  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.
