Hardy’s inequality and the boundary size
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- by Pekka Koskela and Xiao Zhong PDF
- Proc. Amer. Math. Soc. 131 (2003), 1151-1158 Request permission
Abstract:
We establish a self-improving property of the Hardy inequality and an estimate on the size of the boundary of a domain supporting a Hardy inequality.References
- Hiroaki Aikawa and Matts Essén, Potential theory—selected topics, Lecture Notes in Mathematics, vol. 1633, Springer-Verlag, Berlin, 1996. MR 1439503, DOI 10.1007/BFb0093410
- Alano Ancona, On strong barriers and an inequality of Hardy for domains in $\textbf {R}^n$, J. London Math. Soc. (2) 34 (1986), no. 2, 274–290. MR 856511, DOI 10.1112/jlms/s2-34.2.274
- J. Björn, P. MacManus, N. Shanmugalingam, Fat sets and Hardy inequalities in metric spaces, J. Anal. Math. 85 (2001), 339–369.
- Marc Bourdon and Hervé Pajot, Poincaré inequalities and quasiconformal structure on the boundary of some hyperbolic buildings, Proc. Amer. Math. Soc. 127 (1999), no. 8, 2315–2324. MR 1610912, DOI 10.1090/S0002-9939-99-04901-1
- Donatella Danielli, Nicola Garofalo, and Duy-Minh Nhieu, Trace inequalities for Carnot-Carathéodory spaces and applications, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 27 (1998), no. 2, 195–252 (1999). MR 1664688
- Piotr Hajłasz and Pekka Koskela, Sobolev met Poincaré, Mem. Amer. Math. Soc. 145 (2000), no. 688, x+101. MR 1683160, DOI 10.1090/memo/0688
- Juha Heinonen and Pekka Koskela, Quasiconformal maps in metric spaces with controlled geometry, Acta Math. 181 (1998), no. 1, 1–61. MR 1654771, DOI 10.1007/BF02392747
- Juha Kinnunen and Olli Martio, The Sobolev capacity on metric spaces, Ann. Acad. Sci. Fenn. Math. 21 (1996), no. 2, 367–382. MR 1404091
- Alois Kufner, Weighted Sobolev spaces, Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], vol. 31, BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1980. With German, French and Russian summaries. MR 664599
- T. J. Laakso, Ahlfors $Q$-regular spaces with arbitrary $Q>1$ admitting weak Poincaré inequality, Geom. Funct. Anal. 10 (2000), no. 1, 111–123. MR 1748917, DOI 10.1007/s000390050003
- John L. Lewis, Uniformly fat sets, Trans. Amer. Math. Soc. 308 (1988), no. 1, 177–196. MR 946438, DOI 10.1090/S0002-9947-1988-0946438-4
- John L. Lewis, On very weak solutions of certain elliptic systems, Comm. Partial Differential Equations 18 (1993), no. 9-10, 1515–1537. MR 1239922, DOI 10.1080/03605309308820984
- V. G. Maz’ya, Sobolev spaces, Springer-Verlag, Berlin - New York, 1985.
- Jan-Olov Strömberg and Alberto Torchinsky, Weighted Hardy spaces, Lecture Notes in Mathematics, vol. 1381, Springer-Verlag, Berlin, 1989. MR 1011673, DOI 10.1007/BFb0091154
- Andreas Wannebo, Hardy inequalities, Proc. Amer. Math. Soc. 109 (1990), no. 1, 85–95. MR 1010807, DOI 10.1090/S0002-9939-1990-1010807-1
Additional Information
- Pekka Koskela
- Affiliation: Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35, Fin-40351 Jyväskylä, Finland
- MR Author ID: 289254
- Email: pkoskela@math.jyu.fi
- Xiao Zhong
- Affiliation: Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35, Fin-40351 Jyväskylä, Finland
- Email: zhong@math.jyu.fi
- Received by editor(s): May 30, 2001
- Received by editor(s) in revised form: November 5, 2001
- Published electronically: July 26, 2002
- Additional Notes: This research was partially supported by the Academy of Finland, projects 39788 and 41964, and the foundation Vilho, Yrjö ja Kalle Väisälän rahasto (X.Z.). Part of this research was done while the second author was visiting at the Mittag-Leffler Institute. He wishes to thank the Institute for their support and hospitality.
- Communicated by: Juha M. Heinonen
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 131 (2003), 1151-1158
- MSC (2000): Primary 26D10, 31C99, 46E35
- DOI: https://doi.org/10.1090/S0002-9939-02-06711-4
- MathSciNet review: 1948106