$L^p-L^q$ 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 ; Giulini, Saverio ; Meda, Stefano

Annales de l'Institut Fourier, Tome 51 (2001), p. 1047-1069 / Harvested from Numdam

Résumé

Soit $X$ un espace symétrique du type noncompact, soit $-{\mathcal {L}}$ son opérateur Laplace- -Beltrami, et soit $[b, \infty )$ le spectre de ${\mathcal {L}}$ (vu comme opérateur sur $L^2(X)$). Si $\tau \in {\mathbb {C}}$ et ${\rm Re}\,\tau \ge 0$, notons ${\mathcal {P}}_\tau $ l’opérateur ${\rm exp}\,(-\tau ({\mathcal {L}} - b)^{1/2})$ sur $L^2(X)$. On démontre des estimations de la norme de ${\mathcal {P}}_\tau $ de $L^p$ dans $L^q$ pour chaque $\tau $, qui sont optimales si $\vert \tau \vert \le T$ ou $\vert {\rm arg}\,\tau \vert \le \phi < \pi /2$.

Abstract

Let $X$ be a symmetric space of the noncompact type, with Laplace–Beltrami operator $- {\mathcal {L}}$, and let $[b,\infty )$ be the $L^2(X)$-spectrum of ${\mathcal {L}}$. For $\tau $ in ${\mathbb {C}}$ such that ${\rm Re}\,\tau \ge 0$, let ${\mathcal {P}}_\tau $ be the operator on $L^2(X)$ defined formally as ${\rm exp}\,(-\tau ({\mathcal {L}} - b)^{1/2} )$. In this paper, we obtain $L^p-L^q$ operator norm estimates for ${\mathcal {P}}_\tau $ for all $\tau $, and show that these are optimal when $\tau $ is small and when $\vert {\rm arg}\,\tau \vert $ is bounded below $\pi /2$.

Détail

Publié le : 2001-01-01

MR 1849214 | Zbl 0980.43007

DOI : https://doi.org/10.5802/aif.1844
Classification:  22E46,  22E30,  58J35,  58J45
Mots clés: Espace symétrique, équation des ondes, estimations $L^p-L^q$

Bibliographie

[A1] J.-Ph. Anker Sharp estimates for some functions of the Laplacian on noncompact symmetric spaces, Duke Math. J., Tome 65 (1992), pp. 257-297 | MR 1150587 | Zbl 0764.43005

[A2] J.-Ph. Anker $L_p$ Fourier multipliers on Riemannian symmetric spaces of the noncompact type, Ann. Math., Tome 132 (1990), pp. 597-628 | Article | MR 1078270 | Zbl 0741.43009

[AJ] J.-Ph. Anker; L. Ji Heat kernel and Green function estimates on noncompact symmetric spaces I, II (Preprint) | Zbl 0942.43005

[B] P.H. Bérard On the wave equation on a compact Riemannian manifold without conjugate point, Math. Z., Tome 155 (1977), pp. 249-276 | Article | MR 455055 | Zbl 0341.35052

[CGM1] M.G. Cowling; S. Giulini; S. Meda Estimates for functions of the Laplace--Beltrami operator on noncompact symmetric spaces. I, Duke Math. J., Tome 72 (1993), pp. 109-150 | Article | MR 1242882 | Zbl 0807.43002

[CGM2] M.G. Cowling; S. Giulini; S. Meda Estimates for functions of the Laplace--Beltrami operator on noncompact symmetric spaces. II, J. Lie Th., Tome 5 (1995), pp. 1-14 | MR 1362006 | Zbl 0835.43011

[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., Tome 17 (1982), pp. 15-53 | MR 658471 | Zbl 0493.53035

[H1] S. Helgason Groups and Geometric Analysis, Academic Press, New York (1984) | MR 754767 | Zbl 0543.58001

[H2] S. Helgason Wave equations on homogeneous spaces, Lie Group Representations III, Springer-Verlag (Lecture Notes in Math.) Tome 1077 (1984), pp. 254-287 | Zbl 0547.58037

[Hö1] L. Hörmander Estimates for translation invariant operators in $L^p$ spaces, Acta Math., Tome 104 (1960), pp. 93-140 | Article | MR 121655 | Zbl 0093.11402

[Hö2] L. Hörmander The Analysis of Linear Partial Differential Operators. III., Springer-Verlag, Berlin Heidelberg New York, Grundlehren der mathematischen Wissenschaften, Tome 274 (1985) | MR 781536 | Zbl 0601.35001

[T] M. Taylor $L^p$ estimates on functions of the Laplace operator, Duke Math. J., Tome 58 (1989), pp. 773-793 | Article | MR 1016445 | Zbl 0691.58043