Lemma: Reduced kernel separation

Let $w$ be a compact distribution on $M^n$ with reduced species indices. The following are equivalent:

1. $w$ annihilates every tensor tuple of smooth solutions of $\mathbf{P}$;

2. ${\mathbf{E}}_1\cdots {\mathbf{E}}_{n}w=0$;

3. $w={{\sum }_{j=1}}^n{\mathbf{P}}_j{r}_j$ with compact distributions $r_j$.

If $w$ is smooth, the $r_j$ can be smooth. Every such $w$ annihilates every smooth multisolution, not only factorized solution tuples.

Proof. All assertions reduce componentwise to $P=\square +{\mu }^2$. The wave causal operator maps compact smooth tests to smooth spatially compact solutions. Pairing (i) against these solutions and using $E^{\mathsf{T}}=-E$ proves (ii); tensor products of tests separate distributions.

Choose a smooth time function $\chi$ equal to zero sufficiently early and one sufficiently late. For compact $w$, define

Causal support and the bounded transition interval make $r$ compact. Direct differentiation, using the inverses of $P$, gives

Every operator here belongs to $P$; the action causal matrix $\Delta$ is absent.

Apply this construction in each slot. The $T_j$ commute, and

If (ii) holds, this expression vanishes. Telescoping $w-T_1\cdots T_{n}w$ with $\mathrm{id}-T_j={\mathbf{P}}_j{R}_j$ gives (iii), with compact primitives. The construction preserves smoothness. Finally (iii) annihilates every smooth multisolution by transposition. It also proves (i).