Resolvent and Spectral Measure on Non-Trapping Asymptotically Hyperbolic Manifolds II: Spectral Measure, Restriction Theorem, Spectral Multipliers

Chen, Xi; Hassell, Andrew

Annales de l'Institut Fourier, Tome 68 (2018) no. 3, pp. 1011-1075.

Nous considérons le Laplacien $\Delta$ sur une variété asymptotiquement hyperbolique $X$ au sens de Mazzeo et Melrose. Nous donnons des estimations ponctuelles sur le noyau de Schwartz de la mesure spectrale pour l’opérateur ${(\Delta -n^2/4)}_{+}^{1/2}$ sur ces variétés, sous l’hypothèse qu’il n’y ni trajectoires captées dans $X,$ ni résonance au bas du spectre. Nous utilisons la construction de la résolvante par Mazzeo et Melrose, Sá Barreto et Vasy, Wang, et nous-mêmes.

Nous donnons deux applications des estimations de la mesure spectrale. La première, qui prolonge l’étude de Guillarmou et Sikora avec le deuxième auteur dans le cas asymptotiquement conique, est un théorème de restriction : c’est-à-dire une borne sur la norme d’opérateur $L^p\left(X\right)\to L^{p^{\text{'}}}\left(X\right)$ de la mesure spectrale. La seconde est un résultat de type multiplicateur spectral sous l’hypothèse additionnelle que $X$ est à courbure strictement négative partout. Plus précisément, nous donnons une estimation sur les fonctions du laplacien de la forme $F\left({(\Delta -n^2/4)}_{+}^{1/2}\right)$ en termes de normes de la fonction $F$. Par rapport au cas asymptotiquement conique, notre résultat de multiplicateur spectral est plus faible, mais l’estimation de restriction est plus forte.

We consider the Laplacian $\Delta$ on an asymptotically hyperbolic manifold $X$, as defined by Mazzeo and Melrose. We give pointwise bounds on the Schwartz kernel of the spectral measure for the operator ${(\Delta -n^2/4)}_{+}^{1/2}$ on such manifolds, under the assumptions that $X$ is nontrapping and there is no resonance at the bottom of the spectrum. This uses the construction of the resolvent given by Mazzeo and Melrose, Melrose, Sá Barreto and Vasy, the present authors, and Wang.

We give two applications of the spectral measure estimates. The first, following work due to Guillarmou and Sikora with the second author in the asymptotically conic case, is a restriction theorem, that is, a $L^p\left(X\right)\to L^{p^{\text{'}}}\left(X\right)$ operator norm bound on the spectral measure. The second is a spectral multiplier result under the additional assumption that $X$ has negative curvature everywhere, that is, a bound on functions of the Laplacian of the form $F\left({(\Delta -n^2/4)}_{+}^{1/2}\right)$, in terms of norms of the function $F$. Compared to the asymptotically conic case, our spectral multiplier result is weaker, but the restriction estimate is stronger.

Classification : 58J50, 35P25, 47F05
Keywords: Asymptotically hyperbolic manifolds, spectral measure, restriction theorem, spectral multiplier
Mot clés : Variété asymptotiquement hyperbolique, mesure spectrale, théorème de restriction, multiplicateur spectral

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