick sets, maximal function, Wiener criterion
Received by editor(s): February 26, 1996
Received by editor(s) in revised form: May 6, 1997
Additional Notes: This research was carried out while the author stayed in the ICTP in Trieste in 1995. He wishes to thank the ICTP for their hospitality. The author was partially supported by KBN grant no. 2–PO3A–034–08.
Communicated by: Theodore W. Gamelin
Article copyright: © Copyright 1999 American Mathematical Society

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