Lemma: Admission of the three trees

$\mathrm{WF}(D_1)\subset {\Sigma }_2$, $\mathrm{WF}(N_0)\subset {\Sigma }_2$, $\mathrm{WF}(\mathcal{M})\subset {\Sigma }_2$, and $\mathrm{WF}({\mathcal{T}}_{\mathrm{mat}}),\mathrm{WF}({\mathcal{T}}_{\mathrm{self}})\subset {\Sigma }_3$. Hence each of the three kernels of Lemma lem:kernels is compactly supported and satisfies (eq:microcausal), and ${\mathsf{T}}_{\mathrm{mat}},{\mathsf{T}}_{\mathrm{mix}},{\mathsf{T}}_{\mathrm{self}}\in \mathfrak{W}$ at the fixed $\epsilon$.

Proof. The local terms of $D_1$ are differential operators with smooth compactly supported coefficients ($f$, $g$, ${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}f$) applied to $\delta (x-z)$; the nonlocal term $\Lambda$ is a derivative of ${{\mathsf{E}}_0}^{\mathrm{R}}(x,z)$ times the smooth function ${\sigma }_{\epsilon }(z)$. Both lie in ${\Sigma }_2$ by Lemma lem:sumzero(a),(b). $N_0=D_1\circ {{\Delta }_{\mathrm{R}}}^{\mathrm{gr}}$ lies in ${\Sigma }_2$ by (d). $\mathcal{M}$ is a differential operator with smooth coefficients applied to ${\sigma }_{\epsilon }{N}_0$, so (b). The two cubic kernels are differential operators with constant coefficients applied to $(g{N}_0)\delta (y-y^{\prime })$, respectively $({\sigma }_{\epsilon }{N}_0)\delta (y-y^{\prime })$, so (b) and (c). Compact supports are Lemma lem:kernels; (eq:microcausal) is (e). The admission criterion is xe:algebra.