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\(L^p\)-estimates of solutions of the wave equation on Riemannian manifolds, Lie groups and applications. (Estimées \(L^p\) des solutions de l’équation des ondes sur les variétés riemanniennes, les groupes de Lie et applications.) (French) Zbl 0927.58014

Drury, S. W. (ed.) et al., Harmonic analysis and number theory. Papers in honour of Carl S. Herz. Proceedings of the conference, April 15–19, 1996, Montréal, Canada. Providence, RI: American Mathematical Society. CMS Conf. Proc. 21, 103-126 (1997).
\(L^p\)-estimates of solutions for wave equations associated to second order differential operators \(L\) on a manifold \(M\) are obtained. These results can be applied in the following cases:
(i) \(M\) is bounded open in a Riemannian manifold and \(L\) is the restriction to \(M\) of the Laplace-Beltrami operator;
(ii) \(M\) is a complete Riemannian \(n\)-manifold with bounded curvature and \(L\) is the Laplace-Beltrami operator for which the associated semigroup satisfies \(P_t(x,y)\leq ct^{-D/2},t\geq 1\), where \(2D>n \) and \(c\) is a real number;
(iii) \(M\) is a Lie group and \(L\) is a sub-Laplacian of Hörmander type;
(iv) \(M=\mathbb{R}^n\) and \(L\) is a global elliptic operator.
The only complaint to be made about this paper concerns the bibliography, because half of the references (8 from 16) have no date and the references [2] and [16] have no source!
For the entire collection see [Zbl 0873.00020].

MSC:

58J45 Hyperbolic equations on manifolds
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
22E30 Analysis on real and complex Lie groups
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