The Sobolev Wavefront Set of the Causal Propagator in Finite Regularity

Abstract

Given a globally hyperbolic spacetime $M=\mathbb{R}\times Σ$ of dimension four and regularity $C^τ$, 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 $(\tilde{x},\tildeξ,\tilde{y},\tildeη)$ such that $\tildeξ,\tildeη$ are cotangent to a null geodesic $γ$ at $\tilde{x}$ resp. $\tilde{y}$ and parallel transports of each other along $γ$. We show that for $τ>2$, $WF'^{-2+τ-ε}(K_G)\subset C$ for every $ε>0$. Furthermore, in regularity $C^{τ+2}$ with $τ>2$, $C\subset WF'^{-\frac{1}{2}}(K_G)\subset WF'^{τ-ε}(K_G)\subset C$ holds for $0<ε<τ+\frac{1}{2}$. In the ultrastatic case with $Σ$ compact, we show $WF'^{-\frac{3}{2}+τ-ε}(K_G)\subset C$ for $ε>0$ and $τ>2$ and $WF'^{-\frac{3}{2}+τ-ε}(K_G)= C$ for $τ>3$ and $ε<τ-3$. Moreover, we show that the global regularity of the propagator $K_G$ is $H^{-\frac{1}{2}-ε}_{loc}(M\times M)$ as in the smooth case.