Lemma: Smoothing by the causal propagator

If $q$ is a compactly supported tensor distribution with

then $u=Eq$ is a smooth, spacelike-compact solution of $\square u=0$.

Proof. The causal propagator extends to compactly supported distributions. Its distribution kernel has only paired characteristic covectors: both nonzero covectors in each wavefront pair are null. In the kernel-composition wavefront bound, a singularity in $Eq$ would require a null input covector from $\mathrm{WF}(q)$, or a zero input covector in the wavefront set of the propagator kernel. Neither occurs: $\mathrm{WF}(q)$ is timelike, and the causal propagator has no such zero-input wavefront component. Hence $Eq$ is smooth. This is an application of the standard kernel wavefront calculus reviewed in BDHwavefront,FR.

The bisolution identity gives $\square Eq=0$, and the support bound

gives spacelike compactness.