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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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})$.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1992-1113644-4
PII: S 0002-9939(1992)1113644-4
Keywords: Invariant Laplacian, one radius property
Article copyright: © Copyright 1992 American Mathematical Society