Sharp local smoothing for manifolds with smooth inflection transmission

摘要: We consider a family of spherically symmetric, asymptotically Euclidean manifolds with two trapped sets, one which is unstable and semi-stable. The phase space structure that an inflection transmission set. prove sharp local smoothing estimate for the linear Schr\"odinger equation loss depends on how flat manifold near each sets. result interpolates between similar estimates in \cite{ChWu-lsm}. As consequence techniques proof, we also show high energy resolvent polynomial depending