Spectral projectors on hyperbolic surfaces

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 Schrödinger group (Strichartz and smoothing estimates) on hyperbolic surfaces.