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Transactions of the American Mathematical Society
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
MathSciNet review: 610957
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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.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1981-0610957-4
PII: S 0002-9947(1981)0610957-4
Article copyright: © Copyright 1981 American Mathematical Society