Proposition: Joint fixed-width bound

The mixed kernel is compactly supported and satisfies

In particular, ${\mathcal{B}}_f\in {\mathcal{P}}_{\mathrm{mc}}$. Its metric support is contained in

and its scalar support is contained in $K$.

Proof. The retarded wavefront has opposite endpoint covectors: diagonal conormals and the off-diagonal null propagation relation. Smooth multiplication by ${\sigma }_{\epsilon }$ and $f$, followed by differentiation, introduces no new wavefront directions BDH.

If $p$ and $-p$ both belong to the same closed causal cone, then $p=0$, which is excluded. Thus the mixed kernel obeys (eq:microcausal). Lower derivative kernels are obtained by smearing one slot with a smooth configuration. Their wavefront bounds have zero covector in that integrated slot, and hence zero in the remaining slot; they are smooth.

Retarded support gives $z\in J^{-}(x)$ and the ramp removes negative-time dependence. The causal past of a compact observation set within a bounded-time closed slab is compact. Therefore the mixed kernel is compactly supported.