Strichartz inequalities on surfaces with cusps

Abstract

We prove Strichartz inequalities for the wave and Schr\"odinger equations on noncompact surfaces with ends of finite area, i.e. with ends isometric to $ \big( (r_0,\infty) \times {\mathbb S}^1 , dr^2 + e^{- 2 \phi (r)}d \theta^2 \big) $ with $ e^{-\phi} $ integrable. We prove first that all Strichartz estimates, with any derivative loss, fail to be true in such ends. We next show for the wave equation that, by projecting off the zero mode of $ {\mathbb S}^1 $, we recover the same inequalities as on $ {\mathbb R}^2 $. On the other hand, for the Schr\"odinger equation, we prove that even by projecting off the zero angular modes we have to consider additional losses of derivatives compared to the case of closed surfaces; in particular, we show that the semiclassical estimates of Burq-G\'erard-Tzvetkov do not hold in such geometries. Moreover our semiclassical estimates with loss are sharp.