Lemma: Kernels and supports

$N_0$ is a well-defined distribution on $M\times M$, and

All three kernels are compactly supported, with

and $y=y^{\prime }$ on the supports of ${\mathcal{T}}_{\mathrm{mat}}$, ${\mathcal{T}}_{\mathrm{self}}$. The coefficients of $\mathcal{M}$ depend on $h_{\mathrm{cl}}$ through $(h_{\mathrm{cl}},\partial h_{\mathrm{cl}},{\partial }^2{h}_{\mathrm{cl}})$ on $J^{-}(K_f)\cap \{t\ge 0\}$ only, linearly.

Proof. $D_1$ has compact support and $z$ ranges over the compact set $J^{-}(K_f)\cap \{t\ge 0\}$; the composition (eq:N0) exists by Lemma lem:sumzero(d) below. For each tree, ${\hat{A}}_1[{{\Delta }_{\mathrm{R}}}^{\mathrm{gr}}S]=\int \hat{\varphi }(x)\int D_1(x,z)({{\Delta }_{\mathrm{R}}}^{\mathrm{gr}}S)(z)\mathrm{d}z\mathrm{d}x=\iint \hat{\varphi }(x)N_0(x,y)S(y)\mathrm{d}y\mathrm{d}x$, and $S$ is read off from (eq:trees): for ${\mathsf{T}}_{\mathrm{mat}}$, $S=\frac{1}{2}g\Theta$ and $\int \psi \Theta =\iint \mathfrak{T}(-{\partial }_y,-{\partial }_{y^{\prime }})[\psi (y)\delta (y-y^{\prime })]:\hat{\varphi }(y)\hat{\varphi }(y^{\prime }):$; for ${\mathsf{T}}_{\mathrm{self}}$ likewise with $\mathfrak{B}$; for ${\mathsf{T}}_{\mathrm{mix}}$, $\int \psi {\sigma }_{\epsilon }{\mathcal{Q}}^{(1)}\hat{h}=\int ({\mathcal{Q}}^{(1)†}[{\sigma }_{\epsilon }\psi ])\hat{h}$. Supports: $x\in K_f$ from $D_1$; $y\in J^{-}(z)$ from ${{\Delta }_{\mathrm{R}}}^{\mathrm{gr}}$, and $g(y)\ne 0$ or ${\sigma }_{\epsilon }(y)\ne 0$ forces $t_y\ge 0$, so $y\in J^{-}(K_f)\cap \{t\ge 0\}$, compact in Minkowski space; $\delta (y-y^{\prime })$ forces $y^{\prime }=y$. Differential operators do not enlarge supports.