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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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One radius theorem for the eigenfunctions of the invariant Laplacian
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by E. G. Kwon PDF
Proc. Amer. Math. Soc. 116 (1992), 27-34 Request permission

Abstract:

Let $B$ be the open unit ball in ${{\mathbf {C}}^n}$ with its boundary $S$. Suppose that $\alpha \geq \tfrac {1}{2}$ and $u(z) = {(1 - {\left | z \right |^2})^{n(1 - \alpha )}}F(z)$ 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 \[ u(z) = \frac {1}{{{g_\alpha }(r(z))}}\int _S {u \circ {\psi _z}(r(z)\zeta )d\sigma (\zeta )} ,\] 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
  • © Copyright 1992 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 116 (1992), 27-34
  • MSC: Primary 35P05; Secondary 35J05
  • DOI: https://doi.org/10.1090/S0002-9939-1992-1113644-4
  • MathSciNet review: 1113644