Lemma: Residual testing erases z -covectors

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

where $G_{ab}{u}^{ab}={\sum }_{a,b}{G}_{ab}(z)u^{ab}\in {\mathcal{D}}^{\prime }(W)$. In particular, any wavefront of $u$ whose covector has nonzero $z$-component—for instance a conormal to the attachment diagonal $\{a=z\}$ in the $z$-direction—is invisible to every smooth residual test.

Proof. $u[G]={\mathrm{pr}}_{\ast }(G_{ab}{u}^{ab})$ for the projection $\mathrm{pr}(a,b,z,y)=(a,b,y)$, which is proper on $\mathrm{supp}(G_{ab}{u}^{ab})\subseteq {\mathbb{R}}^{12}\times \mathrm{supp}G$. Write ${\mathrm{pr}}_{\ast }$ as the operator with Schwartz kernel $K(x^{\prime };x)=\delta (x^{\prime }-\mathrm{pr}(x))$, whose wavefront is $\{(x^{\prime },x;\theta ,-{}^{t}d\mathrm{pr}\theta ):x^{\prime }=\mathrm{pr}(x)\}$ by Horm (Thm. 8.2.4), with ${}^{t}d\mathrm{pr}({\xi }_a,{\xi }_b,{\xi }_y)=({\xi }_a,{\xi }_b,0,{\xi }_y)$. The set ${\mathrm{WF}}_X(K)$ of Horm (Thm. 8.2.12) is empty since ${}^{t}d\mathrm{pr}$ is injective, and that theorem gives $\mathrm{WF}({\mathrm{pr}}_{\ast }v)\subseteq \{(\mathrm{pr}(x),\theta ):(x,{}^{t}d\mathrm{pr}\theta )\in \mathrm{WF}(v)\}$, which is the display. The last sentence is the case ${\xi }_z\ne 0$.