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Transactions of the American Mathematical Society

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Mean value properties of the Laplacian via spectral theory

Author: Robert S. Strichartz
Journal: Trans. Amer. Math. Soc. 284 (1984), 219-228
MSC: Primary 31C12; Secondary 22E30, 35P99, 43A85
MathSciNet review: 742422
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Abstract: Let $ \phi ({z^2})$ be an even entire function of temperate exponential type, $ L$ a selfadjoint realization of $ - \Delta + c\,(x)$, where $ \Delta $ is the Laplace-Beltrami operator on a Riemannian manifold, and $ \phi \,(L)$ the operator given by spectral theory. A Paley-Wiener theorem on the support of $ \phi \,(L)$ is proved, and is used to show that $ Lu = \lambda u$ on a suitable domain implies $ \phi \,(L)\,u = \phi \,(\lambda)\,u$, as well as a generalization of Àsgeirsson's theorem. A concrete realization of the operators $ \phi \,(L)$ is given in the case of a compact Lie group or a noncompact symmetric space with complex isometry group.

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Keywords: Laplace-Beltrami operator, spectral theory, mean value theorem, Asgeirsson's theorem, functions of the Laplacian, Paley-Wiener theorem, symmetric space, compact Lie group
Article copyright: © Copyright 1984 American Mathematical Society

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