The Sobolev Wavefront Set of the Causal Propagator in Finite Regularity

Abstract

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 $W{F}^{\prime }(K_G)=C$, where $C\subset T^{\ast }(M\times M)$ 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 $\tau >2$, $W{F}^{\prime -2+\tau -\epsilon }(K_G)\subset C$ for every $\epsilon >0$. Furthermore, in regularity $C^{\tau +2}$ with $\tau >2$, $C\subset W{F}^{\prime -\frac{1}{2}}(K_G)\subset W{F}^{\prime \tau -\epsilon }(K_G)\subset C$ holds for $0<\epsilon <\tau +\frac{1}{2}$. In the ultrastatic case with $\Sigma$ compact, we show $W{F}^{\prime -\frac{3}{2}+\tau -\epsilon }(K_G)\subset C$ for $\epsilon >0$ and $\tau >2$ and $W{F}^{\prime -\frac{3}{2}+\tau -\epsilon }(K_G)=C$ for $\tau >3$ and $\epsilon <\tau -3$. Moreover, we show that the global regularity of the propagator $K_G$ is $H_{loc}^{-\frac{1}{2}-\epsilon }(M\times M)$ as in the smooth case.