$L^p-L^q$ 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

@article{AIF_2001__51_4_1047_0,
     author = {Cowling, Michael and Giulini, Saverio and Meda, Stefano},
     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},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {51},
     number = {4},
     year = {2001},
     doi = {10.5802/aif.1844},
     mrnumber = {1849214},
     zbl = {0980.43007},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1844/}
}

TY  - JOUR
AU  - Cowling, Michael
AU  - Giulini, Saverio
AU  - Meda, Stefano
TI  - $L^p-L^q$ estimates for functions of the Laplace-Beltrami operator on noncompact symmetric spaces. III
JO  - Annales de l'Institut Fourier
PY  - 2001
SP  - 1047
EP  - 1069
VL  - 51
IS  - 4
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.1844/
DO  - 10.5802/aif.1844
LA  - en
ID  - AIF_2001__51_4_1047_0
ER  -

%0 Journal Article
%A Cowling, Michael
%A Giulini, Saverio
%A Meda, Stefano
%T $L^p-L^q$ estimates for functions of the Laplace-Beltrami operator on noncompact symmetric spaces. III
%J Annales de l'Institut Fourier
%D 2001
%P 1047-1069
%V 51
%N 4
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.1844/
%R 10.5802/aif.1844
%G en
%F AIF_2001__51_4_1047_0

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/

Bibliographie
Cité par

[A1] J.-Ph. Anker Sharp estimates for some functions of the Laplacian on noncompact symmetric spaces, Duke Math. J., Volume 65 (1992), pp. 257-297 | MR | Zbl

[A2] J.-Ph. Anker $L_{p}$ Fourier multipliers on Riemannian symmetric spaces of the noncompact type, Ann. Math., Volume 132 (1990), pp. 597-628 | DOI | MR | Zbl

[AJ] J.-Ph. Anker; L. Ji Heat kernel and Green function estimates on noncompact symmetric spaces I, II (Preprint) | Zbl

[B] P.H. Bérard On the wave equation on a compact Riemannian manifold without conjugate point, Math. Z., Volume 155 (1977), pp. 249-276 | DOI | MR | Zbl

[CGM1] M.G. Cowling; S. Giulini; S. Meda Estimates for functions of the Laplace--Beltrami operator on noncompact symmetric spaces. I, Duke Math. J., Volume 72 (1993), pp. 109-150 | DOI | MR | Zbl

[CGM2] M.G. Cowling; S. Giulini; S. Meda Estimates for functions of the Laplace--Beltrami operator on noncompact symmetric spaces. II, J. Lie Th., Volume 5 (1995), pp. 1-14 | MR | Zbl

[CGT] J. Cheeger; M. Gromov; M. Taylor Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Diff. Geom., Volume 17 (1982), pp. 15-53 | MR | Zbl

[H1] S. Helgason Groups and Geometric Analysis, Academic Press, New York, 1984 | MR | Zbl

[H2] S. Helgason Wave equations on homogeneous spaces, Lie Group Representations III (Lecture Notes in Math.), Volume 1077 (1984), pp. 254-287 | Zbl

[Hö1] L. Hörmander Estimates for translation invariant operators in $L^{p}$ spaces, Acta Math., Volume 104 (1960), pp. 93-140 | DOI | MR | Zbl

[Hö2] L. Hörmander The Analysis of Linear Partial Differential Operators. III., Grundlehren der mathematischen Wissenschaften, 274, Springer-Verlag, Berlin Heidelberg New York, 1985 | MR | Zbl

[T] M. Taylor $L^{p}$ estimates on functions of the Laplace operator, Duke Math. J., Volume 58 (1989), pp. 773-793 | DOI | MR | Zbl

Cité par Sources :