Theorem: Root-attached complete-source coefficient

Assume Hypothesis ${\mathrm{H}}_{\mathrm{root}}$ ((R1)–(R3)). For every smooth residual metric $h$ and every smooth scalar $\varphi$, the coefficient ${{\mathcal{C}}^{{\mathfrak{R}}_{\mathrm{L}}}}_{\mathrm{L}}(h,\varphi )$ of (eq:CL) is well defined. Moreover:

1. Bilinearity. ${{\mathcal{C}}^{{\mathfrak{R}}_{\mathrm{L}}}}_{\mathrm{L}}(h,\varphi )$ is linear in $h$ and linear in $\varphi$.

2. Joint continuity. The map $(h,\varphi )\mapsto {{\mathcal{C}}^{{\mathfrak{R}}_{\mathrm{L}}}}_{\mathrm{L}}(h,\varphi )$ is jointly continuous for the $C^{\infty }$ topologies: there exist $C_0$, compact sets $K_h$, $K_{\varphi }$, and finite orders $r,s$ with $|{{\mathcal{C}}^{{\mathfrak{R}}_{\mathrm{L}}}}_{\mathrm{L}}(h,\varphi )|\le C_0{p}_{r,K_h}(h)p_{s,K_{\varphi }}(\varphi )$. The orders $r,s$ are not computed here.

3. Off-diagonal agreement. If $\psi \in {C_c}^{\infty }(M^3)$ has $\mathrm{supp}\psi \cap \{y=w\}=\varnothing$, then $⟨{U^{{\mathfrak{R}}_{\mathrm{L}}}}_{\mathrm{L}}(\cdot ;h),\psi ⟩$ equals the same pairing with each ${\mathfrak{R}}_{\mathrm{L}}({q^{\circ }}_{\lambda })$ replaced by the unextended ${q^{\circ }}_{\lambda }$.

4. Compact support. ${U^{{\mathfrak{R}}_{\mathrm{L}}}}_{\mathrm{L}}(\cdot ;h)$ has compact support, by Lemma lem:Lattach(3).

The functional is not asserted to be nonzero.

Proof. Lemma lem:Lattach(2)–(3): ${U^{{\mathfrak{R}}_{\mathrm{L}}}}_{\mathrm{L}}(\cdot ;h)\in {\mathcal{E}}^{\prime }(M^3)$, and (eq:CL) is the pairing against $\zeta (1\otimes \varphi \otimes 1)$ for any cutoff $\zeta \equiv 1$ near the support, hence defined.

(1) Linearity in $\varphi$: pairing. Linearity in $h$: each ${L^{pq}}_{ij}(w;h)$ is linear in the residual-metric jet (XVI-K, Proposition prop:hess(1)), hence so are ${a^{\mu \nu }}_{\lambda }$ (Definition def:La); ${\pi }^{\ast }{B}_{f,g}$ and ${\Phi }_3^{\ast }{\mathfrak{R}}_{\mathrm{L}}({q^{\circ }}_{\lambda })$ are independent of $h$; Hörmander products are bilinear.

(2) As in XVI-K, Theorem thm:main(2): write $T_{\lambda ,\mu \nu }:={\pi }^{\ast }{B}_{f,g,\mu \nu }\cdot {\Phi }_3^{\ast }{\mathfrak{R}}_{\mathrm{L}}({q^{\circ }}_{\lambda })$, restrict to a fixed compact neighbourhood of the support, of some finite order $m$; Leibniz expands ${\partial }^{\alpha }(\zeta a_{\lambda }\varphi )$ into finitely many derivatives of $\varphi$ and of $a_{\lambda }(\cdot ;h)$, the latter controlled by finitely many derivatives of $h$ on the $w$-projection. The distributional order $m$ is not computed, so $r,s$ are not read off from the pointwise jet bounds of Proposition prop:Ljet.

(3) On $\{y\ne w\}$, ${\Phi }_3^{\ast }{\mathfrak{R}}_{\mathrm{L}}({q^{\circ }}_{\lambda })={\Phi }_3^{\ast }{q^{\circ }}_{\lambda }$ by Proposition prop:RL(1); the product is local.