Mean value properties of the Laplacian via spectral theory
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- by Robert S. Strichartz PDF
- Trans. Amer. Math. Soc. 284 (1984), 219-228 Request permission
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.References
- Leifur Ásgeirsson, Über eine Mittelwertseigenschaft von Lösungen homogener linearer partieller Differentialgleichungen 2. Ordnung mit konstanten Koeffizienten, Math. Ann. 113 (1937), no. 1, 321–346 (German). MR 1513094, DOI 10.1007/BF01571637
- Paul R. Chernoff, Essential self-adjointness of powers of generators of hyperbolic equations, J. Functional Analysis 12 (1973), 401–414. MR 0369890, DOI 10.1016/0022-1236(73)90003-7
- Mogens Flensted-Jensen, Spherical functions of a real semisimple Lie group. A method of reduction to the complex case, J. Functional Analysis 30 (1978), no. 1, 106–146. MR 513481, DOI 10.1016/0022-1236(78)90058-7
- Harish-Chandra, Spherical functions on a semisimple Lie group. I, Amer. J. Math. 80 (1958), 241–310. MR 94407, DOI 10.2307/2372786
- Sigurđur Helgason, Partial differential equations on Lie groups, Treizième congrès des mathématiciens scandinaves, tenu à Helsinki 18-23 août 1957, Mercators Tryckeri, Helsinki, 1958, pp. 110–115. MR 0106357
- Sigurđur Helgason, Differential operators on homogeneous spaces, Acta Math. 102 (1959), 239–299. MR 117681, DOI 10.1007/BF02564248
- Sigurdur Helgason, Analysis on Lie groups and homogeneous spaces, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 14, American Mathematical Society, Providence, R.I., 1972. MR 0316632 —, Wave equations on homogeneous spaces, preprint.
- Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR 0304972
- Robert S. Strichartz, Multiplier transformations on compact Lie groups and algebras, Trans. Amer. Math. Soc. 193 (1974), 99–110. MR 357688, DOI 10.1090/S0002-9947-1974-0357688-2
- Reiji Takahashi, Sur les représentations unitaires des groupes de Lorentz généralisés, Bull. Soc. Math. France 91 (1963), 289–433 (French). MR 179296
Additional Information
- © Copyright 1984 American Mathematical Society
- 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