Resolvent and spectral measure on non-trapping asymptotically hyperbolic manifolds III: Global-in-time Strichartz estimates without loss

Abstract

In the present paper, we investigate global-in-time Strichartz estimates without loss on non-trapping asymptotically hyperbolic manifolds. Due to the hyperbolic nature of such manifolds, the set of admissible pairs for Strichartz estimates is much larger than usual. These results generalize the works on hyperbolic space due to Anker-Pierfelice and Ionescu-Staffilani. However, our approach is to employ the spectral measure estimates, obtained in the author's joint work with Hassell, to establish the dispersive estimates for truncated / microlocalized Schrödinger propagators as well as the corresponding energy estimates. Compared with hyperbolic space, the crucial point here is to cope with the conjugate points on the manifold. Additionally, these Strichartz estimates are applied to the L 2 well-posedness and L 2 scattering for nonlinear Schrödinger equations with power-like nonlinearity and small Cauchy data. Résumé. Dans cet article, nous examinons les estimations de Strichartz sans perte et globales en temps, définies sur des variétés non-captives asymptotiquement hyperboliques. De par la nature hyperbolique de ces variétés, l'ensemble des paires admissibles pour les estimations de Strichartz est beaucoup plus grand que d'ordinaire. Ces résultats généralisent les travaux menés par Anker-Pierfelice et Ionescu-Staffilani sur les espaces hyperboliques. Toutefois, notre approche utilise ici les estimations de mesures spectrales obtenues par l'auteur en collaboration avec Hassell afin d'établir les estimations de dispersion pour des propagateurs de Schrödinger tronqués ou micro-localisés ainsi que les estimations dénergie correspondantes. À la différence des espaces hyperboliques, l'élément crucial est ici de gérer les points conjugués de la variété. Enfin, ces estimations de Strichartz sont appliquées au caractère bien posé dans L 2 et à la diffusion L 2 pour les équations de Schrödinger avec des non-linearités de type puissance et des données initiales petites.

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

“…This is equal to 2x max when α = 1, and depends continuously on α, hence is close to 2x max for α close to 1, that is, when is sufficiently small 8 . It follows that if d(y, y ) ≥ 4 , the geodesic between p and p must enter the region {x ≥ } (provided is sufficiently small).…”

“…From this we see that if κ(r) is smooth and decays as e −nr/2 , then convolution with κ maps L p to L p for all p ∈ [1, 2). Additionally, this non-Euclidean feature also affects the range of valid Strichartz estimates on (asymptotically) hyperbolic manifolds -see [1,27,8].…”

“…Strichartz estimates on asymptotically hyperbolic manifolds. In the third paper in this series, [8], the first author will prove global-in-time Strichartz type estimates without loss on non-trapping asymptotically hyperbolic manifolds. Namely, for solutions of the inhomogeneous Schrödinger equation,…”

“…Compared to the asymptotically conic case, our spectral multiplier result is weaker, but the restriction estimate is stronger. In both cases, the difference can be traced to the exponential volume growth at infinity for asymptotically hyperbolic manifolds, as opposed to polynomial growth in the asymptotically conic setting.The pointwise bounds on the spectral measure established here will also be applied to Strichartz estimates in [8].…”

“…The pointwise bounds on the spectral measure established here will also be applied to Strichartz estimates in [8].…”