Lossless Strichartz and spectral projection estimates on unbounded manifolds

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.

💡 Research Summary

The paper establishes new loss‑free Strichartz estimates and spectral projection bounds on a class of non‑compact manifolds with negative curvature, in particular on even asymptotically hyperbolic surfaces and all convex‑cocompact hyperbolic surfaces. The authors also prove log‑scale loss‑free estimates on manifolds of uniformly bounded geometry whose sectional curvatures are non‑positive (or strictly negative). These results extend recent work of the first two authors on compact manifolds and remove the ε‑losses, logarithmic losses, or dimension restrictions that appeared in earlier literature.

Theorem 1.1 (Loss‑free Strichartz).
Let (M,g) be an even asymptotically hyperbolic surface with negative curvature. For any admissible pair (p,q) satisfying the Keel–Tao scaling 2/p+ (n‑1)(1/2‑1/q)=1 (here n=2) and (p,q)≠(2,∞), one has \