Pointwise Hardy inequalities
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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.References
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Additional Information
- Piotr Hajłasz
- Affiliation: Instytut Matematyki, Uniwersytet Warszawski, Banacha 2, 02–097 Warszawa, Poland
- MR Author ID: 332316
- Email: hajlasz@mimuw.edu.pl
- 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
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 127 (1999), 417-423
- MSC (1991): Primary 31C15, 46E35; Secondary 42B25
- DOI: https://doi.org/10.1090/S0002-9939-99-04495-0
- MathSciNet review: 1458875