Lebesgue points via the Poincaré inequality

详细信息    查看全文

• 作者：Nijjwal Karak ; Pekka Koskela
• 关键词：Lebesgue point ; Poincaré inequality ; Q ; doubling space ; 46E35 ; 28A78 ; 28A15
• 刊名：SCIENCE CHINA Mathematics
• 出版年：2015
• 出版时间：August 2015
• 年：2015
• 卷：58
• 期：8
• 页码：1697-1706
• 全文大小：203 KB
• 参考文献：1.Adams D R, Hedberg L I. Function Spaces and Potential Theory. Berlin: Springer-Verlag, 1996View Article
  2.Bj?rn J, Onninen J. Orlicz capacities and Hausdorff measures on metric spaces. Math Z, 2005, 251: 131-46MathSciNet View Article
  3.Cheeger J. Differentiability of Lipschitz functions on metric measure spaces. Geom Funct Anal, 1999, 9: 428-17MathSciNet View Article
  4.Federer H. Geometric Measure Theory. New York: Springer-Verlag, 1969
  5.Giusti E. Precisazione delle funzioni di H 1,p e singolarit`a delle soluzioni deboli di sistemi ellittici non lineari. Boll Unione Mat Ital (4), 1969, 2: 71-6MathSciNet
  6.Haj?asz P. Sobolev spaces on an arbitrary metric space. Potential Anal, 1996, 5: 403-15MathSciNet
  7.Haj?asz P, Koskela P. Sobolev met Poincaré. Mem Amer Math Soc, 2000, 145: x+101
  8.Heinonen J. Lectures on Analysis on Metric Spaces. New York: Springer-Verlag, 2001View Article
  9.Heinonen J, Kilpel?inen T, Martio O. Nonlinear Potential Theory of Degenerate Elliptic Equations. Mineola: Dover Publications, 2006
  10.Heinonen J, Koskela P. Quasiconformal maps in metric spaces with controlled geometry. Acta Math, 1998, 181: 1-1MathSciNet View Article
  11.Heinonen J, Koskela P, Shanmugalingam N, et al. Sobolev Spaces on Metric Measure Spaces: An Approach Based On Upper Gradients. Cambridge: Cambridge University Press, 2015View Article
  12.Keith S, Zhong X. The Poincaré inequality is an open ended condition. Ann of Math (2), 2008, 167: 575-99MathSciNet View Article
  13.Khavin V P, Maz’ya V G. Non-linear potential theory. Russian Math Surveys, 1972, 27: 71-48
  14.Kinnunen J, Latvala V. Lebesgue points for Sobolev functions on metric spaces. Rev Mat Iberoamericana, 2002, 18: 685-00MathSciNet View Article
  15.Korte R. Geometric implications of the Poincaré inequality. Results Math, 2007, 50: 93-07MathSciNet View Article
  16.Koskela P. Removable sets for Sobolev spaces. Ark Mat, 1999, 37: 291-04MathSciNet View Article
  17.Koskela P, MacManus P. Quasiconformal mappings and Sobolev spaces. Studia Math, 1998, 131: 1-7MathSciNet
  18.Maly J, Ziemer W P. Fine Regularity of Solutions of Elliptic Partial Differential Equations. Providence, RI: Amer Math Soc, 1997View Article
  19.Müller D, Yang D. A difference characterization of Besov and Triebel-Lizorkin spaces on RD-spaces. Forum Math, 2009, 21: 259-98MathSciNet View Article
  20.Rauch H E. Harmonic and analytic functions of several variables and the maximal theorem of Hardy and Littlewood. Canad J Math, 1956, 8: 171-83MathSciNet View Article
  21.Rogers C A. Hausdorff Measures. Cambridge: Cambridge University Press, 1998
  22.Shanmugalingam N. Newtonian spaces: An extension of Sobolev spaces to metric measure spaces. Rev Mat Iberoamericana, 2000, 16: 243-79MathSciNet View Article
  23.Smith K T. A generalization of an inequality of Hardy and Littlewood. Canad J Math, 1956, 8: 157-70MathSciNet View Article
  24.Yang D, Zhou Y. New properties of Besov and Triebel-Lizorkin spaces on RD-spaces. Manuscripta Math, 2011, 134: 59-0MathSciNet View Article
  25.Ziemer W P. Weakly Differentiable Functions. New York: Springer-Verlag, 1989View Article
  
• 作者单位：Nijjwal Karak (1)
  Pekka Koskela (1)
  
  1. Department of Mathematics and Statistics, University of Jyv?skyl?, Jyv?skyl?, 40014, Finland
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Chinese Library of Science
  Applications of Mathematics
  
• 出版者：Science China Press, co-published with Springer
• ISSN：1869-1862

文摘

We show that in a Q-doubling space (X, d, μ), Q > 1, which satisfies a chain condition, if we have a Q-Poincaré inequality for a pair of functions (u, g) where g ?L Q (X), then u has Lebesgue points \(\mathcal{H}^h \)-a.e. for \(h(t) = \log ^{1 - Q - \varepsilon } (1/t)\). We also discuss how the existence of Lebesgue points follows for u ?W 1,Q (X) where (X, d, μ) is a complete Q-doubling space supporting a Q-Poincaré inequality for Q > 1.