Abstract
We studyL q-Liouville properties of nonnegativep-superharmonic and, respectively,p-subharmonic functions on a complete Riemannian manifoldM. In particular, we prove that everyp-harmonic functionu ∈L q (M) is constant ifq>p−1.
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[CY] S. Y. Cheng and S. T. Yau,Differential equations on Riemannian manifolds and their geometric applications, Communications on Pure and Applied Mathematics28 (1975), 333–354.
[CS] T. Coulhon and L. Saloff-Coste,Harnack inequality and hyperbolicity for the p-Laplacian with applications to quasiregular mappings, preprint.
[GW] R. Greene and H. Wu,Integrals of subharmonic functions on manifolds of nonnegative curvature, Inventiones Mathematicae27 (1974), 265–298.
[G1] A. Grigor'yan,On stochastically complete manifolds, Soviet Mathematics Doklady34 (1987), 310–313.
[G2] A. Grigor'yan,Stochastically complete manifolds and summable harmonic functions, Mathematics of the USSR-Izvestiya33 (1989), 425–432.
[G3] A. Grigor'yan,Analytic and geometric background of recurrence and nonexplosion of the Brownian motion on Riemannian manifolds, Bulletin of the American Mathematical Society36 (1999), 135–249.
[HKM] J. Heinonen, T. Kilpeläinen and O. Martio,Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Mathematical Monographs Clarendon Press, Oxford-New York-Tokyo, 1993.
[H1] I. Holopainen,Nonlinear potential theory and quasiregular mappings on Riemannian manifolds, Annales Academiae Scientiarum Fennicae. Series A I. Mathematica Dissertationes74 (1990), 1–45.
[H2] I. Holopainen,Positive solutions of quasilinear elliptic equations on Riemannian manifolds, Proceedings of the London Mathematical Society (3)65 (1992), 651–672.
[H3] I. Holopainen,Rough isometries and p-harmonic functions with finite Dirichlet integral, Revista Matemática Iberoamericana10 (1994), 143–176.
[H4] I. Holopainen,Volume growth, Green's functions, and parabolicity of ends, Duke Mathematical Journal (to appear).
[HR] I. Holopainen and S. Rickman,Ricci curvature, Harnack functions, and Picard type theorems for quasiregular mappings, inAnalysis and Topology (C. Andreian-Cazacu, O. Lehto and Th. M. Rassias, ed.), World Scientific, Singapore, 1998, pp. 315–326.
[K1] L. Karp,Asymptotic behavior of solutions of elliptic equations I, II, Journal d'Analyse Mathématique39 (1981), 75–102, 103–115.
[K2] L. Karp,Subharmonic functions on real and complex manifolds, Mathematische Zeitschrift179 (1982), 535–554.
[K3] L. Karp,Subharmonic functions, harmonic mappings, and isometric immersions, inSeminar on Differential Geometry (S. T. Yau, ed.), Annals of Mathematics Studies, Vol. 102, Princeton University Press, Princeton, N.J., 1982, pp. 133–142.
[L] P. Li,Curvature and function theory on Riemannian manifolds, preprint.
[LS] P. Li and R. Schoen,L p and mean value properties of subharmonic functions on Riemannian manifolds, Acta Mathematica153 (1984), 279–301.
[RSV] M. Rigoli, M. Salvatori and M. Vignati,A note on p-subharmonic functions on complete manifolds, Manuscripta mathematica92 (1997), 339–359.
[S] J. Serrin,Local behavior of solutions of-quasilinear equations, Acta Mathematica111 (1964), 247–302.
[St] K.-T. Sturm,Analysis on local Dirichlet spaces I. Recurrence, conservativeness, and L p -Liouville properties, Journal für die Reine und Angewandte Mathematik456 (1994), 173–196.
[Y1] S. T. Yau,Harmonic functions on complete Riemannian manifolds, Communications on Pure and Applied Mathematics,28 (1975), 201–228.
[Y2] S. T. Yau,Some function-theoretic properties of complete Riemannian manifolds and their applications to geometry, Indiana University Mathematics Journal25 (1976), 659–670.
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Supported by the Academy of Finland, Project 6355.
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Holopainen, I. A sharpL q-Liouville theorem forp-harmonic functions. Isr. J. Math. 115, 363–379 (2000). https://doi.org/10.1007/BF02810597
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DOI: https://doi.org/10.1007/BF02810597