A Strichartz inequality for the Schroedinger equation on non-trapping asymptotically conic manifolds

We obtain an $L^4$ space-time Strichartz inequality for any smooth three-dimensional Riemannian manifold $(M,g)$ which is asymptotically conic at infinity and non-trapping, where $u$ is a solution to the Schr"odinger equation $i{u}_t+1\mathrm{/}2{\mathrm{\Delta }}_{M}u=0$. The exponent $H^{1\mathrm{/}4}(M)$ is sharp, by scaling considerations. In particular our result covers asymptotically flat non-trapping manifolds. Our argument is based on the interaction Morawetz inequality introduced by Colliander et al., interpreted here as a positive commutator inequality for the tensor product $U(t,z^{\mathrm{\prime }},z^{\mathrm{\prime }\mathrm{\prime }}):=u(t,z^{\mathrm{\prime }})u(t,z^{\mathrm{\prime }\mathrm{\prime }})$ of the solution with itself. We also use smoothing estimates for Schr"odinger solutions including a new one proved here with weight $r^{-1}$ at infinity and with the gradient term involving only one angular derivative.