$L^p-L^q$ estimates for functions of the Laplace-Beltrami operator on noncompact symmetric spaces. III

Cowling, Michael; Giulini, Saverio; Meda, Stefano

doi:10.5802/aif.1844

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]

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)

Annales de l'Institut Fourier, Tome 51 (2001) no. 4, pp. 1047-1069.

Résumé
Abstract

Soit $X$ un espace symétrique du type noncompact, soit $- ℒ$ son opérateur Laplace- -Beltrami, et soit $[b, \infty)$ le spectre de $ℒ$ (vu comme opérateur sur $L^{2} (X)$ ). Si $τ \in ℂ$ et $Re τ \geq 0$ , notons $𝒫_{τ}$ l’opérateur $\exp (- τ {(ℒ - b)}^{1 / 2})$ 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 $| τ | \leq T$ ou $| \arg τ | \leq φ < π / 2$ .

Let $X$ be a symmetric space of the noncompact type, with Laplace–Beltrami operator $- ℒ$ , and let $[b, \infty)$ be the $L^{2} (X)$ -spectrum of $ℒ$ . For $τ$ in $ℂ$ such that $Re τ \geq 0$ , let $𝒫_{τ}$ be the operator on $L^{2} (X)$ defined formally as $\exp (- τ {(ℒ - b)}^{1 / 2})$ . 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 $π / 2$ .

MR Zbl

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 $L^p-L^q$

Affiliations des auteurs :

Cowling, Michael ¹ ; Giulini, Saverio ² ; Meda, Stefano ³

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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},
     pages = {1047--1069},
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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://archive.numdam.org/articles/10.5802/aif.1844/

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