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

Cowling, Michael; Giulini, Saverio; Meda, Stefano

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

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 $L^2\left(X\right)$). Si $\tau \in ℂ$ et $\mathrm{Re}\tau \ge 0$, notons ${𝒫}_{\tau }$ l’opérateur $\exp (-\tau {(ℒ-b)}^{1/2})$ sur $L^2\left(X\right)$. On démontre des estimations de la norme de ${𝒫}_{\tau }$ de $L^p$ dans $L^q$ pour chaque $\tau$, qui sont optimales si $\left|\tau \right|\le T$ ou $\left|\arg \tau \right|\le \phi <\pi /2$.

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

[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