Lemma: Detection by spatially compact solutions

For $d\in \mathcal{D}$ the following are equivalent:

1. $d\in P_m\mathcal{D}$, i.e.\ $\hat{\varphi }(d)=0$;

2. ${\Delta }_{S}d=0$;

3. $⟨d,u⟩=0$ for every $u\in {\Delta }_S\mathcal{D}$ (every smooth spatially compact solution);

4. $⟨d,u⟩=0$ for every smooth solution $u$ of $P_{m}u=0$.

Proof. (a)$\Rightarrow$(d): $⟨P_{m}v,u⟩=⟨v,P_{m}u⟩=0$, the integration by parts being justified by compactness of $\mathrm{supp}v$. (d)$\Rightarrow$(c) is trivial. (c)$\Rightarrow$(b): for $e\in \mathcal{D}$, $⟨d,{\Delta }_{S}e⟩=⟨{{\Delta }_S}^{\mathsf{T}}d,e⟩=-⟨{\Delta }_{S}d,e⟩$; vanishing for all $e$ forces the smooth function ${\Delta }_{S}d$ to vanish. (b)$\Rightarrow$(a): if ${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}d={{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}d=:v$, then $\mathrm{supp}v\subset J^{+}(\mathrm{supp}d)\cap J^{-}(\mathrm{supp}d)$, which is compact in Minkowski space (global hyperbolicity), so $v\in \mathcal{D}$ and $P_{m}v=P_m{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}d=d$. The parenthetical identification in (c) is BGP (Thm. 3.4.7).