Sharp local smoothing for manifolds with smooth inflection transmission

TL;DR

This paper establishes sharp local smoothing and high energy resolvent estimates for the linear Schrödinger equation on certain manifolds with trapped sets, revealing how geometric flatness influences these estimates.

Contribution

It introduces a novel sharp local smoothing estimate for manifolds with inflection transmission sets, extending previous results and linking geometric flatness to analytical bounds.

Findings

Sharp local smoothing estimate with loss depending on manifold flatness

High energy resolvent estimate with polynomial loss

Interpolation between existing smoothing estimates

Abstract

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