Cowling, Michael; Giulini, Saverio; Meda, Stefano
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 , p. 1047-1069
Zbl 0980.43007 | MR 1849214
doi : 10.5802/aif.1844
URL stable : http://www.numdam.org/item?id=AIF_2001__51_4_1047_0

Classification:  22E46,  22E30,  58J35,  58J45
Mots clés: Espace symétrique, équation des ondes, estimations L p -L q
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.

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