The Sobolev Wavefront Set of the Causal Propagator in Finite Regularity

Given a globally hyperbolic spacetime $M=\mathbb{R}\times \Sigma$ of dimension four and regularity $C^\tau$, we estimate the Sobolev wavefront set of the causal propagator $K_G$ of the Klein-Gordon operator. In the smooth case, the propagator satisfies $WF'(K_G)=C$, where $C\subset T^*(M\times M)$ consists of those points $(\tilde{x},\tilde{\xi},\tilde{y},\tilde{\eta})$ such that $\tilde{\xi},\tilde{\eta}$ are cotangent to a null geodesic $\gamma$ at $\tilde{x}$ resp. $\tilde{y}$ and parallel transports of each other along $\gamma$. We show that for $\tau>2$, $WF'^{-2+\tau-{\epsilon}}(K_G)\subset C$ for every ${\epsilon}>0$. Furthermore, in regularity $C^{\tau+2}$ with $\tau>2$, $C\subset WF'^{-\frac{1}{2}}(K_G)\subset WF'^{\tau-\epsilon}(K_G)\subset C$ holds for $0<\epsilon<\tau+\frac{1}{2}$. In the ultrastatic case with $\Sigma$ compact, we show $WF'^{-\frac{3}{2}+\tau-\epsilon}(K_G)\subset C$ for $\epsilon >0$ and $\tau>2$ and $WF'^{-\frac{3}{2}+\tau-\epsilon}(K_G)= C$ for $\tau>3$ and $\epsilon<\tau-3$. Moreover, we show that the global regularity of the propagator $K_G$ is $H^{-\frac{1}{2}-\epsilon}_{loc}(M\times M)$ as in the smooth case.