Proposition: Support and continuity

Let ${\mathcal{D}}_f=J^{-}(\mathrm{supp}f)\cap M_T$, a compact set.

1. ${{\mathcal{O}}^{(j)}}_{f,g,\epsilon }[h,k,\varphi ]$, $j=0,1,2$, depends on $(h,k,\varphi )$ only through their restrictions to every neighbourhood of ${\mathcal{D}}_f$; the same holds for the on-shell evaluation, with ${\varphi }_0$ in place of $\varphi$.

2. For fixed $\epsilon$, $h\mapsto {\mathsf{X}}_1[h]$ is a continuous linear map and $(h,k)\mapsto {\mathsf{X}}_2[h,k]$ a continuous polynomial map of $C^{\infty }$ into $C^{\infty }$, with respect to the smooth topologies on compact sets; $(h,k,\varphi )\mapsto {{\mathcal{O}}^{(j)}}_{f,g,\epsilon }$ is a continuous polynomial functional, and $f\mapsto {{\mathcal{O}}^{(j)}}_{f,g,\epsilon }[h,k,\varphi ]$ is a continuous linear functional on ${C_c}^{\infty }({M_T}^{\circ })$.

Proof. (a) By Lemma lem:cauchy, ${\mathsf{X}}_j(z)$ depends on $(h,k)$ in $J^{-}(z)\cap M_T$; the local differential operators in (eq:X1eq)–(eq:X2eq) do not enlarge support, and nested terms obey $J^{-}(z^{\prime })\cap M_T\subset J^{-}(z)\cap M_T$ for $z^{\prime }\in J^{-}(z)$. The relational coefficients at $z$ involve $\varphi$ and its derivatives at $z$. Taking the union over $z\in \mathrm{supp}f$ gives ${\mathcal{D}}_f$. The same argument applies to ${\rho }_{j,g}$. (b) The estimate in Lemma lem:cauchy and the continuity of products and derivatives in $C^{\infty }$ give the continuity of the coefficient maps; pairing with $f$ is continuous.