Sharp Strichartz estimates on nontrapping asymptotically conic manifolds


We obtain the Strichartz inequalities ║u║LqtLrx ([0,1]×M) ≥ C║u(0)L2║(M) for any smooth n-dimensional Riemannian manifold M which is asymptotically conic at infinity (with either short-range or long-range metric perturbation) and nontrapping, where u is a solution to the Schrödinger equation iut + 1/2 ΔMu = 0, and 2 < q, r ≥ ∞ are admissible Strichartz exponents (2/q + n/r = n/2). This corresponds with the estimates available for Euclidean space (except for the endpoint (q, r) = (2, 2n/n-2) when n > 2). These estimates imply existence theorems for semi-linear Schrödinger equations on M, by adapting arguments from Cazenave and Weissler and Kato.

This result improves on our previous result, which was an L4t,x Strichartz estimate in three dimensions. It is closely related to results of Staffilani-Tataru, Burq, Robbiano-Zuily and Tataru, who consider the case of asymptotically flat manifolds.