Sharp local smoothing for manifolds with smooth inflection transmission Hans ChristiansonJason Metcalfe 35J1035B34local smoothingtrappingresolvent estimates 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. Indiana University Mathematics Journal 2014 text pdf 10.1512/iumj.2014.63.5323 10.1512/iumj.2014.63.5323 en Indiana Univ. Math. J. 63 (2014) 969 - 992 state-of-the-art mathematics http://iumj.org/access/