Strichartz and Spectral Projection Estimates on Asymptotically Conic Manifolds

Technical Summary: Strichartz and Spectral Projection Estimates on Asymptotically Conic Manifolds

Problem Statement
This paper addresses the derivation of Strichartz estimates and spectral projection estimates for the Schrödinger and wave equations on asymptotically conic manifolds. Specifically, the author investigates two-dimensional surfaces with Euclidean ends and nonpositive curvature, as well as higher-dimensional ($n \ge 3$) asymptotically Euclidean manifolds. The central challenge lies in obtaining "lossless" estimates (estimates without derivative loss) on the unit time interval for Strichartz inequalities and on logarithmic or unit spectral windows for spectral projections, particularly in the presence of a trapped set. While previous works (e.g., Burq, Guillarmou, Hassell, Huang, Sogge, Tao) established these results under specific conditions like hyperbolic trapped sets with pressure conditions or even asymptotically hyperbolic geometries, this work aims to extend these results to broader classes of asymptotically conic manifolds, specifically relaxing the "evenness" assumption and handling non-trapping regions with specific curvature constraints.

Methodology
The methodology relies on a combination of geometric construction, microlocal analysis, and the comparison of the target manifold with a background model manifold.

1. Construction of Background Manifolds:
   For the two-dimensional case, the author constructs an "even asymptotically hyperbolic" background manifold $(\tilde{M}, \tilde{g})$ that agrees with the original manifold $(M, g)$ on a compact core $M_0$ and in a neighborhood of the trapped set. This background manifold is engineered to have strictly negative Gaussian curvature everywhere. This construction allows the author to transfer known results from the negative curvature setting (where Strichartz and spectral projection estimates are well-understood) to the original asymptotically conic setting.

2. Decomposition and Localization:
   The proofs involve decomposing the spectral projection operator using Littlewood-Paley theory and cutoff functions. The manifold is partitioned into regions:

   • Compact Core ($M_0$): Where the geometry is handled by the background manifold construction.
   • Trapped Set Neighborhood: Where local smoothing estimates are applied.
   • Euclidean/Non-trapped Ends: Where the geometry is either Euclidean or asymptotically conic with convex geodesic flow.
     The commutator terms arising from these cutoffs are analyzed using the finite propagation speed of the wave equation and the Hadamard parametrix.

3. Resolvent and Propagator Estimates:
   The core technical work involves establishing localized resolvent estimates and Strichartz estimates for the background manifold. The author utilizes:

   • Local Smoothing Estimates: Specifically, estimates with at most logarithmic loss on the trapped set for $n \ge 3$.
   • Oscillatory Integral Bounds: Utilizing the Hadamard parametrix and stationary phase arguments to bound the kernels of propagators and resolvents.
   • Geodesic Flow Analysis: The assumption of negative curvature on a neighborhood of the trapped set ensures that geodesic flow is convex in terms of the radial coordinate $r$, preventing geodesics from remaining trapped indefinitely in the non-compact region.

4. Restriction to Geodesics:
   For Theorem 3, the methodology extends to restrictions of spectral projections onto non-trapped geodesics. This involves lifting calculations to the universal cover (using the Cartan-Hadamard theorem) and analyzing the interaction between the geodesic flow and the cutoff functions.

Key Contributions and Results

• Theorem 1 (Strichartz Estimates on Surfaces):
  The paper proves a lossless unit-interval Strichartz estimate for 2-dimensional asymptotically conic surfaces. The result holds under the assumption that the compact core $M_0$ has negative sectional curvature and the metric satisfies specific decay and convexity conditions (equations (2) and (4)) at infinity. Crucially, this result is achieved without assuming the "pressure condition" required in earlier works by Burq, Guillarmou, and Hassell, provided the trapped set has a neighborhood of negative curvature.

• Theorem 2 (Spectral Projection on Surfaces):
  The author establishes sharp spectral projection estimates on 2-dimensional surfaces with Euclidean ends and nonpositive curvature. The estimates are given in $L^q$ norms for spectral windows of size $\delta \in (0, 1]$. The bounds match the Stein-Tomas restriction theorem for $\mathbb{R}^2$ and extend previous logarithmic-scale results to the Euclidean end setting, assuming negative curvature on a large compact set.

• Theorem 3 (Restriction to Non-trapped Geodesics):
  The paper proves supercritical spectral projection estimates restricted to non-trapped geodesics on surfaces with Euclidean ends. This extends previous results on compact manifolds and asymptotically hyperbolic surfaces to the asymptotically conic setting.

• Theorem 4 (Spectral Projection in Higher Dimensions):
  For $n \ge 3$, the paper proves spectral projection estimates on asymptotically Euclidean manifolds with nonpositive sectional curvature. The result assumes that the trapped set is either:

  1. Surrounded by a neighborhood of negative sectional curvature satisfying the pressure condition; or
  2. Smooth and normally hyperbolic.
     The proof relies on a local smoothing estimate with logarithmic loss on the trapped set (Equation 13). Under these conditions, the author derives lossless spectral projection estimates.

Significance and Claims
The paper claims to unify and extend existing spectral and dispersive estimates by removing the "evenness" assumption on the asymptotic expansion of the metric, which was previously necessary for certain resolvent constructions. By constructing a specific background manifold with strictly negative curvature, the author demonstrates that the favorable properties of hyperbolic geometry (specifically regarding the trapped set) can be leveraged to control the dispersive behavior of the Schrödinger flow on more general asymptotically conic manifolds.

The author notes that the restriction to surfaces with Euclidean ends in Theorem 2 is currently due to the difficulty in inverting the model operator at the "zf face" (zero face) for non-trapped asymptotically conic surfaces in the construction of resolvents. The paper suggests that if the 2-dimensional non-trapped case can be further explored, Theorem 2 could potentially be generalized to all asymptotically conic surfaces.

Furthermore, the work highlights that the local smoothing estimate with logarithmic loss (Equation 13) is a sufficient condition for obtaining lossless spectral projections in dimensions $n \ge 3$. This condition is shown to be satisfied by manifolds with negatively curved trapped neighborhoods (satisfying the pressure condition) or those with normally hyperbolic trapped sets, thereby broadening the class of manifolds for which these sharp estimates are known to hold.