Spectral projectors on hyperbolic surfaces

TL;DR

This paper establishes $L^2 o L^p$ bounds for spectral projectors on a broad class of hyperbolic surfaces, combining resolvent bounds and Schrödinger estimates to achieve optimal results in the convex cocompact case.

Contribution

It provides new $L^2 o L^p$ spectral projector estimates on hyperbolic surfaces, extending previous bounds and incorporating advanced resolvent and Schrödinger group techniques.

Findings

Optimal bounds for spectral projectors in convex cocompact case

Extension of $L^2 o L^p$ estimates to geometrically finite hyperbolic surfaces

Integration of resolvent bounds with Schrödinger estimates for spectral analysis

Abstract

In this paper, we prove $L^2 \to L^p$ estimates, where $p>2$, for spectral projectors on a wide class of hyperbolic surfaces. More precisely, we consider projections in small spectral windows $[\lambda-\eta,\lambda+\eta]$ on geometrically finite hyperbolic surfaces of infinite volume. In the convex cocompact case, we obtain optimal bounds with respect to $\lambda$ and $\eta$, up to subpolynomial losses. The proof combines the resolvent bound of Bourgain-Dyatlov and improved estimates for the Schr\"odinger group (Strichartz and smoothing estimates) on hyperbolic surfaces.