Proposition: Smooth first-derivative kernels

Fix a configuration $(h,k,\varphi )$. The first functional derivatives of the observable coefficients are:

where $(f\mathrm{d}{\varphi }_0{)}_{\mu }=f{\partial }_{\mu }{\varphi }_0$, and, for a smooth symmetric direction $\tilde{h}$,

with

Every one of these derivatives is represented by a smooth compactly supported function or tensor on $M_T$, supported in ${\mathcal{D}}_f$. In particular the metric derivatives have no hypersurface delta term.

Proof. Equations (eq:kern1)–(eq:kern4) follow from (eq:Phi1)–(eq:Phi2) by differentiation and integration by parts in the $\varphi$ slots; (eq:X2deriv) follows from (eq:Xsol) and the polarization of $a_2$. The $h$-derivative of ${\mathcal{O}}^{(1)}$ and the $k$-derivative of ${\mathcal{O}}^{(2)}$ are $-{\mathsf{J}}_{\epsilon }(f\mathrm{d}{\varphi }_0)$, whose kernel is given by Lemma lem:kernel. For (eq:kern4), each term is of the form ${\mathsf{J}}_{\epsilon }(c)[\tilde{h}]$ for a smooth compactly supported covector $c$ built from $f$, $\varphi$, ${\mathsf{X}}_1[h]$, and their derivatives, or of the form $\int ({\mathsf{E}}_{\mathrm{A}}c)\cdot ({\mathcal{V}}_{\epsilon }[\tilde{h}]{\mathsf{X}}_1[h]+2{\partial }_{\alpha }({\sigma }_{\epsilon }{a_2}^{\alpha \mu }[h,\tilde{h}]))$ by Lemma lem:transpose with $c=f\mathrm{d}{\varphi }_0$; after integrating by parts, the latter is the pairing of $\tilde{h}$ with a smooth compactly supported tensor built from ${\sigma }_{\epsilon }$, ${\mathsf{E}}_{\mathrm{A}}c$, ${\mathsf{X}}_1[h]$, $h$, and derivatives. The term $\int ({\mathsf{E}}_{\mathrm{A}}c)\cdot {\mathcal{V}}_{\epsilon }[h]{\mathsf{X}}_1[\tilde{h}]$ equals ${\mathsf{J}}_{\epsilon }(c^{\prime })[\tilde{h}]$ with ${c^{\prime }}_{\mu }={\partial }_{\beta }({\partial }_{\alpha }({\mathsf{E}}_{\mathrm{A}}c{)}_{\mu }{\sigma }_{\epsilon }{a_1}^{\alpha \beta }[h])$ smooth and compactly supported, hence has kernel ${\sigma }_{\epsilon }{C}^{\ast }({\mathsf{E}}_{\mathrm{A}}{c}^{\prime })$. Supports follow from Proposition prop:continuity(a).