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Volume growth, capacity estimates, p-parabolicity and sharp integrability properties of p-harmonic Green functions

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Abstract

In a complete metric space equipped with a doubling measure supporting a p-Poincaré inequality, we prove sharp growth and integrability results for p-harmonic Green functions and their minimal p-weak upper gradients. We show that these properties are determined by the growth of the underlying measure near the singularity. Corresponding results are obtained also for more general p-harmonic functions with poles, as well as for singular solutions of elliptic differential equations in divergence form on weighted Rn and on manifolds.

The proofs are based on a new general capacity estimate for annuli, which implies precise pointwise estimates for p-harmonic Green functions. The capacity estimate is valid under considerably milder assumptions than above. We also use it, under these milder assumptions, to characterize singletons of zero capacity and the p-parabolicity of the space. This generalizes and improves earlier results that have been important especially in the context of Riemannian manifolds.

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References

  1. T. Adamowicz and N. Shanmugalingam, Non-conformal Loewner type estimates for modulus of curve families, Ann. Acad. Sci. Fenn. Math. 35 (2010), 609–626.

    Article  MathSciNet  MATH  Google Scholar 

  2. A. Björn, Characterizations of p-superharmonic functions on metric spaces, Studia Math. 169 (2005), 45–62.

    Article  MathSciNet  MATH  Google Scholar 

  3. A. Björn, A weak Kellogg property for quasiminimizers, Comment. Math. Helv. 81 (2006), 809–825.

    Article  MathSciNet  MATH  Google Scholar 

  4. 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.

    MathSciNet  MATH  Google Scholar 

  5. A. Björn and J. Björn, Nonlinear Potential Theory on Metric Spaces, European Mathematical Society, Zürich, 2011.

    Book  MATH  Google Scholar 

  6. A. Björn and J. Björn, Power-type quasiminimizers, Ann. Acad. Sci. Fenn. Math. 36 (2011), 301–319.

    Article  MathSciNet  MATH  Google Scholar 

  7. A. Björn and J. Björn, The variational capacity with respect to nonopen sets in metric spaces, Potential Anal. 40 (2014), 57–80.

    Article  MathSciNet  MATH  Google Scholar 

  8. A. Björn and J. Björn, Local and semilocal Poincaré inequalities on metric spaces, J. Math. Pures Appl. 119 (2018), 158–192.

    Article  MathSciNet  MATH  Google Scholar 

  9. A. Björn, J. Björn and J. Lehrbäck, The annular decay property and capacity estimates for thin annuli, Collect. Math. 68 (2017), 229–241.

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Björn, J. Björn and J. Lehrbäck, Sharp capacity estimates for annuli in weighted Rnand metric spaces, Math. Z. 286 (2017), 1173–1215.

    Article  MathSciNet  MATH  Google Scholar 

  11. A. Björn, J. Björn and J. Lehrbäck, Existence and almost uniqueness for p-harmonic Green functions in metric spaces, J. Differential Equations 269 (2020), 6602–6640.

    Article  MathSciNet  MATH  Google Scholar 

  12. A. Björn, J. Björn and A. Mwasa, Resolutivity and invariance for the Perron method for degenerate equations of divergence type, J. Math. Anal. Appl. 509 (2022), Article no. 125937.

  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.

    Article  MathSciNet  MATH  Google Scholar 

  14. J. Björn, Poincaré inequalities for powers and products of admissible weights, Ann. Acad. Sci. Fenn. Math. 26 (2001), 175–188.

    MathSciNet  MATH  Google Scholar 

  15. 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.

    Article  MathSciNet  MATH  Google Scholar 

  16. L. Capogna, D. Danielli and N. Garofalo, Capacitary estimates and the local behavior of solutions of nonlinear subelliptic equations, Amer. J. Math. 118 (1996), 1153–1196.

    Article  MathSciNet  MATH  Google Scholar 

  17. S. Chanillo and R. L. Wheeden, Existence and estimates of Green’s function for degenerate elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 15 (1988), 309–340.

    MathSciNet  MATH  Google Scholar 

  18. T. Coulhon, I. Holopainen and L. Saloff-Coste, Harnack inequality and hyperbolicity for subelliptic p-Laplacians with applications to Picard type theorems, Geom. Funct. Anal. 11 (2001), 1139–1191.

    Article  MathSciNet  MATH  Google Scholar 

  19. D. Danielli, N. Garofalo and N. Marola, Local behavior of p-harmonic Green functions in metric spaces, Potential Anal. 32 (2010), 343–362.

    Article  MathSciNet  MATH  Google Scholar 

  20. E. B. Fabes, D. Jerison and C. E. Kenig, The Wiener test for degenerate elliptic equations, Ann. Inst. Fourier (Grenoble) 32 (1982), no. 3, 151–182.

    Article  MathSciNet  MATH  Google Scholar 

  21. N. Garofalo and N. Marola, Sharp capacitary estimates for rings in metric spaces, Houston J. Math. 36 (2010), 681–695.

    MathSciNet  MATH  Google Scholar 

  22. M. Giaquinta and E. Giusti, On the regularity of the minima of variational integrals, Acta Math. 148 (1982), 31–46.

    Article  MathSciNet  MATH  Google Scholar 

  23. M. Giaquinta and E. Giusti, Quasi-minima, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), 79–107.

    Article  MathSciNet  MATH  Google Scholar 

  24. A. Grigor’yan, Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. Amer. Math. Soc. 36 (1999), 135–249.

    Article  MathSciNet  MATH  Google Scholar 

  25. P. Hajłasz and P. Koskela, Sobolev met Poincaré, Mem. Amer. Math. Soc. 145 (2000), no. 688.

    MATH  Google Scholar 

  26. T. Hara, The Wolff potential estimate for solutions to elliptic equations with signed data, Manuscripta Math. 150 (2016), 45–58.

    Article  MathSciNet  MATH  Google Scholar 

  27. T. Hara, Wolff potential estimates for Cheeger p-harmonic functions, Collect. Math. 69 (2018), 407–426.

    Article  MathSciNet  MATH  Google Scholar 

  28. J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, 2nd ed., Dover, Mineola, NY, 2006.

    MATH  Google Scholar 

  29. J. Heinonen and P. Koskela, Quasiconformal maps in metric spaces with controlled geometry, Acta Math. 181 (1998), 1–61.

    Article  MathSciNet  MATH  Google Scholar 

  30. J. Heinonen, P. Koskela, N. Shanmugalingam and J. T. Tyson, Sobolev Spaces on Metric Measure Spaces, Cambridge University Press, Cambridge, 2015.

    Book  MATH  Google Scholar 

  31. I. Holopainen, Nonlinear potential theory and quasiregular mappings on Riemannian manifolds, Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes 74 (1990).

  32. I. Holopainen, Positive solutions of quasilinear elliptic equations on Riemannian manifolds, Proc. Lond. Math. Soc. (3) 65 (1992), 651–672.

    Article  MathSciNet  MATH  Google Scholar 

  33. I. Holopainen, Volume growth, Green’s functions, and parabolicity of ends, Duke Math. J. 97 (1999), 319–346.

    Article  MathSciNet  MATH  Google Scholar 

  34. I. Holopainen and P. Koskela, Volume growth and parabolicity, Proc. Amer. Math. Soc. 129 (2001), 3425–3435.

    Article  MathSciNet  MATH  Google Scholar 

  35. I. Holopainen and N. Shanmugalingam, Singular functions on metric measure spaces, Collect. Math. 53 (2002), 313–332.

    MathSciNet  MATH  Google Scholar 

  36. S. Keith and X. Zhong, The Poincaré inequality is an open ended condition, Ann. of Math. (2) 167 (2008), 575–599.

    Article  MathSciNet  MATH  Google Scholar 

  37. V. M. Kesel’man and V. A. Zorich, On the conformal type of a Riemannian manifold, Funktsional. Anal. i Prilozhen. 30 (1996), no. 2, 40–55, 96; English translation in Funct. Anal. Appl. 30 (1996), 106–117.

    MathSciNet  MATH  Google Scholar 

  38. S. Kichenassamy and L. Véron, Singular solutions of the p-Laplace equation, Math. Ann. 275 (1986), 599–615; Erratum, ibid. 277 (1987), 352.

    Article  MathSciNet  MATH  Google Scholar 

  39. T. Kilpeläinen and J. Malý, Degenerate elliptic equations with measure data and nonlinear potentials, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 19 (1992), 591–613.

    MathSciNet  MATH  Google Scholar 

  40. T. Kilpeläinen and J. Malý, The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math. 172 (1994), 137–161.

    Article  MathSciNet  MATH  Google Scholar 

  41. J. Kinnunen and O. Martio, Nonlinear potential theory on metric spaces, Illinois J. Math. 46 (2002), 857–883.

    Article  MathSciNet  MATH  Google Scholar 

  42. J. Kinnunen and O. Martio, Sobolev space properties of superharmonic functions on metric spaces, Results Math. 44 (2003), 114–129.

    Article  MathSciNet  MATH  Google Scholar 

  43. J. Kinnunen and N. Shanmugalingam, Regularity of quasi-minimizers on metric spaces, Manuscripta Math. 105 (2001), 401–423.

    Article  MathSciNet  MATH  Google Scholar 

  44. J. Kinnunen and N. Shanmugalingam, Polar sets on metric spaces, Trans. Amer. Math. Soc. 358 (2006), 11–37.

    Article  MathSciNet  MATH  Google Scholar 

  45. P. Koskela and P. MacManus, Quasiconformal mappings and Sobolev spaces, Studia Math. 131 (1998), 1–17.

    MathSciNet  MATH  Google Scholar 

  46. T. Kuusi and G. Mingione, Guide to nonlinear potential estimates, Bull. Math. Sci. 4 (2014), 1–82.

    Article  MathSciNet  MATH  Google Scholar 

  47. T. Kuusi and G. Mingione, Vectorial nonlinear potential theory, J. Eur. Math. Soc. (JEMS) 20 (2018), 929–1004.

    Article  MathSciNet  MATH  Google Scholar 

  48. P. Lindqvist, On the definition and properties of p-superharmonic functions, J. Reine Angew. Math. 365 (1986), 67–79.

    MathSciNet  MATH  Google Scholar 

  49. W. Littman, G. Stampacchia and H. F. Weinberger, Regular points for elliptic equations with discontinuous coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 17 (1963), 43–77.

    MathSciNet  MATH  Google Scholar 

  50. V. G. Maz’ya, On the continuity at a boundary point of solutions of quasi-linear elliptic equations, Vestnik Leningrad. Univ. Mat. Mekh. Astronom. 25 (1970), no. 13, 42–55; English translation in Vestnik Leningrad Univ. Math. 3 (1976), 225–242.

    MathSciNet  MATH  Google Scholar 

  51. V. G. Maz’ya and V. P. Havin, A nonlinear analogue of the Newtonian potential, and metric properties of (p, l)-capacity, Dokl. Akad. Nauk SSSR 194 (1970), 770–773; English translation in Soviet Math. Dokl. 11 (1970), 1294–1298}.

    MathSciNet  Google Scholar 

  52. V. G. Maz’ya and V. P. Havin, A nonlinear potential theory, Uspekhi Mat. Nauk 27 (1972), no. 6, 67–138; English translation in Russian Math. Surveys 27 (1974), 71–148.

    MathSciNet  Google Scholar 

  53. P. Mikkonen, On the Wolff potential and quasilinear elliptic equations involving measures, Ann. Acad. Sci. Fenn. Math. Diss. 104 (1996).

  54. J. Serrin, Local behavior of solutions of quasi-linear equations, Acta Math. 111 (1964), 247–302.

    Article  MathSciNet  MATH  Google Scholar 

  55. N. Shanmugalingam, Newtonian spaces: An extension of Sobolev spaces to metric measure spaces, Rev. Mat. Iberoam. 16 (2000), 243–279.

    Article  MathSciNet  MATH  Google Scholar 

  56. H. Svensson, Radial weights in Rnand local dimensions, Master’s thesis, Linköping University, Linköping, 2014, http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-107173.

    Google Scholar 

  57. S. Svensson, Local dimensions and radial weights in Rn, Bachelor’s thesis, Linköping University, Linköping, 2017, http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-138114.

    Google Scholar 

  58. N. T. Varopoulos, Potential theory and diffusion on Riemannian manifolds, in Conference on Harmonic Analysis in Honor of Antoni Zygmund (Chicago, IL, 1981), Wadsworth, Belmont, CA, 1983, Vol. II, pp. 821–837.

    MATH  Google Scholar 

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Acknowledgement

A. B. and J. B. were supported by the Swedish Research Council, grants 2016-03424 resp., 621-2014-3974 and 2018-04106. Part of this research was done during several visits of J. L. to Linköping University; he is grateful for the support and hospitality.

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Authors and Affiliations

  1. Department of Mathematics, Linköping University, SE-581 83, Linköping, Sweden

    Anders Björn & Jana Björn

  2. Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35 (MaD), FI-40014, Jyväskylä, Finland

    Juha Lehrbäck

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  1. Anders Björn
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Björn, A., Björn, J. & Lehrbäck, J. Volume growth, capacity estimates, p-parabolicity and sharp integrability properties of p-harmonic Green functions. JAMA 150, 159–214 (2023). https://doi.org/10.1007/s11854-023-0273-4

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