Lemma: Smooth causal propagation of the linear boundary kernel

For $q_a=L^{\ast }a$, the causal solution $u_a=E_0{q}_a$ is smooth and spacelike compact.

Proof. Proposition prop:boundary gives $\mathrm{WF}(q_a)\subset N^{\ast }\Sigma$, consisting of timelike covectors. The causal propagator has only paired nonzero characteristic covectors, which are null. Its kernel-composition relation cannot transport a timelike input singularity. Thus the output has empty wavefront set. Its support is contained in $J(\mathrm{supp}{q}_a)$, giving spacelike compactness BGP,Hormander.