Lossless Strichartz and spectral projection estimates on unbounded manifolds

TL;DR

The paper proves lossless Strichartz and spectral projection estimates for the Schrödinger evolution on negatively curved, asymptotically hyperbolic surfaces, including convex cocompact hyperbolic cases, and extends log-scale Strichartz and spectral-projection bounds to general manifolds with uniformly bounded geometry. The authors combine nontrapping region analysis with trapping-gluing via a background manifold $\tilde{M}$ and employ dyadic time localization of size $\lambda^{-1}\log\lambda$, along with half-localized resolvent estimates and local smoothing bounds. They also develop log-scale Strichartz and refined spectral projection bounds on manifolds of bounded geometry with curvature pinching, using a sophisticated second microlocalization and bilinear harmonic analysis, plus Littlewood–Paley theory for spectral multipliers. The results advance dispersive and spectral theory on noncompact negatively curved spaces and have potential applications to nonlinear Schrödinger equations in geometric settings as well as to spectral analysis on convex cocompact and related manifolds.

Abstract

We prove new lossless Strichartz and spectral projection estimates on asymptotically hyperbolic surfaces, and, in particular, on all convex cocompact hyperbolic surfaces. In order to do this, we also obtain log-scale lossless Strichartz and spectral projection estimates on manifolds of uniformly bounded geometry with nonpositive and negative sectional curvatures, extending the recent works of the first two authors for compact manifolds. We are able to use these along with known $L^2$-local smoothing and new $L^2 \to L^q$ half-localized resolvent estimates to obtain our lossless bounds.

Figures (1)

• Figure 1: Convex cocompact hyperbolic surfaces

Theorems & Definitions (46)

• Theorem 1
• Theorem 2
• Theorem 3
• Corollary 4
• Theorem 5
• Lemma 1
• proof
• Proposition 2
• proof
• Proposition 3