article

Title: Sharp local smoothing for manifolds with smooth inflection transmission

Authors: Hans Christianson and Jason Metcalfe

Issue: Volume 63 (2014), Issue 4, 969-992

Abstract:

$We consider a family of rotationally symmetric, asymptotically Euclidean manifolds with two trapped sets, one of which is unstable and one of which is semistable. We prove a sharp local smoothing estimate for the linear Schr\"odinger equation with a loss that depends on how flat the manifold is near each of the trapped sets. The result interpolates amongst the members of the family of similar estimates in [Hans Christianson and Jared Wunsch, \textit{Local smoothing for the Schr\"odinger equation with a prescribed loss}, Amer. J. Math. \textbf{135} (2013), no. 6, 1601--1632]. As a consequence of the techniques used in our proof, we also show a sharp high-energy resolvent estimate with a polynomial loss that depends on how flat the manifold is near each of the trapped sets.$