The Sobolev Wavefront Set of the Causal Propagator in Finite Regularity

TL;DR

This paper investigates the Sobolev wavefront set of the causal propagator for the Klein-Gordon operator on spacetimes with finite regularity, extending known smooth case results to less regular settings and establishing precise inclusions.

Contribution

It provides new estimates for the Sobolev wavefront set of the causal propagator in finite regularity spacetimes, generalizing smooth case characterizations and analyzing ultrastatic scenarios.

Findings

For $ au>2$, $WF'^{-2+ au- ext{epsilon}}(K_G) ot i C$

In $C^{ au+2}$ regularity, $WF'^{-rac{1}{2}}(K_G)$ contains $C$

In ultrastatic case, $WF'^{-rac{3}{2}+ au- ext{epsilon}}(K_G)= C$ for $ au>3$

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 $\tau>2$, $WF'^{-2+\tau-{\epsilon}}(K_G)\subset C$ 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.