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Sobolev Extensions of Hölder Continuous and Characteristic Functions on Metric Spaces

Published online by Cambridge University Press:  20 November 2018

Anders Björn ,
Jana Björn  and
Nageswari Shanmugalingam
Anders Björn
Affiliation:
Department of Mathematics, Linköping University, SE-581 83 Linköping, Sweden email: anbjo@mai.liu.se, jabjo@mai.liu.se
Jana Björn
Affiliation:
Department of Mathematics, Linköping University, SE-581 83 Linköping, Sweden email: anbjo@mai.liu.se, jabjo@mai.liu.se
Nageswari Shanmugalingam
Affiliation:
Department of Mathematical Sciences, University of Cincinnati, P.O. Box 210025 Cincinnati, OH 45221-0025, U.S.A. email: nages@math.uc.edu
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Abstract

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We study when characteristic and Hölder continuous functions are traces of Sobolev functions on doubling metric measure spaces. We provide analytic and geometric conditions sufficient for extending characteristic and Hölder continuous functions into globally defined Sobolev functions.

Keywords

46E3531C45Characteristic functionNewtonian functionmetric spaceresolutivityHölder continuousPerron solutionp-harmonicSobolev extensionWhitney covering
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 59 , Issue 6 , 01 December 2007 , pp. 1135 - 1153
Copyright
Copyright © Canadian Mathematical Society 2007

References

[1] Björn, A. and Björn, J., Boundary regularity for p-harmonic functions and solutions of the obstacle problem. J. Math. Soc. Japan 58(2006), no. 4, 1211–1232.Google Scholar
[2] Björn, A., Björn, J., and Shanmugalingam, N., The Dirichlet problem for p-harmonic functions on metric spaces. J. Reine Angew.Math. 556(2003), 173–203.Google Scholar
[3] Björn, A., Björn, J., and Shanmugalingam, N., The Perron method for p-harmonic functions in metric spaces. J. Differential Equations 195(2003), no. 2, 398–429.Google Scholar
[4] Björn, A., Björn, J., and Shanmugalingam, N., A problem of Baernstein on the equality of the p-harmonic measure of a set and its closure. Proc. Amer. Math. Soc. 134(2006), no. 2, 509–519.Google Scholar
[5] Björn, A., Björn, J., and Shanmugalingam, N., Quasicontinuity of Newton–Sobolev functions and density of Lipschitz functions on metric spaces. To appear in Houston, J. Math.Google Scholar
[6] Buckley, S., Is the maximal function of a Lipschitz function continuous. Ann. Acad. Sci. Fenn. Math. 24(1999), no. 2, 519–528.Google Scholar
[7] Cheeger, J., Differentiability of Lipschitz functions on metric spaces. Geom. Funct. Anal. 9(1999), no. 3, 428–517.Google Scholar
[8] Coifman, R. R. and Weiss, G., Analyse harmonique non-commutative sur certains espaces homogènes. Lecture Notes in Mathematics 242, Springer-Verlag, Berlin, 1971.Google Scholar
[9] Danielli, D., Garofalo, N., and Nhieu, D. M., Non-doubling Ahlfors measures, perimeter measures, and the characterization of the trace spaces of Sobolev functions in Carnot-Carathéodory spaces. Mem. Amer. Math. Soc. 182(2006), no. 857.Google Scholar
[10] Gagliardo, E., Caratterizzazioni delle tracce sulla frontiera relative ad alcune classi di funzioni in n variabili. Rend. Sem. Mat. Univ. Padova 27(1957), 284–305.Google Scholar
[11] Hajłasz, P., Sobolev spaces on an arbitrary metric space. Potential Anal. 5(1996), no. 4, 403–415.Google Scholar
[12] Hajłasz, P. and Koskela, P., Sobolev met Poincaré. Mem. Amer. Math. Soc. 145(2000), no. 688.Google Scholar
[13] Hajłasz, P. and Martio, O., Traces of Sobolev functions on fractal type sets and characterization of extension domains. J. Funct. Anal. 143(1997), no. 1, 221–246.Google Scholar
[14] Heinonen, J., Lectures on Analysis on Metric Spaces. Springer-Verlag, New York, 2001.Google Scholar
[15] Heinonen, J., Kilpeläinen, T., and Martio, O., Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford University Press, Oxford, 1993.Google Scholar
[16] Heinonen, J. and Koskela, P., Quasiconformal maps in metric spaces with controlled geometry. Acta Math. 181(1998), no. 1, 1–61.Google Scholar
[17] Herron, D. and Koskela, P., Continuity of Sobolev functions and Dirichlet finite harmonic measures. Potential Anal. 6(1997), no. 4, 347–353.Google Scholar
[18] Jonsson, A. and Wallin, H., Function spaces on subsets of R n. Math. Rep. 2(1984) no. 1.Google Scholar
[19] Keith, S. and Zhong, X., The Poincaré inequality is an open ended condition. To appear in Ann. Math.Google Scholar
[20] Kilpeläinen, T., Kinnunen, J., and Martio, O., Sobolev spaces with zero boundary values on metric spaces. Potential Anal. 12(2000), no. 3, 233–247.Google Scholar
[21] Kinnunen, J. and Martio, O., The Sobolev capacity on metric spaces. Ann. Acad. Sci. Fenn. Math. 21(1996), no. 2, 367–382.Google Scholar
[22] Kinnunen, J. and Martio, O., Choquet property for the Sobolev capacity in metric spaces. In: Proceedings on Analysis and Geometry (Russian). Izdat. Ross. Akad. Nauk Sib. Otd. Inst. Mat., Novosibirsk, 2000, pp. 285–290.Google Scholar
[23] Kinnunen, J. and Martio, O., Nonlinear potential theory on metric spaces. Illinois Math. J. 46(2002), no. 3, 857–883.Google Scholar
[24] Koskela, P. and MacManus, P., Quasiconformal mappings and Sobolev spaces. Studia Math. 131(1998), no. 1, 1–17.Google Scholar
[25] Kufner, A., John, O., and Fučík, S., Function Spaces. Academia, Prague, 1977.Google Scholar
[26] Martio, O. and Vuorinen, M., Whitney cubes, p-capacity and Minkowski content. Exposition. Math. 5(1987), no. 1, 17–40.Google Scholar
[27] Mattila, P., Geometry of Sets and Measures in Euclidean Spaces. Cambridge Studies in Advanced Mathematics 44, Cambridge University Press, Cambridge, 1995.Google Scholar
[28] Shanmugalingam, N., Newtonian spaces: An extension of Sobolev spaces to metric measure spaces. Rev. Mat. Iberoamericana 16(2000), no. 2, 243–279.Google Scholar
[29] Shanmugalingam, N., Harmonic functions on metric spaces. Illinois J. Math. 45(2001), no. 3, 1021–1050.Google Scholar