Lemma: Partial pairing with XIV-G's test

For $u\in {\mathcal{D}}^{\prime }(W;E_{\mathrm{res}})$ and $G\in {C^{\infty }}_c(Z;{\mathrm{Sym}}^2{T}^{\ast }M)$, the prescription

defines $u[G]\in {\mathcal{D}}^{\prime }({\mathbb{R}}^{12})$; the map $G\mapsto u[G]$ is linear and continuous from ${C^{\infty }}_c(Z;{\mathrm{Sym}}^2{T}^{\ast }M)$ to ${\mathcal{D}}^{\prime }({\mathbb{R}}^{12})$.

Proof. $\varphi \mapsto G\otimes \varphi$ is linear and continuous $\mathcal{D}({\mathbb{R}}^{12})\to {\Gamma }_c$ (supports stay in ${\mathrm{pr}}_{(a,b,y)}^{-1}(\mathrm{supp}\varphi )\cap {\mathrm{pr}}_z^{-1}(\mathrm{supp}G)$, compact, and seminorms are multiplicative). Composition with the continuous $u$ gives $u[G]\in {\mathcal{D}}^{\prime }$. Continuity in $G$ is the same estimate read in the other factor.