Abstract
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.
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References
Adams D R, Hedberg L I. Function Spaces and Potential Theory. Berlin: Springer-Verlag, 1996
Björn J, Onninen J. Orlicz capacities and Hausdorff measures on metric spaces. Math Z, 2005, 251: 131–146
Cheeger J. Differentiability of Lipschitz functions on metric measure spaces. Geom Funct Anal, 1999, 9: 428–517
Federer H. Geometric Measure Theory. New York: Springer-Verlag, 1969
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: 71–76
Hajłasz P. Sobolev spaces on an arbitrary metric space. Potential Anal, 1996, 5: 403–415
Hajłasz P, Koskela P. Sobolev met Poincaré. Mem Amer Math Soc, 2000, 145: x+101
Heinonen J. Lectures on Analysis on Metric Spaces. New York: Springer-Verlag, 2001
Heinonen J, Kilpeläinen T, Martio O. Nonlinear Potential Theory of Degenerate Elliptic Equations. Mineola: Dover Publications, 2006
Heinonen J, Koskela P. Quasiconformal maps in metric spaces with controlled geometry. Acta Math, 1998, 181: 1–61
Heinonen J, Koskela P, Shanmugalingam N, et al. Sobolev Spaces on Metric Measure Spaces: An Approach Based On Upper Gradients. Cambridge: Cambridge University Press, 2015
Keith S, Zhong X. The Poincaré inequality is an open ended condition. Ann of Math (2), 2008, 167: 575–599
Khavin V P, Maz’ya V G. Non-linear potential theory. Russian Math Surveys, 1972, 27: 71–148
Kinnunen J, Latvala V. Lebesgue points for Sobolev functions on metric spaces. Rev Mat Iberoamericana, 2002, 18: 685–700
Korte R. Geometric implications of the Poincaré inequality. Results Math, 2007, 50: 93–107
Koskela P. Removable sets for Sobolev spaces. Ark Mat, 1999, 37: 291–304
Koskela P, MacManus P. Quasiconformal mappings and Sobolev spaces. Studia Math, 1998, 131: 1–17
Malý J, Ziemer W P. Fine Regularity of Solutions of Elliptic Partial Differential Equations. Providence, RI: Amer Math Soc, 1997
Müller D, Yang D. A difference characterization of Besov and Triebel-Lizorkin spaces on RD-spaces. Forum Math, 2009, 21: 259–298
Rauch H E. Harmonic and analytic functions of several variables and the maximal theorem of Hardy and Littlewood. Canad J Math, 1956, 8: 171–183
Rogers C A. Hausdorff Measures. Cambridge: Cambridge University Press, 1998
Shanmugalingam N. Newtonian spaces: An extension of Sobolev spaces to metric measure spaces. Rev Mat Iberoamericana, 2000, 16: 243–279
Smith K T. A generalization of an inequality of Hardy and Littlewood. Canad J Math, 1956, 8: 157–170
Yang D, Zhou Y. New properties of Besov and Triebel-Lizorkin spaces on RD-spaces. Manuscripta Math, 2011, 134: 59–90
Ziemer W P. Weakly Differentiable Functions. New York: Springer-Verlag, 1989
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Karak, N., Koskela, P. Lebesgue points via the Poincaré inequality. Sci. China Math. 58, 1697–1706 (2015). https://doi.org/10.1007/s11425-015-5001-9
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DOI: https://doi.org/10.1007/s11425-015-5001-9