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 \(WF'(K_G)=C\), 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 \(\tau >2\),

$$\begin{aligned} WF'^{-2+\tau -{\epsilon }}(K_G)\subset C \end{aligned}$$

for every \({\epsilon }>0\). Furthermore, in regularity \(C^{\tau +2}\) with \(\tau >2\),

$$\begin{aligned} C\subset WF'^{-\frac{1}{2}}(K_G)\subset WF'^{\tau -\epsilon }(K_G)\subset C \end{aligned}$$

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.