Propagation of singularities for rough metrics

Abstract

We use a wave packet transform and weighted norm estimates in phase space to establish propagation of singularities for solutions to time-dependent scalar hyperbolic equations that have coefficients of limited regularity. It is assumed that the second order derivatives of the principal coefficients belong to $L_t^1{L}_x^{\infty }$, and that $u$ is a solution to the homogeneous equation of global Sobolev regularity $s_0=0$ or 1. It is then proven that the $H^{s_0+1}$ wavefront set of $u$ is a union of maximally extended null bicharacteristic curves.