Manuscript XI-G@1 · XI-G.tex · lemma · f029bbcd

Lemma: Pairings with solutions are the trisolution

G2: the separation lemma › Solutions and their pairings · lines 907–933 · proof included or linked

XI-G@1#lem:pairings

For $w \in E^{'}$ the following are equivalent.

$⟨ w, γ_{1} \otimes γ_{2} \otimes ϕ ⟩ = 0$ for all smooth solutions $□ γ_{1} = □ γ_{2} = 0$ , $P_{m} ϕ = 0$ on $M$ .
The same for all spatially compact smooth solutions.
$W_{w} = 0$ .

Proof. (i) $\Rightarrow$ (ii) is trivial. (ii) $\Rightarrow$ (i): $w$ is compactly supported, so the pairing depends only on the solutions near the slot projections $K_{i}$ of $supp w$ ; replace each solution by the spatially compact one of Lemma lem:spcompact(c). (ii) $\Leftrightarrow$ (iii): by Lemma lem:spcompact(a),(b) the spatially compact solutions are exactly $E_{0} w_{1}$ , $E_{0} w_{2}$ , $E_{m} w_{3}$ with $w_{i} \in C_{c}^{\infty} (M)$ , and by three slot transpositions,

⟨ w, E_{0} w_{1} \otimes E_{0} w_{2} \otimes E_{m} w_{3} ⟩ = (- 1)^{3} ⟨ W_{w}, w_{1} \otimes w_{2} \otimes w_{3} ⟩ .

Finite sums of products $w_{1} \otimes w_{2} \otimes w_{3}$ are dense in $C_{c}^{\infty} (M^{3})$ , so (ii) holds iff $W_{w} = 0$ .

Displayed formulas in this passage

unlabeled · line 927
$⟨ w, E_{0} w_{1} \otimes E_{0} w_{2} \otimes E_{m} w_{3} ⟩ = (- 1)^{3} ⟨ W_{w}, w_{1} \otimes w_{2} \otimes w_{3} ⟩ .$

Nearby in this section

lemma: Lemma: Spatially compact solutions