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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory, Springer, Berlin–Heidelberg, 1996.
D. R. Adams and J. L. Lewis, Fine and quasiconnectedness in nonlinear potential theory, Ann. Inst. Fourier (Grenoble) 35 (1985), 57–73.
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.
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.
A. Björn, Characterizations of p-superharmonic functions on metric spaces, Studia Math. 169 (2005), 45–62.
A. Björn, A weak Kellogg property for quasiminimizers, Comment. Math. Helv. 81 (2006), 809–825.
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.
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.
A. Björn and J. Björn, Nonlinear Potential Theory on Metric Spaces, European Math. Soc., Zürich, 2011.
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.
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.
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.
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.
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.
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.
A. Björn and N. Marola, Moser iteration for (quasi)minimizers on metric spaces, Manuscripta Math. 121 (2006), 339–366.
J. Björn, Boundary continuity for quasiminimizers on metric spaces, Illinois J. Math. 46 (2002), 383–403.
J. Björn, Wiener criterion for Cheeger p-harmonic functions on metric spaces, Potential Theory in Matsue, Math. Soc. Japan, Tokyo, 2006, pp. 103–115.
J. Björn, Fine continuity on metric spaces, Manuscripta Math. 125 (2008), 369–381.
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.
M. Brelot, Sur la théorie moderne du potentiel, C. R. Acad. Sci. Paris 209 (1939), 828–830.
M. Brelot, Points irréguliers et transformations continues en théorie du potentiel J. Math. Pures Appl. (9) 19 (1940), 319–337.
M. Brelot, Sur les ensembles effilés, Bull. Sci. Math. (2) 68 (1944), 12–36.
M. Brelot On Topologies and Boundaries in Potential Theory, Springer, Berlin–Heidelberg, 1971.
H. Cartan, Théorie générale du balayage en potentiel newtonien Ann. Univ. Grenoble Sect. Sci. Math. Phys. (N.S.) 22 (1946), 221–280.
J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal. 9 (1999), 428–517.
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.
B. Franchi, P. Hajlasz and P. Koskela, Definitions of Sobolev classes on metric spaces, Ann. Inst. Fourier (Grenoble) 49 (1999), 1903–1924.
B. Fuglede, The quasi topology associated with a countably subadditive set function, Ann. Inst. Fourier (Grenoble) 21 (1971), 123–169.
B. Fuglede, Finely Harmonic Functions, Springer, Berlin–New York, 1972
P. Hajlasz and P. Koskela, Sobolev met Poincaré, Mem. Amer. Math. Soc. 145 (2000), no. 688.
L. I. Hedberg, Non-linear potentials and approximation in the mean by analytic functions, Math. Z. 129 (1972), 299–319.
L. I. Hedberg and T. H. Wolff, Thin sets in nonlinear potential theory, Ann. Inst. Fourier (Grenoble) 33 (1983), 161–187.
J. Heinonen, Lectures on Analysis on Metric Spaces, Springer, New York, 2001.
J. Heinonen, T Kilpeläinen, and J. Malý, Connectedness in fine topologies, Ann. Acad. Sci. Fenn. Ser. A I Math. 15 (1990), 107–123.
J. Heinonen, T. Kilpeläinen, and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, 2nd ed., Dover, Mineola, NY, 2006.
J. Heinonen and P. Koskela, Quasiconformal maps in metric spaces with controlled geometry, Acta Math. 181 (1998), 1–61.
J. Heinonen, P. Koskela, N. Shanmugalingam, and J. T. Tyson, Sobolev Spaces on Metric Measure Spaces, Cambridge Univ. Press, Cambridge, 2015.
S. Keith, Measurable differentiable structures and the Poincaré inequality, Indiana Univ. Math. J. 53 (2004), 1127–1150.
T. Kilpeläinen and J. Malý, The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math. 172 (1994), 137–161.
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.
J. Kinnunen and O. Martio, Nonlinear potential theory on metric spaces, Illinois J. Math. 46 (2002), 857–883.
J. Kinnunen and N. Shanmugalingam, Regularity of quasi-minimizers on metric spaces, Manuscripta Math. 105 (2001), 401–423.
R. Korte, A Caccioppoli estimate and fine continuity for superminimizers on metric spaces, Ann. Acad. Sci. Fenn. Math. 33 (2008), 597–604.
P. Koskela and P. MacManus, Quasiconformal mappings and Sobolev spaces, Studia Math. 131 (1998), 1–17.
V. Latvala, Finely superharmonic functions of degenerate elliptic equations, Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes 96 (1994).
J. Lukeš, J. Malý, and L. Zajíček, Fine Topology Methods in Real Analysis and Potential Theory, Springer, Berlin–Heidelberg, 1986
J. Malý and W. P. Ziemer, Fine Regularity of Solutions of Elliptic Partial Differential Equations, Amer. Math. Soc., Providence, RI, 1997.
P. Mikkonen, On the Wolff potential and quasilinear elliptic equations involving measures, Ann. Acad. Sci. Fenn. Math. Diss. 104 (1996).
N. Shanmugalingam, Newtonian spaces: An extension of Sobolev spaces to metric measure spaces, Rev. Mat. Iberoam. 16 (2000), 243–279.
N. Shanmugalingam, Harmonic functions on metric spaces, Illinois J. Math. 45 (2001), 1021–1050.
N. Wiener, The Dirichlet problem, J. Math. Phys. 3 (1924), 127–146.
Author information
Authors and Affiliations
Corresponding author
Additional information
The first two authors were supported by the Swedish Research Council.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
About this article
Cite this article
Björn, A., Björn, J. & Latvala, V. The Cartan, Choquet and Kellogg properties for the fine topology on metric spaces. JAMA 135, 59–83 (2018). https://doi.org/10.1007/s11854-018-0029-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11854-018-0029-8