The Sobolev Wavefront Set of the Causal Propagator in Finite Regularity

Abstract

AbstractGiven a globally hyperbolic spacetime $$M={\mathbb {R}}\times \Sigma $$ M = R × Σ of dimension four and regularity $$C^\tau $$ C τ , we estimate the Sobolev wavefront set of the causal propagator $$K_G$$ K G of the Klein–Gordon operator. In the smooth case, the propagator satisfies $$WF'(K_G)=C$$ W F ′ ( K G ) = C , where $$C\subset T^*(M\times M)$$ C ⊂ T ∗ ( M × M ) consists of those points $$(\tilde{x},\tilde{\xi },\tilde{y},\tilde{\eta })$$ ( x ~ , ξ ~ , y ~ , η ~ ) such that $$\tilde{\xi },\tilde{\eta }$$ ξ ~ , η ~ are cotangent to a null geodesic $$\gamma $$ γ at $$\tilde{x}$$ x ~ resp. $$\tilde{y}$$ y ~ and parallel transports of each other along $$\gamma $$ γ . We show that for $$\tau >2$$ τ > 2 , $$\begin{aligned} WF'^{-2+\tau -{\epsilon }}(K_G)\subset C \end{aligned}$$ W F ′ - 2 + τ - ϵ ( K G ) ⊂ C for every $${\epsilon }>0$$ ϵ > 0 . Furthermore, in regularity $$C^{\tau +2}$$ C τ + 2 with $$\tau >2$$ τ > 2 , $$\begin{aligned} C\subset WF'^{-\frac{1}{2}}(K_G)\subset WF'^{\tau -\epsilon }(K_G)\subset C \end{aligned}$$ C ⊂ W F ′ - 1 2 ( K G ) ⊂ W F ′ τ - ϵ ( K G ) ⊂ C holds for $$0<\epsilon <\tau +\frac{1}{2}$$ 0 < ϵ < τ + 1 2 . In the ultrastatic case with $$\Sigma $$ Σ compact, we show $$WF'^{-\frac{3}{2}+\tau -\epsilon }(K_G)\subset C$$ W F ′ - 3 2 + τ - ϵ ( K G ) ⊂ C for $$\epsilon >0$$ ϵ > 0 and $$\tau >2$$ τ > 2 and $$WF'^{-\frac{3}{2}+\tau -\epsilon }(K_G)= C$$ W F ′ - 3 2 + τ - ϵ ( K G ) = C for $$\tau >3$$ τ > 3 and $$\epsilon <\tau -3$$ ϵ < τ - 3 . Moreover, we show that the global regularity of the propagator $$K_G$$ K G is $$H^{-\frac{1}{2}-\epsilon }_{loc}(M\times M)$$ H loc - 1 2 - ϵ ( M × M ) as in the smooth case.