Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Fine convergence and admissible convergence for symmetric spaces of rank one


Authors: Adam Korányi and J. C. Taylor
Journal: Trans. Amer. Math. Soc. 263 (1981), 169-181
MSC: Primary 32M15; Secondary 31C05, 43A85
DOI: https://doi.org/10.1090/S0002-9947-1981-0590418-1
MathSciNet review: 590418
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The connections between fine convergence in the sense of potential theory and admissible convergence to the boundary for quotients of eigenfunctions of the Laplace-Beltrami operator are investigated. This leads to a version of the local Fatou theorem on symmetric spaces of rank one which is considerably stronger than previously known results.

The appendix establishes the relationship between harmonic measures on the intersection of the Martin boundaries of a domain and a subdomain.


References [Enhancements On Off] (What's this?)

  • [1] N. Bourbaki, Eléménts de mathématiques. Topologie générale, Chap. 9, 2$ ^{e}$ ed., Hermann, Paris, 1958. MR 0173226 (30:3439)
  • [2] M. Brelot and J. L. Doob, Limites angulaires et limites fines, Ann. Inst. Fourier (Grenoble) 13 (2) (1963), 395-415. MR 0196107 (33:4299)
  • [3] A. P. Calderón, On the behaviour of harmonic functions at the boundary, Trans. Amer. Math. Soc. 68 (1950), 47-54. MR 0032863 (11:357e)
  • [4] L. Carleson, On the existence of boundary values for harmonic functions in several variables, Ark. Mat. 4 (1961), 393-399. MR 0159013 (28:2232)
  • [5] E. A. Coddington and N. Levinson, Theory of ordinary differential equations, McGraw-Hill, New York, 1955. MR 0069338 (16:1022b)
  • [6] R. Courant and D. Hilbert, Methods of mathematical physics. II, Interscience, New York, 1962.
  • [7] A. Debiard, Comparison des espaces $ {H^p}$ géométrique et probabilistes au dessus de l'espace hermitien hyperbolique, Bull. Sci. Math. 103 (1979), 305-351. MR 539352 (81f:32044)
  • [8] J. L. Doob, Conditional Brownian motion and the boundary limits of harmonic functions, Bull. Soc. Math. France 85 (1957), 431-458. MR 0109961 (22:844)
  • [9] -, A non-probabilistic proof of the relative Fatou theorem, Ann. Inst. Fourier (Grenoble) 9 (1959), 293-300. MR 0117454 (22:8233)
  • [10] M. Flensted-Jensen, Paley-Wiener type theorems for a differential operator connected with symmetric spaces, Ark. Mat. 10 (1972), 143-162. MR 0318974 (47:7520)
  • [11] H. Föllmer, Feine Topologie am Martinrand eines Standardprozesses, Z. Wahrsch. Verw. Gebiete 12 (1969), 127-144. MR 0245092 (39:6404)
  • [12] K. Gowrisankaran, Extremal harmonic functions and boundary value problems, Ann. Inst. Fourier (Grenoble) 13 (2) (1963), 307-356. MR 0164051 (29:1350)
  • [13] S. Helgason, Differential geometry and symmetric spaces, Academic Press, New York, 1969. MR 0145455 (26:2986)
  • [14] Mme. R. M. Hervé, Recherches axiomatiques sur la théorie des fonctions surharmoniques et du potentiel, Ann. Inst. Fourier (Grenoble) 12 (1962), 415-571. MR 0139756 (25:3186)
  • [15] F. I. Karpelevič, The geometry of geodesics and the eigenfunctions of the Beltrami-Laplace operator on symmetric spaces, Trans. Moscow Math. Soc. 14 (1965), 51-199. MR 0231321 (37:6876)
  • [16] A. Korányi, Harmonic functions on hermitian hyperbolic space, Trans. Amer. Math. Soc. 135 (1969), 507-516. MR 0277747 (43:3480)
  • [17] -, Harmonic functions on symmetric spaces, Symmetric Spaces (W. M. Boothby and G. Weiss, eds.), Marcel Dekker, New York, 1972. MR 0407541 (53:11314)
  • [18] -, A survey of harmonic functions on symmetric spaces, Proc. Sympos. Pure Math., vol. 35, Amer. Math. Soc., Providence, R. I., 1979. MR 545272 (80k:43012)
  • [19] A. Korányi and R. B. Putz, Local Fatou theorem and area theorem for symmetric spaces of rank one, Trans. Amer. Math. Soc. 224 (1976), 157-168. MR 0492068 (58:11223)
  • [20] R. S. Martin, Minimal positive harmonic functions, Trans. Amer. Math. Soc. 49 (1941), 137-172. MR 0003919 (2:292h)
  • [21] H. L. Michelson, Fatou theorems for eigenfunctions of the invariant differential operators on symmetric spaces, Trans. Amer. Math. Soc. 177 (1973), 257-274. MR 0319113 (47:7659)
  • [22] L. Naïm, Sur le role de la frontière de R. S. Martin dans la théorie du potentiel, Ann. Inst. Fourier (Grenoble) 7 (1957), 183-281. MR 0100174 (20:6608)
  • [23] J. Serrin, On the Harnack inequality for linear elliptic equations, J. Analyse Math. 4 (1955/56), 292-308. MR 0081415 (18:398f)
  • [24] J. C. Taylor, Martin boundaries of equivalent sheaves, Ann. Inst. Fourier (Grenoble) 20 (1970), 433-456. MR 0267120 (42:2022)
  • [25] -, Fine and admissible convergence for the unit ball in $ {C^n}$ (Proc. 1979 Copenhagen Potential Theory Colloquium, 1979), Lecture Notes in Math., vol. 787, Springer-Verlag, Berlin and New York (to appear).
  • [26] G. Warner, Harmonic analysis on semi-simple Lie groups. I, Die Grundlehren der Math. Wissenschaften, Band 188, Springer-Verlag, Berlin, 1972. MR 0498999 (58:16979)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 32M15, 31C05, 43A85

Retrieve articles in all journals with MSC: 32M15, 31C05, 43A85


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1981-0590418-1
Keywords: Rank one symmetric space, admissible convergence, eigenfunction of Laplace-Beltrami operator, fine convergence, Martin boundary, local Fatou theorem
Article copyright: © Copyright 1981 American Mathematical Society

American Mathematical Society