Theorem: Separation and the admission criterion

Let $u$ be the frozen kernel.

1. $v\in [u{]}_{\mathrm{sol}}$ if and only if $v-u\in {\mathcal{I}}_{\mathrm{sol}}$ if and only if ${\mathsf{W}}_v={\mathsf{W}}_u$. In particular every pairing of $u$ with a triple of free solutions is a class invariant, and no bulk equation term changes it.

2. $[u{]}_{\mathrm{sol}}$ meets ${\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$ if and only if

   $$\mathrm{WF}({\mathsf{W}}_u)\cap ({\tilde{V}}_{+}^3\cup {\tilde{V}}_{-}^3)=\varnothing .$$

   When (eq:criterion) holds, an explicit representative is $\tilde{u}={\square }_1{\square }_2{P}_{m,3}[{\chi }_1{\chi }_2{\chi }_3{\mathsf{W}}_u]$.

Proof. (a) is Lemma lem:pairings applied to $w=v-u$ together with Lemma lem:kernel. (b) “Only if”: if $v\in [u{]}_{\mathrm{sol}}\cap {\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$ then ${\mathsf{W}}_u={\mathsf{W}}_v$ by (a), and by Lemma lem:transport every covector triple of $\mathrm{WF}({\mathsf{W}}_v)$ is a covector triple of $\mathrm{WF}(v)$, none of which lies in ${\tilde{V}}_{+}^3\cup {\tilde{V}}_{-}^3$ by (eq:mc). “If”: $\tilde{u}$ is compactly supported and lies in $u+{\mathcal{I}}_{\mathrm{sol}}$ by (eq:decomp), hence in $[u{]}_{\mathrm{sol}}$ by (a); it is symmetric in the metric slots because $u$ is and ${\chi }_1={\chi }_2$. Multiplication by smooth functions and differential operators do not enlarge wavefront sets Hormander ((8.1.11)), so $\mathrm{WF}(\tilde{u})\subset \mathrm{WF}({\mathsf{W}}_u)$, and (eq:criterion) gives (eq:mc). Lemma lem:reduction converts $\tilde{u}$ into an element of ${\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$ restricting to ${\mathcal{C}}_3(f,g)$ on ${\mathcal{C}}_{\mathrm{sol}}$.