Weighted Hardy inequalities beyond Lipschitz domains
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Abstract:
It is a well-known fact that in a Lipschitz domain $\Omega \subset \mathbb {R}^n$, a $p$-Hardy inequality, with weight $\operatorname {dist}(x,\partial \Omega )^\beta$, holds for all $u\in C_0^\infty (\Omega )$ whenever $\beta <p-1$. We show that actually the same is true under the sole assumption that the boundary of the domain satisfies a uniform density condition with the exponent $\lambda =n-1$. Corresponding results also hold for smaller exponents, and, in fact, our methods work in general metric spaces satisfying standard structural assumptions.References
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Additional Information
- Juha Lehrbäck
- Affiliation: Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35 (MaD), University of Jyväskylä, FIN-40014, Finland
- Email: juha.lehrback@jyu.fi
- Received by editor(s): March 12, 2012
- Received by editor(s) in revised form: June 21, 2012
- Published electronically: February 7, 2014
- Additional Notes: The author was supported in part by the Academy of Finland, grant no. 120972
- Communicated by: Jeremy Tyson
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 142 (2014), 1705-1715
- MSC (2010): Primary 26D15, 46E35
- DOI: https://doi.org/10.1090/S0002-9939-2014-11904-6
- MathSciNet review: 3168477