A Strichartz Inequality for the Schrödinger Equation on Nontrapping Asymptotically Conic Manifolds

Abstract

We obtain the Strichartz inequality for any smooth three-dimensional Riemannian manifold (M, g) which is asymptotically conic at infinity and nontrapping, where u is a solution to the Schrödinger equation iu t + (1/2)Δ M u = 0. The exponent H 1/4(M) is sharp, by scaling considerations. In particular our result covers asymptotically flat nontrapping 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′, z′′): = u(t, z′)u(t, z′′) of the solution with itself. We also use smoothing estimates for Schrödinger solutions including one (proved here) with weight r −1 at infinity and with the gradient term involving only one angular derivative.