Abstract
We study harmonic functions on general weighted graphs which allow for a compatible intrinsic metric. We prove an \(L^{p}\) Liouville type theorem which is a quantitative integral \(L^{p}\) estimate of harmonic functions analogous to Karp’s theorem for Riemannian manifolds. As corollaries we obtain Yau’s \(L^{p}\)-Liouville type theorem on graphs, identify the domain of the generator of the semigroup on \(L^{p}\) and get a criterion for recurrence. As a side product, we show an analogue of Yau’s \(L^{p}\) Caccioppoli inequality. Furthermore, we derive various Liouville type results for harmonic functions on graphs and harmonic maps from graphs into Hadamard spaces.
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Burago, D., Burago, Yu., Ivanov, S.: A Course in Metric Geometry. Number 33 in Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (2001)
Bridson, M.R., Haefliger, A.: Metric Spaces of Non-Positive Curvature. Number 319 in Grundlehren der Mathematischen Wissenschaften. Springer, Berlin (1999)
Bauer, F., Hua, B., Keller, M.: On the \(l^p\) spectrum of Laplacians on graphs. Adv. Math. 248, 717–735 (2013)
Bauer, F., Keller, M., Wojciechowski, R.: Cheeger inequalities for unbounded graph Laplacians. J. Eur. Math. Soc. (to appear)
Coulhon, T., Grigor’yan, A.: Random walks on graphs with regular volume growth. Geom. Funct. Anal. 8(4), 656–701 (1998)
Cheng, S.Y.: Liouville theorem for harmonic maps. In: Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979), number XXXVI in Proceedings of Symposia in Pure Mathematics, pp. 147–151. Amer. Math, Soc, Providence, RI (1980)
Chen, M.: From Markov Chains to Non-Equilibrium Particle Systems. World Scientific Publishing Co. Inc., River Edge (2004)
Chung, L.O.: Existence of harmonic \(L^1\) functions in complete Riemannian manifolds. Proc. Am. Math. Soc. 88(3), 531–532 (1983)
Cheng, S.Y., Tam, L.F., Wan, T.Y.H.: Harmonic maps with finite total energy. Proc. Am. Math. Soc. 124(1), 275–284 (1996)
Dodziuk, J., Karp, L.: Spectral and function theory for combinatorial Laplacians. In: Geometry of Random Motion, Volume 73 of Contemporary Mathematics, pp. 25–40. Amer. Math, Soc, Ithaca (1987)
Eells, J., Sampson, J.H.: Harmonic mappings of Riemannian manifolds. Am. J. Math. 86, 109–160 (1964)
Frank, R.L., Lenz, D., Wingert, D.: Intrinsic metrics for non-local symmetric Dirichlet forms and applications to spectral theory. J. Funct. Anal. (to appear)
Folz, M.: Gaussian upper bounds for heat kernels of continuous time simple random walks. Electron. J. Probab. 16(62), 1693–1722 (2011)
Folz, M.: Volume growth and stochastic completeness of graphs. Trans. Am. Math. Soc. (2012) (to appear)
Grigor’yan, A., Huang, X., Masamune, J.: On stochastic completeness of jump processes. Math. Z. 271(3–4), 1211–1239 (2012)
Grigor’yan, A.: Stochastically complete manifolds and summable harmonic functions (translation in Math. USSR-Izv. 33 (1989), no. 2, 425–432). Izv. Akad. Nauk SSSR Ser. Mat., 52 (5), 1102–1108 (1988)
Grigor’yan, A.: Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds. Bull. Am. Math. Soc. (N.S.), 36(2), 135–249 (1999)
Grigor’yan, A.: Analysis on Graphs. University Bielefeld, Lecture Notes (2009)
Gromov, M., Schoen, R.: Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one. Inst. Hautes Études Sci. Publ. Math. 76, 165–246 (1992)
Hua, B., Jost, J.: \(l^q\) harmonic functions on graphs. arXiv:1301.3403 (2013)
Hildebrandt, S., Jost, J., Widman, K.O.: Harmonic mappings and minimal submanifolds. Invent. Math., 62(2), 269–298 (1980/1981)
Huang, H., Kendall, W.S.: Correction note to ‘martingales on manifolds and harmonic maps’. Stoch. Stoch. Rep. 37(4), 253–257 (1991)
Haeseler, S., Keller, M., Lenz, D., Wojciechowski, R.: Laplacians on infinite graphs: Dirichlet and Neumann boundary conditions. J. Spectr. Theory 2(4), 397–432 (2012)
Huang, X., Keller, M., Masamune, J., Wojciechowski, R.K.: A note on self-adjoint extensions of the Laplacian on weighted graphs. J. Funct. Anal. 265(8), 1556–1578 (2013)
Holopainen, I., Soardi, P.M.: A strong Liouville theorem for \(p\)-harmonic functions on graphs. Ann. Acad. Sci. Fenn. Math. 22(1), 205–226 (1997)
Izeki, H., Nayatani, S.: Combinatorial harmonic maps and discrete-group actions on hadamard spaces. Geom. Dedic. 114, 147–188 (2005)
Jost, J.: Equilibrium maps between metric spaces. Calc. Var. Partial Differ. Equ. 2(2), 173–204 (1994)
Jost, J.: Generalized Dirichlet forms and harmonic maps. Calc. Var. Partial Differ. Equ. 5(1), 1–19 (1997)
Jost, J.: Nonpositive Curvature: Geometric and Analytic Aspects. Lectures in Mathematics, ETH Zürich. Birkhäuser, Basel (1997)
Jost, J.: Nonlinear Dirichlet Forms. New Directions in Dirichlet Forms. AMS and International Press, pp. 1–47 (1998)
Jost, J., Todjihounde, L.: Harmonic nets in metric spaces. Pac. J. Math. 231(2), 437–444 (2007)
Karp, L.: Subharmonic functions on real and complex manifolds. Math. Z. 179(4), 535–554 (1982)
Kendall, W.S.: Martingales on manifolds and harmonic maps. In: The Geometry of Random Motion (Ithaca, N.Y., 1987), Number 73 in Contemporary Mathematics, pp. 121–157. American Mathematical Society, Providence, RI (1988)
Kendall, W.S.: Probability, convexity, and harmonic maps with small image. I. Uniqueness and fine existence. Proc. Lond. Math. Soc. (3), 61(2), 371–406 (1990)
Keller, M., Lenz, D.: Dirichlet forms and stochastic completeness of graphs and subgraphs. J. Reine Angew. Math. 666, 189–223 (2012)
Keller, M., Lenz, D., Wojciechowski, R.K.: Volume growth, spectrum and stochastic completeness of infinite graphs. Math. Z. 274(3), 905–932 (2013)
Korevarr, N.J., Schoen, R.M.: Sobolev spaces and harmonic maps for metric space targets. Comm. Anal. Geom. 1, 561–659 (1993)
Korevarr, N.J., Schoen, R.M.: Global existence theorems for harmonic maps to non-locally compact spaces. Comm. Anal. Geom. 5, 333–387 (1997)
Kotani, M., Sunada, T.: Standard realizations of crystal lattices via harmonic maps. Trans. Am. Math. Soc. 353(1), 1–20 (2001)
Kuwae, K., Sturm, K.T.: On a Liouville theorem for harmonic maps to convex spaces via Markov chains. In: Proceedings of RIMS Workshop on Stochastic Analysis and Applications, number B6 in RIMS Kokyuroku Bessatsu, pp. 177–191. Res. Inst. Math. Sci. (RIMS), Kyoto (2008)
Li, P., Schoen, R.: Lp and mean value properties of subharmonic functions on Riemannian manifolds. Acta Math. 153(3–4), 279–301 (1984)
Li, P., Wang, J.: Convex hull properties of harmonic maps. J. Differ. Geom. 48(3), 497–530 (1998)
Lin, Y., Xi, L.: Lipschitz property of harmonic function on graphs. J. Math. Anal. Appl. 366(2), 673–678 (2010)
Masamune, J.: A Liouville Property and its application to the Laplacian of an infinite graph. In: Spectral Analysis in Geometry and Number Theory, Volume 484 of Contemporary Mathematics, pp. 103–115. Amer. Math, Soc, Providence, RI (2009)
Masamune, J., Uemura, T., Wang, J.: On the conservativeness and the recurrence of symmetric jump-diffusions. J. Funct. Anal. 263(12), 3984–4008 (2012)
Rigoli, M., Salvatori, M., Vignati, M.: Subharmonic functions on graphs. Israel J. Math. 99, 1–27 (1997)
Schmidt, M.: Global properties of Dirichlet forms on discrete spaces. Diplomarbeit, arXiv:1201.3474 (2012)
Soardi, P.: Potential Theory on Infinite Networks, Volume 1590 of Lecture Notes in Mathematics. Springer, Berlin (1994)
Sturm, K.-T.: Analysis on local Dirichlet spaces. I. Recurrence, conservativeness and \(L^p\)-Liouville properties. J. Reine Angew. Math. 456, 173–196 (1994)
Sturm, K.T.: Nonlinear markov operators associated with symmetric markov kernels and energy minimizing maps between singular spaces. Calc. Var. Partial Differ. Equ. 12(4), 317–357 (2001)
Sturm, K.T.: Probability measures on metric spaces of nonpositive curvature. In: Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces (Paris, 2002), Number 338 in Contemporary Mathematics, pp. 357–390. Amer. Math, Soc, Providence, RI (2003)
Sturm, K.T.: A semigroup approach to harmonic maps. Potential Anal. 23(3), 225–277 (2005)
Schoen, R., Yau, S.T.: Harmonic maps and the topology of stable hypersurfaces and manifolds with non-negative Ricci curvature. Comment. Math. Helv. 51(3), 333–341 (1976)
Tam, L.F.: Liouville properties of harmonic maps. Math. Res. Lett. 2(6), 719–735 (1995)
Woess, W.: Random Walks on Infinite Graphs and Groups. Number 138 in Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (2000)
Wojciechowski, R.: Heat kernel and essential spectrum of infinite graphs. Indiana Univ. Math. J. 58(3), 1419–1441 (2009)
Wojciechowski, R.: Stochastically incomplete manifolds and graphs. Random Walks Bound. Spectra Prog. Probab. 64, 163–179 (2011)
Yau, S.T.: Some function-theoretic properties of complete Riemannian manifold and their applications to geometry. Indiana Univ. Math. J. 25(7), 659–670 (1976)
Acknowledgments
BH thanks Jürgen Jost for inspiring discussions on \(L^p\) Liouville theorem and constant support, and acknowledges the financial support from the funding of the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no 267087. MK enjoyed discussions with Gabor Lippner, Dan Mangoubi, Marcel Schmidt and Radosław Wojciechowski on the subject and acknowledges the financial support of the German Science Foundation (DFG), Golda Meir Fellowship, the Israel Science Foundation (grant no. 1105/10 and no. 225/10) and BSF grant no. 2010214.
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Communicated by J. Jost.