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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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
DOI: https://doi.org/10.1090/S0002-9947-1984-0742422-0
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