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A converse to the mean value property on homogeneous trees


Authors: Massimo A. Picardello and Wolfgang Woess
Journal: Trans. Amer. Math. Soc. 311 (1989), 209-225
MSC: Primary 31C20; Secondary 31C35, 60J15, 60J50
DOI: https://doi.org/10.1090/S0002-9947-1989-0974775-7
MathSciNet review: 974775
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Abstract: The homogeneous tree $ {\mathbf{T}}$ of degree $ q + 1\quad (q \geq 2)$ may be considered as a discrete analogue of the open unit disc $ {\mathbf{D}}$. On $ {\mathbf{D}}$, every harmonic function satisfies the mean value property (MVP) at every point. Conversely, positive functions on $ {\mathbf{D}}$ having the MVP with respect to a ball with specified radius at each point of $ {\mathbf{D}}$ are harmonic under certain assumptions concerning the radius function: results of this type are due to J. R. Baxter, W. Veech and others. Here we consider harmonic functions on $ {\mathbf{T}}$ with respect to a natural choice of a discrete Laplacian: the analogous MVP is true in this setting. We present a Lipschitz-type condition on the radius function (which now has integer values and refers to the discrete metric of $ {\mathbf{T}}$) under which harmonicity holds for positive functions whose value at each point is the mean of its values over the ball of the radius assigned to this point. The method is based upon our previous results concerning the geometrical realization of Martin boundaries of certain transition operators as the space of ends of the underlying graph.


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  • [Ba1] J. R. Baxter, Restricted mean values and harmonic functions, Trans. Amer. Math. Soc. 167 (1972), 451-463. MR 0293112 (45:2191)
  • [Ba2] -, Harmonic functions and mass cancellation, Trans. Amer. Math. Soc. 245 (1978), 375-384. MR 511416 (80a:60085)
  • [Ca1] P. Cartier, Fonctions harmoniques sur un arbre, Symposia Math. 9 (1972), 203-270. MR 0353467 (50:5950)
  • [Ca2] -, Harmonic analysis on trees, Proc. Sympos. Pure Math., vol. 26, Amer. Math. Soc., Providence, R.I., 1972, pp. 419-424. MR 0338272 (49:3038)
  • [De] Y. Derriennic, Marche aléatoire sur le groupe libre et frontière de Martin, Z. Wahrsch. Verw. Gebiete 32 (1975), 261-276. MR 0388545 (52:9381)
  • [Do] J. Dodziuk, Difference equations, isoperimetric inequality and transience of certain random walks, Trans. Amer. Math. Soc. 284 (1984), 787-794. MR 743744 (85m:58185)
  • [DK] J. Dodziuk and W. S. Kendall. Combinatorial Laplacians and isoperimetric inequality, From Local Times to Global Geometry, K. D. Ellworthy, ed., Pitman Res. Notes Math. Ser. 150, 1986, pp. 68-74. MR 894523 (88h:58118)
  • [DM] E. B. Dynkin and M. B. Malyutov, Random walks on groups with a finite number of generators, Soviet Math. Dokl. 2 (1961), 399-402.
  • [Fe] W. Feller, Boundaries induced by nonnegative matrices, Trans. Amer. Math. Soc. 83 (1956), 19-54. MR 0090927 (19:892a)
  • [FP] A. Figà-Talamanca and M. A. Picardello, Harmonic analysis on free groups, Lecture Notes in Pure and Appl. Math., vol. 87, Dekker, New York and Basel, 1987. MR 710827 (85j:43001)
  • [Ge] P. Gerl, Random walks on graphs with a strong isoperimetric property, J. Theoret. Probab. 1 (1988), 171-187. MR 938257 (89g:60216)
  • [He] D. Heath, Functions possessing restricted mean value properties, Proc. Amer. Math. Soc. 41 (1973), 588-595. MR 0333213 (48:11538)
  • [Iv] A. A. Ivanov, Bounding the diameter of a distance-regular graph, Soviet Math. Dokl. 28 (1983), 149-152. MR 719819 (84m:05045)
  • [Ke] O. D. Kellogg, Converses of Gauss' theorem on the arithmetic mean, Trans. Amer. Math. Soc. 36 (1934), 227-242. MR 1501739
  • [KSK] J. G. Kemeny, J. L. Snell and A. W. Knapp, Denumerable Markov chains, 2nd ed., Springer, New York, Heidelberg and Berlin, 1976. MR 0407981 (53:11748)
  • [Mo] B. Mohar, Isoperimetric inequalities, growth and the spectrum of graphs, Simon Fraser Univ., 1987, preprint. MR 943998 (89k:05071)
  • [PW1] M. A. Picardello and W. Woess, Martin boundaries of random walks: ends of trees and groups, Trans. Amer. Math. Soc. 302 (1987), 185-205. MR 887505 (89a:60177)
  • [PW2] -, Harmonic functions and ends of graphs, Proc. Edinburgh Math. Soc. (in print).
  • [Ve1] W. Veech, A zero-one law for a class of random walks and a converse to Gauss' mean value theorem, Ann. of Math. 97 (1973), 189-216. MR 0310269 (46:9370)
  • [Ve2] -, A converse to the mean value theorem for harmonic functions, Amer. J. Math. 97 (1975), 1007-1027. MR 0393521 (52:14330)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1989-0974775-7
Keywords: Homogeneous tree, harmonic functions, mean value property, Martin boundary
Article copyright: © Copyright 1989 American Mathematical Society

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