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)

 
 

 

A tangential convergence for bounded harmonic functions on a rank one symmetric space


Author: Jacek Cygan
Journal: Trans. Amer. Math. Soc. 265 (1981), 405-418
MSC: Primary 43A85; Secondary 32M15, 43A20
DOI: https://doi.org/10.1090/S0002-9947-1981-0610957-4
MathSciNet review: 610957
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ u$ be a bounded harmonic function on a noncompact rank one symmetric space $ M = G/K \approx {N^ - }A,{N^ - }AK$ being a fixed Iwasawa decomposition of $ G$. We prove that if for an $ {a_0} \in A$ there exists a limit $ u(n{a_0}) \equiv {c_0}$, as $ n \in {N^ - }$ goes to infinity, then for any $ a \in A$, $ u(na) = {c_0}$. For $ M = SU(n,1)/S(U(n) \times U(1)) = {B^n}$, the unit ball in $ {{\mathbf{C}}^n}$ with the Bergman metric, this is a result of Hulanicki and Ricci, and in this case it reads (via the Cayley transformation) as a theorem on convergence of a bounded harmonic function to a boundary value at a fixed boundary point, along appropriate, tangent to $ \partial {B^n}$, surfaces.


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

  • [1] P. Bernat et al., Représentations des groupes de Lie résolubles, Dunod, Paris, 1972.
  • [2] D. Geller, Fourier analysis on the Heisenberg group, Proa Nat. Acad. Sci. U.S.A. 74 (1977), 1328-1331. MR 0486312 (58:6069)
  • [3] S. Helgason, Differential geometry and symmetric spaces, Academic Press, New York and London, 1962. MR 0145455 (26:2986)
  • [4] -, Application of the Radon transform to representations of semisimple Lie groups, Proc. Nat. Acad. Sci. U.S.A. 63 (1969), 643-647. MR 0263987 (41:8586)
  • [5] A. Hulanicki and F. Ricci, A Tauberian theorem and tangential convergence for bounded harmonic functions on balls in $ {{\mathbf{C}}^n}$, Invent. Math. 62 (1980), 325-331. MR 595591 (82e:32008)
  • [6] A. Kaplan and R. Putz, Boundary behavior of harmonic forms on a rank one symmetric space, Trans. Amer. Math. Soc. 231 (1977), 369-384. MR 0477174 (57:16715)
  • [7] A. Korányi, Boundary behavior of Poisson integrals on symmetric spaces, Trans. Amer. Math. Soc. 140 (1969), 393-409. MR 0245826 (39:7132)
  • [8] H. Leptin, Harmonische Analyse auf gewissen nilpotenten Lieschen Gruppen, Studia Math. 48 (1973), 201-205. MR 0349900 (50:2393)
  • [9] E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Univ. Press, Princeton, N. J., 1971. MR 0304972 (46:4102)
  • [10] R. Takahashi, Quelques résultats sur l'analyse harmonique dans l'espace symétrique non compact de rang un du type exceptionnel, Analyse Harmonique sur les Groupes de Lie. II, Lecture Notes in Math., vol. 739, Springer-Verlag, Berlin and New York, 1979. MR 0384973 (52:5843)
  • [11] G. Warner, Harmonic analysis on semi-simple Lie groups. I, Springer-Verlag, Berlin and New York, 1972. MR 0498999 (58:16979)
  • [12] G. N. Watson, A treatise on the theory of Bessel functions, 2nd ed., Cambridge Univ. Press, Cambridge, 1966. MR 1349110 (96i:33010)
  • [13] D. P. Zhelobenko, Kompaktnye gruppy Li i ikh predstavleniya, Nauka, Moskva, 1970. MR 0473097 (57:12776a)
  • [14] A. Korányi, Some applications of Gelfand pairs in classical analysis, Harmonic Analysis and Group Representations (C.I.M.E., Cortona, 1980), Liguori, Naples (to appear). MR 777343 (86i:22015)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 43A85, 32M15, 43A20

Retrieve articles in all journals with MSC: 43A85, 32M15, 43A20


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1981-0610957-4
Article copyright: © Copyright 1981 American Mathematical Society

American Mathematical Society