Proposition: Linear smoothing and the sharp limit

For each fixed $a$:

1. $q_{a,ϵ}$ is a smooth compactly supported test tensor, and ${\ell }_{a,ϵ}[h]=⟨q_{a,ϵ},h⟩$ is a regular linear element of ${\mathcal{P}}_{\mathrm{mc}}$;

2. the supports of $q_{a,ϵ}$ are contained in the fixed compact set $J^{-}(\mathrm{supp}a)\cap M_{+}$;

3. $q_{a,ϵ}\to q_a$ in the strong topology of compactly supported distributions;

4. if $b{|}_{{\Sigma }_0}\ne 0$, this sequence cannot converge to an element of any fixed prescribed-wavefront space ${{\mathcal{E}}^{\prime }}_{\Gamma }$ with $\Gamma \subset {\Xi }_1$.

Proof. Smoothness and support follow directly from (eq:smooth-edge). For a smooth test tensor $h$,

The integrand is supported in a slab of time thickness $ϵ$, inside a common spatial compact set. It follows that

The estimate is uniform on bounded sets of smooth test tensors, which proves strong distributional convergence.

If convergence held in ${{\mathcal{E}}^{\prime }}_{\Gamma }$ to an element of that space, continuity of its inclusion into distributions would identify the limit with $q_a$. But Lemma lem:wavefront gives a timelike wavefront component of $q_a$, incompatible with $\Gamma \subset {\Xi }_1$.