Conformal Perturbations and Local Smoothing

The purpose of this paper is to study the effect of conformal perturbations on the local smoothing effect for the Schr\"odinger equation on surfaces of revolution. The paper \cite{ChWu-lsm} studied the Schr\"odinger equation on surfaces of revolution with one trapped orbit. The dynamics near this trapping were unstable, but degenerately so. Beginning from the metric $g$ from this paper, we consider the perturbed metric $g_s = e^{sf}g$, where $f$ is a smooth, compactly supported function. If $s$ is small enough and finitely many derivatives of $f$ satisfy appropriate symbolic estimates, then we show that a local smoothing estimate still holds.