Advertisement

Journal d'Analyse Mathématique

, Volume 135, Issue 1, pp 59–83 | Cite as

The Cartan, Choquet and Kellogg properties for the fine topology on metric spaces

Open Access
Article

Abstract

We prove the Cartan and Choquet properties for the fine topology on a complete metric space equipped with a doubling measure supporting a p-Poincaré inequality, 1 < p < ∞. We apply these key tools to establish a fine version of the Kellogg property, characterize finely continuous functions by means of quasicontinuous functions, and show that capacitary measures associated with Cheeger supersolutions are supported by the fine boundary of the set.

References

  1. [1]
    D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory, Springer, Berlin–Heidelberg, 1996.CrossRefMATHGoogle Scholar
  2. [2]
    D. R. Adams and J. L. Lewis, Fine and quasiconnectedness in nonlinear potential theory, Ann. Inst. Fourier (Grenoble) 35 (1985), 57–73.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    L. Ambrosio, M. Colombo, and S. Di Marino, Sobolev spaces in metric measure spaces: reflexivity and lower semicontinuity of slope, Variational Methods for Evolving Objects, Math. Soc. Japan, Tokyo, 2015, pp. 1–58.CrossRefGoogle Scholar
  4. [4]
    L. Ambrosio, N. Gigli, and G. Savaré, Density of Lipschitz functions and equivalence of weak gradients in metric measure spaces, Rev. Mat. Iberoam. 29 (2013), 969–996.MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    A. Björn, Characterizations of p-superharmonic functions on metric spaces, Studia Math. 169 (2005), 45–62.MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    A. Björn, A weak Kellogg property for quasiminimizers, Comment. Math. Helv. 81 (2006), 809–825.MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    A. Björn, Removable singularities for bounded p-harmonic and quasi(super)harmonic functions on metric spaces, Ann. Acad. Sci. Fenn. Math. 31 (2006), 71–95.MathSciNetMATHGoogle Scholar
  8. [8]
    A. Björn and J. Björn, Boundary regularity for p-harmonic functions and solutions of the obstacle problem on metric spaces, J. Math. Soc. Japan 58 (2006), 1211–1232.MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    A. Björn and J. Björn, Nonlinear Potential Theory on Metric Spaces, European Math. Soc., Zürich, 2011.CrossRefMATHGoogle Scholar
  10. [10]
    A. Björn and J. Björn, Obstacle and Dirichlet problems on arbitrary nonopen sets in metric spaces, and fine topology, Rev. Mat. Iberoam. 31 (2015), 161–214.MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    A. Björn, J. Björn, and V. Latvala, The weak Cartan property for the p-fine topology on metric spaces, Indiana Univ. Math. J. 64 (2015), 915–941.MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    A. Björn, J. Björn, and V. Latvala, Sobolev spaces, fine gradients and quasicontinuity on quasiopen sets in Rn and metric spaces, Ann. Acad. Sci. Fenn. Math. 41 (2016), 551–560.MATHGoogle Scholar
  13. [13]
    A. Björn, J. Björn, and M. Parviainen, Lebesgue points and the fundamental convergence theorem for superharmonic functions on metric spaces, Rev. Mat. Iberoam. 26 (2010), 147–174.MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    A. Björn, J. Björn, and N. Shanmugalingam, The Dirichlet problem for p-harmonic functions on metric spaces, J. Reine Angew. Math. 556 (2003), 173–203.MathSciNetMATHGoogle Scholar
  15. [15]
    A. Björn, J. Björn, and N. Shanmugalingam, Quasicontinuity of Newton–Sobolev functions and density of Lipschitz functions on metric spaces, Houston J. Math. 34 (2008), 1197–1211.MathSciNetMATHGoogle Scholar
  16. [16]
    A. Björn and N. Marola, Moser iteration for (quasi)minimizers on metric spaces, Manuscripta Math. 121 (2006), 339–366.MathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    J. Björn, Boundary continuity for quasiminimizers on metric spaces, Illinois J. Math. 46 (2002), 383–403.MathSciNetMATHGoogle Scholar
  18. [18]
    J. Björn, Wiener criterion for Cheeger p-harmonic functions on metric spaces, Potential Theory in Matsue, Math. Soc. Japan, Tokyo, 2006, pp. 103–115.Google Scholar
  19. [19]
    J. Björn, Fine continuity on metric spaces, Manuscripta Math. 125 (2008), 369–381.MathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    J. Björn, P. MacManus, and N. Shanmugalingam, Fat sets and pointwise boundary estimates for p-harmonic functions in metric spaces, J. Anal. Math. 85 (2001), 339–369.MathSciNetCrossRefMATHGoogle Scholar
  21. [21]
    M. Brelot, Sur la théorie moderne du potentiel, C. R. Acad. Sci. Paris 209 (1939), 828–830.MathSciNetMATHGoogle Scholar
  22. [22]
    M. Brelot, Points irréguliers et transformations continues en théorie du potentiel J. Math. Pures Appl. (9) 19 (1940), 319–337.MathSciNetMATHGoogle Scholar
  23. [23]
    M. Brelot, Sur les ensembles effilés, Bull. Sci. Math. (2) 68 (1944), 12–36.MathSciNetMATHGoogle Scholar
  24. [24]
    M. Brelot On Topologies and Boundaries in Potential Theory, Springer, Berlin–Heidelberg, 1971.CrossRefMATHGoogle Scholar
  25. [25]
    H. Cartan, Théorie générale du balayage en potentiel newtonien Ann. Univ. Grenoble Sect. Sci. Math. Phys. (N.S.) 22 (1946), 221–280.MathSciNetMATHGoogle Scholar
  26. [26]
    J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal. 9 (1999), 428–517.MathSciNetCrossRefMATHGoogle Scholar
  27. [27]
    G. Choquet, Sur les points d’effilement d’un ensemble. Application à l’étude de la capacité, Ann. Inst. Fourier (Grenoble) 9 (1959), 91–101.MathSciNetCrossRefMATHGoogle Scholar
  28. [28]
    B. Franchi, P. Hajlasz and P. Koskela, Definitions of Sobolev classes on metric spaces, Ann. Inst. Fourier (Grenoble) 49 (1999), 1903–1924.MathSciNetCrossRefMATHGoogle Scholar
  29. [29]
    B. Fuglede, The quasi topology associated with a countably subadditive set function, Ann. Inst. Fourier (Grenoble) 21 (1971), 123–169.MathSciNetCrossRefMATHGoogle Scholar
  30. [30]
    B. Fuglede, Finely Harmonic Functions, Springer, Berlin–New York, 1972CrossRefMATHGoogle Scholar
  31. [31]
    P. Hajlasz and P. Koskela, Sobolev met Poincaré, Mem. Amer. Math. Soc. 145 (2000), no. 688.Google Scholar
  32. [32]
    L. I. Hedberg, Non-linear potentials and approximation in the mean by analytic functions, Math. Z. 129 (1972), 299–319.MathSciNetCrossRefMATHGoogle Scholar
  33. [33]
    L. I. Hedberg and T. H. Wolff, Thin sets in nonlinear potential theory, Ann. Inst. Fourier (Grenoble) 33 (1983), 161–187.MathSciNetCrossRefMATHGoogle Scholar
  34. [34]
    J. Heinonen, Lectures on Analysis on Metric Spaces, Springer, New York, 2001.CrossRefMATHGoogle Scholar
  35. [35]
    J. Heinonen, T Kilpeläinen, and J. Malý, Connectedness in fine topologies, Ann. Acad. Sci. Fenn. Ser. A I Math. 15 (1990), 107–123.MathSciNetCrossRefMATHGoogle Scholar
  36. [36]
    J. Heinonen, T. Kilpeläinen, and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, 2nd ed., Dover, Mineola, NY, 2006.MATHGoogle Scholar
  37. [37]
    J. Heinonen and P. Koskela, Quasiconformal maps in metric spaces with controlled geometry, Acta Math. 181 (1998), 1–61.MathSciNetCrossRefMATHGoogle Scholar
  38. [38]
    J. Heinonen, P. Koskela, N. Shanmugalingam, and J. T. Tyson, Sobolev Spaces on Metric Measure Spaces, Cambridge Univ. Press, Cambridge, 2015.CrossRefMATHGoogle Scholar
  39. [39]
    S. Keith, Measurable differentiable structures and the Poincaré inequality, Indiana Univ. Math. J. 53 (2004), 1127–1150.MathSciNetCrossRefMATHGoogle Scholar
  40. [40]
    T. Kilpeläinen and J. Malý, The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math. 172 (1994), 137–161.MathSciNetCrossRefMATHGoogle Scholar
  41. [41]
    J. Kinnunen and V. Latvala, Fine regularity of superharmonic functions on metric spaces, Future Trends in Geometric Function Theory, Univ. Jyväskylä, Jyväskylä, 2003, pp. 157–167.Google Scholar
  42. [42]
    J. Kinnunen and O. Martio, Nonlinear potential theory on metric spaces, Illinois J. Math. 46 (2002), 857–883.MathSciNetMATHGoogle Scholar
  43. [43]
    J. Kinnunen and N. Shanmugalingam, Regularity of quasi-minimizers on metric spaces, Manuscripta Math. 105 (2001), 401–423.MathSciNetCrossRefMATHGoogle Scholar
  44. [44]
    R. Korte, A Caccioppoli estimate and fine continuity for superminimizers on metric spaces, Ann. Acad. Sci. Fenn. Math. 33 (2008), 597–604.MathSciNetMATHGoogle Scholar
  45. [45]
    P. Koskela and P. MacManus, Quasiconformal mappings and Sobolev spaces, Studia Math. 131 (1998), 1–17.MathSciNetMATHGoogle Scholar
  46. [46]
    V. Latvala, Finely superharmonic functions of degenerate elliptic equations, Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes 96 (1994).Google Scholar
  47. [47]
    J. Lukeš, J. Malý, and L. Zajíček, Fine Topology Methods in Real Analysis and Potential Theory, Springer, Berlin–Heidelberg, 1986CrossRefMATHGoogle Scholar
  48. [48]
    J. Malý and W. P. Ziemer, Fine Regularity of Solutions of Elliptic Partial Differential Equations, Amer. Math. Soc., Providence, RI, 1997.CrossRefMATHGoogle Scholar
  49. [49]
    P. Mikkonen, On the Wolff potential and quasilinear elliptic equations involving measures, Ann. Acad. Sci. Fenn. Math. Diss. 104 (1996).Google Scholar
  50. [50]
    N. Shanmugalingam, Newtonian spaces: An extension of Sobolev spaces to metric measure spaces, Rev. Mat. Iberoam. 16 (2000), 243–279.MathSciNetCrossRefMATHGoogle Scholar
  51. [51]
    N. Shanmugalingam, Harmonic functions on metric spaces, Illinois J. Math. 45 (2001), 1021–1050.MathSciNetMATHGoogle Scholar
  52. [52]
    N. Wiener, The Dirichlet problem, J. Math. Phys. 3 (1924), 127–146.CrossRefMATHGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Department of MathematicsLinköpings Universitet LinköpingSweden
  2. 2.Department of Physics and MathematicsUniversity of Eastern FinlandJoensuuFinland

Personalised recommendations