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Proceedings of the American Mathematical Society

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One radius theorem for the eigenfunctions of the invariant Laplacian

Author: E. G. Kwon
Journal: Proc. Amer. Math. Soc. 116 (1992), 27-34
MSC: Primary 35P05; Secondary 35J05
MathSciNet review: 1113644
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Abstract: Let $ B$ be the open unit ball in $ {{\mathbf{C}}^n}$ with its boundary $ S$. Suppose that $ \alpha \geq \tfrac{1}{2}$ and $ u(z) = {(1 - {\left\vert z \right\vert^2})^{n(1 - \alpha )}}F(z)$ for some $ F(z) \in C(\overline B )$. If for every $ z \in B$ there corresponds an $ r(z):0 < r(z) < 1$ and an automorphism $ {\psi _z}$ with $ {\psi _z}(0) = z$ such that

$\displaystyle u(z) = \frac{1}{{{g_\alpha }(r(z))}}\int_S {u \circ {\psi _z}(r(z)\zeta )d\sigma (\zeta )} ,$

then $ \tilde \Delta u(z) = - 4{n^2}\alpha (1 - \alpha )u(z),z \in B$. Here $ \tilde \Delta $ is the invariant Laplacian and $ {g_\alpha }(r)$ is the hypergeometric function $ F(n - n\alpha ,n - n\alpha ,n;{r^2})$.

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

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  • [2] Oliver D. Kellog, Converses of Gauss's theorem of the arithmetic mean, Trans. Amer. Math. Soc. 30 (1934), 227-242. MR 1501739
  • [3] Lucy John Slater, Generalized hypergeometric functions, Cambridge Univ. Press, London and New York, 1966. MR 0201688 (34:1570)
  • [4] Walter Rudin, Function theory in the unit ball of $ {{\mathbf{C}}^n}$, Springer-Verlag, New York, 1980. MR 601594 (82i:32002)

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Keywords: Invariant Laplacian, one radius property
Article copyright: © Copyright 1992 American Mathematical Society

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