The Sobolev Wavefront Set of the Causal Propagator in Finite Regularity

In this paper we estimate the Sobolev wavefront set for the causal propagator $K_G$ of the Klein-Gordon equation in an ultrastatic spacetime when the regularity of the metric is $C^\tau$ with $\tau>2$ and $C^{1,1}$. Our main tools are a propagation of singularities result for non-smooth pseudodifferential operators and eigenvalue asymptotics for elliptic operators of low regularity.