L p -L q estimates for functions of the Laplace-Beltrami operator on noncompact symmetric spaces. III
[Estimations L p -L q de fonctions de l’opérateur de Laplace-Beltrami pour les espaces symétriques non compacts. III]
Annales de l'Institut Fourier, Tome 51 (2001) no. 4, pp. 1047-1069.

Soit X un espace symétrique du type noncompact, soit - son opérateur Laplace- -Beltrami, et soit [b,) le spectre de (vu comme opérateur sur L 2 (X)). Si τ et Re τ0, notons 𝒫 τ l’opérateur exp (-τ(-b) 1/2 ) sur L 2 (X). On démontre des estimations de la norme de 𝒫 τ de L p dans L q pour chaque τ, qui sont optimales si |τ|T ou | arg τ|φ<π/2.

Let X be a symmetric space of the noncompact type, with Laplace–Beltrami operator -, and let [b,) be the L 2 (X)-spectrum of . For τ in such that Re τ0, let 𝒫 τ be the operator on L 2 (X) defined formally as exp (-τ(-b) 1/2 ). In this paper, we obtain L p -L q operator norm estimates for 𝒫 τ for all τ, and show that these are optimal when τ is small and when | arg τ| is bounded below π/2.

DOI : 10.5802/aif.1844
Classification : 22E46, 22E30, 58J35, 58J45
Keywords: symmetric space, wave equation, $L^p-L^q$ estimates
Mot clés : Espace symétrique, équation des ondes, estimations $L^p-L^q$
Cowling, Michael 1 ; Giulini, Saverio 2 ; Meda, Stefano 3

1 University of New South Wales, School of Mathematics, Sydney NSW 2052 (Australie)
2 Università di Genova, Dipartimento di Matematica, Via Dodescano 35, 16146 Genova (Italie)
3 Università di Milano-Bicocca, Dipartimento di Statistica, Via Bicocca degli Arcimboldi 8, 20126 Milano (Italie)
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     title = {$L^p-L^q$ estimates for functions of the {Laplace-Beltrami} operator on noncompact symmetric spaces. {III}},
     journal = {Annales de l'Institut Fourier},
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Cowling, Michael; Giulini, Saverio; Meda, Stefano. $L^p-L^q$ estimates for functions of the Laplace-Beltrami operator on noncompact symmetric spaces. III. Annales de l'Institut Fourier, Tome 51 (2001) no. 4, pp. 1047-1069. doi : 10.5802/aif.1844. http://www.numdam.org/articles/10.5802/aif.1844/

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