Skip to content
BY 4.0 license Open Access Published by De Gruyter Open Access November 28, 2022

Potential Theory on Gromov Hyperbolic Spaces

  • Matthias Kemper and Joachim Lohkamp EMAIL logo

Abstract

Gromov hyperbolic spaces have become an essential concept in geometry, topology and group theory. Herewe extend Ancona’s potential theory on Gromov hyperbolic manifolds and graphs of bounded geometry to a large class of Schrödinger operators on Gromov hyperbolic metric measure spaces, unifying these settings in a common framework ready for applications to singular spaces such as RCD spaces or minimal hypersurfaces. Results include boundary Harnack inequalities and a complete classification of positive harmonic functions in terms of the Martin boundary which is identified with the geometric Gromov boundary.

MSC 2010: 31C25; 31C35; 35J10; 51M10; 53A10

References

[1] H. Aikawa, Boundary Harnack principle and Martin boundary for a uniform domain, J. Math. Soc. Japan 53 no. 1 (2001), 119–145. MR 1800526. Available at https://mathscinet.ams.org/mathscinet-getitem?mr=1800526.Search in Google Scholar

[2] H. Aikawa, Potential analysis on nonsmooth domains—Martin boundary and boundary Harnack principle, in Complex Analysis and Potential Theory, CRM Proc. Lecture Notes 55, Amer. Math. Soc., Providence, RI, 2012, pp. 235–253. MR 2986906. Available at https://mathscinet.ams.org/mathscinet-getitem?mr=2986906.10.1090/crmp/055/18Search in Google Scholar

[3] S. Albeverio, Z.-M. Ma, and M. Röckner, Partitions of unity in Sobolev spaces over infinite-dimensional state spaces, J. Funct. Anal. 143 no. 1 (1997), 247–268. MR 1428125. 10.1006/jfan.1996.2968.10.1006/jfan.1996.2968Search in Google Scholar

[4] L. Ambrosio and S. Honda, Local spectral convergence in RCD*(K, N) spaces, Nonlinear Anal. 177 no. part A (2018), 1–23. MR 3865185. 10.1016/j.na.2017.04.003.Search in Google Scholar

[5] A. Ancona, Théorie du potentiel sur les graphes et les variétés, in École d’été de Probabilités de Saint-Flour XVIII—1988, Lecture Notes in Math. 1427, Springer, Berlin, 1990, pp. 1–112. MR 1100282. 10.1007/BFb0103041.Search in Google Scholar

[6] A. Ancona, On strong barriers and an inequality of Hardy for domains in ℝn, J. London Math. Soc. (2) 34 no. 2 (1986), 274–290. MR 856511. 10.1112/jlms/s2-34.2.274.10.1112/jlms/s2-34.2.274Search in Google Scholar

[7] A. Ancona, Negatively curved manifolds, elliptic operators, and the Martin boundary, Ann. of Math. 125 no. 3 (1987), 495–536. MR 890161. 10.2307/1971409.10.2307/1971409Search in Google Scholar

[8] A. Ancona, Positive harmonic functions and hyperbolicity, in Potential Theory Surveys and Problems (J. Král, J. Lukeš, I. Netuka, and J. Veselý, eds.), 1344, Springer, Berlin, Heidelberg, 1988, pp. 1–23. 10.1007/BFb0103341.Search in Google Scholar

[9] A. Ancona, On positive harmonic functions in cones and cylinders, Rev. Mat. Iberoam. 28 no. 1 (2012), 201–230. MR 2904138. 10.4171/RMI/674.10.4171/RMI/674Search in Google Scholar

[10] M. T. Anderson and R. Schoen, Positive harmonic functions on complete manifolds of negative curvature, Ann. of Math. 121 no. 3 (1985), 429–461. MR 794369. 10.2307/1971181.10.2307/1971181Search in Google Scholar

[11] M. T. Barlow and M. Murugan, Boundary Harnack principle and elliptic Harnack inequality, J. Math. Soc. Japan 71 no. 2 (2019), 383–412. MR 3943443. 10.2969/jmsj/77057705.Search in Google Scholar

[12] H. Bauer, Harmonische Räume und ihre Potentialtheorie, Lecture Notes in Mathematics, Vol. 22, Springer, Berlin-New York, 1966. MR 0210916. Available at https://mathscinet.ams.org/mathscinet-getitem?mr=0210916.10.1007/BFb0075360Search in Google Scholar

[13] M. Biroli, TheWiener test for Poincaré-Dirichlet forms, in Classical andModern Potential Theory and Applications (Chateau de Bonas, 1993), NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci. 430, Kluwer Acad. Publ., Dordrecht, 1994, pp. 93–104. MR 1321609. Available at https://mathscinet.ams.org/mathscinet-getitem?mr=1321609.10.1007/978-94-011-1138-6_9Search in Google Scholar

[14] M. Biroli and U. Mosco, A Saint-Venant type principle for Dirichlet forms on discontinuous media, Ann. Mat. Pura Appl. (4) 169 (1995), 125–181. MR 1378473. 10.1007/BF01759352.10.1007/BF01759352Search in Google Scholar

[15] M. Biroli and S. Marchi,Wiener criterion at the boundary related to p-homogeneous strongly local Dirichlet forms, Matematiche (Catania) 62 no. 2 (2007), 37–52. MR 2401177. Available at https://mathscinet.ams.org/mathscinet-getitem?mr=2401177.Search in Google Scholar

[16] A. Björn and J. Björn, Nonlinear Potential Theory on Metric Spaces, EMS Tracts in Mathematics 17, European Mathematical Society, Zürich, 2011. MR 2867756. 10.4171/099.10.4171/099Search in Google Scholar

[17] A. Björn and J. Björn, Local and semilocal Poincaré inequalities on metric spaces, J. Math. Pures Appl. (9) 119 (2018), 158–192. MR 3862146. 10.1016/j.matpur.2018.05.005.10.1016/j.matpur.2018.05.005Search in Google Scholar

[18] 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. MR 1869615. 10.1007/BF02788087.10.1007/BF02788087Search in Google Scholar

[19] S. Blachère and S. Brofferio, Internal diffusion limited aggregation on discrete groups having exponential growth, Probab. Theory Related Fields 137 no. 3-4 (2007), 323–343. MR 2278460. 10.1007/s00440-006-0009-2.Search in Google Scholar

[20] S. Blachère, P. Haïssinsky, and P. Mathieu, Harmonic measures versus quasiconformal measures for hyperbolic groups, Ann. Sci. Éc. Norm. Supér. (4) 44 no. 4 (2011), 683–721. MR 2919980. 10.24033/asens.2153.10.24033/asens.2153Search in Google Scholar

[21] J. Bliedtner and W. Hansen, Potential Theory, Universitext, Springer, Berlin, Heidelberg, 1986. MR 850715. 10.1007/978-3-642-71131-2.10.1007/978-3-642-71131-2_1Search in Google Scholar

[22] M. Bonk, Quasi-geodesic segments and Gromov hyperbolic spaces, Geom. Dedicata 62 no. 3 (1996), 281–298.MR1406442. 10.1007/BF00181569.Search in Google Scholar

[23] M. Bonk, J. Heinonen, and P. Koskela, Uniformizing Gromov Hyperbolic Spaces, Astérisque 270, Société mathématique de France, 2001. MR 1829896. Available at http://www.numdam.org/item/AST_2001__270__R1_0.Search in Google Scholar

[24] A. Borel and L. Ji, Compactifications of Symmetric and Locally Symmetric Spaces, Mathematics: Theory & Applications, Birkhäuser, Boston, 2006. MR 2189882. Available at https://mathscinet.ams.org/mathscinet-getitem?mr=2189882.Search in Google Scholar

[25] M. Brelot, Lectures on Potential Theory, second ed., Tata Institute of Fundamental Research Lectures on Mathematics 19, Tata Institute of Fundamental Research, Bombay, 1967. MR 0259146. Available at https://mathscinet.ams.org/mathscinetgetitem?mr=0259146.Search in Google Scholar

[26] M. R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Grundlehren der Mathematischen Wissenschaften 319, Springer, Berlin, 1999. MR 1744486. 10.1007/978-3-662-12494-9.10.1007/978-3-662-12494-9Search in Google Scholar

[27] S. Buyalo and V. Schroeder, Elements of Asymptotic Geometry, EMSMonographs inMathematics, EuropeanMathematical Society, Zürich, 2007. MR 2327160. 10.4171/036.10.4171/036Search in Google Scholar

[28] G. Choquet, Lectures on Analysis. Vol. I: Integration and Topological Vector Spaces, W. A. Benjamin, New York-Amsterdam, 1969. MR 0250011. Available at https://mathscinet.ams.org/mathscinet-getitem?mr=0250011.Search in Google Scholar

[29] C. Constantinescu and A. Cornea, Potential Theory on Harmonic Spaces, Springer-Verlag, New York-Heidelberg, 1972. MR 0419799. Available at https://mathscinet.ams.org/mathscinet-getitem?mr=0419799.Search in Google Scholar

[30] F. W. Gehring and B. G. Osgood, Uniform domains and the quasihyperbolic metric, J. Analyse Math. 36 (1979), 50–74. MR 581801. Available at https://mathscinet.ams.org/mathscinet-getitem?mr=581801.Search in Google Scholar

[31] É. Ghys and P. de la Harpe (eds.), Sur les groupes hyperboliques d’après Mikhael Gromov, Progress in Mathematics 83, Birkhäuser, Boston, 1990. MR 1086648.10.1007/978-1-4684-9167-8Search in Google Scholar

[32] N. Gigli, On the differential structure of metricmeasure spaces and applications, Mem. Amer.Math. Soc. 236 no. 1113 (2015), vi+91. MR 3381131. 10.1090/memo/1113.Search in Google Scholar

[33] N. Gigli, A. Mondino, and T. Rajala, Euclidean spaces as weak tangents of infinitesimally Hilbertian metric measure spaces with Ricci curvature bounded below, J. Reine Angew. Math. 705 (2015), 233–244. MR 3377394. 10.1515/crelle-2013-0052.Search in Google Scholar

[34] S. Giulini and W. Woess, The Martin compactification of the Cartesian product of two hyperbolic spaces, J. Reine Angew. Math. 444 (1993), 17–28. MR 1241792. 10.1515/crll.1993.444.17.Search in Google Scholar

[35] A. Grigoryan and J. Hu, Off-diagonal upper estimates for the heat kernel of the Dirichlet forms on metric spaces, Invent. Math. 174 no. 1 (2008), 81–126. MR 2430977. 10.1007/s00222-008-0135-9.10.1007/s00222-008-0135-9Search in Google Scholar

[36] A. Grigoryan and J. Hu, Heat kernels and Green functions on metric measure spaces, Canad. J. Math 66 no. 3 (2014), 641–699. Available at https://www.researchgate.net/profile/Jiaxin_Hu2/publication/264892240_Heat_Kernels_and_Green_Functions_on_Metric_Measure_Spaces/links/53f620a00cf2888a7492c12b.pdf.10.4153/CJM-2012-061-5Search in Google Scholar

[37] M. Gromov, Hyperbolic groups, in Essays in Group Theory, Math. Sci. Res. Inst. Publ. 8, Springer, New York, 1987, pp. 75–263. MR 919829. Available at https://mathscinet.ams.org/mathscinet-getitem?mr=919829.10.1007/978-1-4613-9586-7_3Search in Google Scholar

[38] P. Gyrya and L. Saloff-Coste, Neumann and Dirichlet heat kernels in inner uniform domains, Astérisque no. 336 (2011). MR 2807275. Available at http://www.ams.org/mathscinet-getitem?mr=2807275.Search in Google Scholar

[39] L. L. Helms, Introduction to Potential Theory, Pure and Applied Mathematics, Vol. XXII, Wiley, 1969. MR 0261018. Available at https://mathscinet.ams.org/mathscinet-getitem?mr=0261018.Search in Google Scholar

[40] R.-M. Hervé, Recherches axiomatiques sur la théorie des fonctions surharmoniques et du potentiel, Ann. Inst. Fourier (Grenoble) 12 (1962), 415–571. MR 139756. 10.5802/aif.125.10.5802/aif.125Search in Google Scholar

[41] J. T. Kemper, A boundary Harnack principle for Lipschitz domains and the principle of positive singularities, Comm. Pure Appl. Math. 25 (1972), 247–255. MR 293114. 10.1002/cpa.3160250303.10.1002/cpa.3160250303Search in Google Scholar

[42] J. Lehrbäck, Hardy inequalities and Assouad dimensions, J. Anal.Math. 131 (2017), 367–398.MR3631460. 10.1007/s11854-017-0013-8.10.1007/s11854-017-0013-8Search in Google Scholar

[43] J. L. Lewis, Uniformly fat sets, Trans. Amer. Math. Soc. 308 no. 1 (1988), 177–196. MR 946438. 10.2307/2000957.10.1090/S0002-9947-1988-0946438-4Search in Google Scholar

[44] J. Lierl and L. Saloff-Coste, Scale-invariant boundary Harnack principle in inner uniform domains, Osaka J. Math. 51 no. 3 (2014), 619–656. MR 3272609. Available at https://mathscinet.ams.org/mathscinet-getitem?mr=3272609.Search in Google Scholar

[45] P. A. Loeb and B. Walsh, The equivalence of Harnack’s principle and Harnack’s inequality in the axiomatic system of Brelot, Ann. Inst. Fourier (Grenoble) 15 no. fasc. 2 (1965), 597–600. MR 190360. Available at https://mathscinet.ams.org/mathscinet-getitem?mr=190360.Search in Google Scholar

[46] J. Lohkamp, Hyperbolic unfoldings of minimal hypersurfaces, Anal. Geom. Metr. Spaces 6 (2018), 96–128. MR 3849619. 10.1515/agms-2018-0006.10.1515/agms-2018-0006Search in Google Scholar

[47] J. Lohkamp, Potential theory on minimal hypersurfaces I: Singularities as Martin boundaries, Potential Analysis 53 no. 4 (2020), 1493–1528. MR 4159389. 10.1007/s11118-019-09815-6.10.1007/s11118-019-09815-6Search in Google Scholar

[48] J. Lohkamp, Potential theory on minimal hypersurfaces II: Hardy structures and Schrödinger operators, Potential Analysis 55 no. 4 (2021), 563–602. MR 4341062. 10.1007/s11118-020-09869-x.10.1007/s11118-020-09869-xSearch in Google Scholar

[49] Z. M. Ma and M. Röckner, Introduction to the Theory of (Nonsymmetric) Dirichlet Forms, Universitext, Springer-Verlag, Berlin, 1992. MR 1214375. 10.1007/978-3-642-77739-4.Search in Google Scholar

[50] J. R. Munkres, Topology, second ed., Prentice Hall, Upper Saddle River, NJ, 2000. MR 3728284. Available at https://mathscinet.ams.org/mathscinet-getitem?mr=3728284.Search in Google Scholar

[51] R. G. Pinsky, Positive Harmonic Functions and Diffusion, Cambridge Studies in Advanced Mathematics 45, Cambridge University Press, Cambridge, 1995. MR 1326606. 10.1017/CBO9780511526244.Search in Google Scholar

[52] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095. Available at https://mathscinet.ams.org/mathscinet-getitem?mr=0290095.Search in Google Scholar

[53] K. T. Sturm, Analysis on local Dirichlet spaces. III. The parabolic Harnack inequality, J. Math. Pures Appl. (9) 75 no. 3 (1996), 273–297. MR 1387522. Available at https://mathscinet.ams.org/mathscinet-getitem?mr=1387522.Search in Google Scholar

Received: 2022-04-05
Accepted: 2022-10-18
Published Online: 2022-11-28

© 2022 Matthias Kemper et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Downloaded on 10.6.2024 from https://www.degruyter.com/document/doi/10.1515/agms-2022-0147/html
Scroll to top button