On refined local smoothing estimates for the Schrödinger equation in exterior domains

Abstract: We consider refinements of the local smoothing estimates for the Schr\"odinger equation in domains which are exterior to a strictly convex obstacle in $\RR<sup>n$. By restricting the solution to small, frequency dependent collars of the boundary, it is expected that taking its square integral in space-time should exhibit a larger gain in regularity when compared to the usual gain of half a derivative. By a result of Ivanovici, these refined local smoothing estimates are satisfied by solutions in the exterior of a ball. We show that when such estimates are valid, they can be combined with wave packet parametrix constructions to yield Strichartz estimates. This provides an avenue for obtaining these bounds when Neumann boundary conditions are imposed.