Lemma: Spatially compact solutions

Let $\mu \in \{0,m\}$, $P=\square +{\mu }^2$, and let $\chi \in C^{\infty }(\mathbb{R})$ with $\chi (t)=0$ for $t\le -T_0$ and $\chi (t)=1$ for $t\ge T_0$.

1. For $w\in {C_c}^{\infty }(M)$, ${\mathsf{E}}_{\mu }w$ is a smooth solution of $P\Phi =0$ with $\mathrm{supp}{\mathsf{E}}_{\mu }w\subset J(\mathrm{supp}w)$, hence spatially compact.

2. Every smooth solution $\Phi$ of $P\Phi =0$ with $\mathrm{supp}\Phi \subset J(K)$, $K$ compact, is $\Phi ={\mathsf{E}}_{\mu }w$ with $w:=P(\chi \Phi )\in {C_c}^{\infty }(M)$.

3. For every compact $K\subset M$ and every smooth solution $\Phi$ there is a spatially compact smooth solution ${\Phi }_K$ with ${\Phi }_K=\Phi$ on a neighbourhood of $K$.

Proof. (a) is the support property of ${{\mathsf{E}}_{\mu }}^{\mathrm{R}/A}$ and $P{{\mathsf{E}}_{\mu }}^{\mathrm{R}/A}w=w$. (b): $P(\chi \Phi )=[P,\chi ]\Phi$ is supported in $\{|t|\le T_0\}\cap J(K)$, which is compact. $\chi \Phi$ has past-compact support and solves $P(\chi \Phi )=w$, so by uniqueness of the retarded solution ${{\mathsf{E}}_{\mu }}^{\mathrm{R}}w=\chi \Phi$; likewise ${{\mathsf{E}}_{\mu }}^{\mathrm{A}}w=(\chi -1)\Phi$. Subtracting, ${\mathsf{E}}_{\mu }w=\Phi$. (c): choose a ball $B\subset \Sigma$ whose domain of dependence contains a neighbourhood of $K$, and $\vartheta \in {C_c}^{\infty }({\mathbb{R}}^3)$ with $\vartheta =1$ on $B$. Let ${\Phi }_K$ solve $P{\Phi }_K=0$ with Cauchy data $\vartheta \cdot (\Phi ,{\partial }_t\Phi ){|}_{\Sigma }$. By finite propagation XA,BGP, ${\Phi }_K=\Phi$ on the domain of dependence of $B$ and $\mathrm{supp}{\Phi }_K\subset J(\mathrm{supp}\vartheta )$.