2025/09/11 by Sanchez Sanchez, Yafet E., Schrohe, Elmar
#510 #530
paper · doi:10.15488/19586
Given a globally hyperbolic spacetime M=R×Σ of dimension four and regularity Cτ, we estimate the Sobolev wavefront set of the causal propagator KG of the Klein–Gordon operator. In the smooth case, the propagator satisfies WF′(KG)=C, where C⊂T∗(M×M) consists of those points (x~,ξ~,y~,η~) such that ξ~,η~ are cotangent to a null geodesic γ at x~ resp. y~ and parallel transports of each other along γ. We show that for τ>2, (Formula presented.) for every ϵ>0. Furthermore, in regularity Cτ+2 with τ>2, (Formula presented.) holds for 0<ϵ<τ+12. In the ultrastatic case with Σ compact, we show WF′-32+τ-ϵ(KG)⊂C for ϵ>0 and τ>2 and WF′-32+τ-ϵ(KG)=C for τ>3 and ϵ<τ-3. Moreover, we show that the global regularity of the propagator KG is Hloc-12-ϵ(M×M) as in the smooth case.