Proposition: Recalled

{XVD} ${\mathsf{Rec}}_T(a)={\sum }_{\gamma }{\mathsf{E}}_{\gamma }({\beta }_{\gamma }(a)T)$; ${\sum }_{\gamma }{\mathsf{E}}_{\gamma }{b}_{\gamma }=0$ iff all $b_{\gamma }=0$; $\mathrm{supp}{\sum }_{\gamma }{\mathsf{E}}_{\gamma }{b}_{\gamma }=({\bigcup }_{\gamma }\mathrm{supp}{b}_{\gamma })\times \{0\}$; $r{F}_0^T={\mathsf{Rec}}^{\circ }$; $H_T\subseteq H$; $\ker {d_0}^T=H_T$.

{XVE} $\beta$ is surjective with right inverse $\sigma (g{)}_{\alpha }(x,q)=g_{\alpha }(x)$; $H={\beta }^{-1}(\mathrm{Ann}(T^{\circ }{)}^{M_r})$, $H_T={\beta }^{-1}(\mathrm{Ann}(T{)}^{M_r})$; ${d_0}^T=0$ iff $\mathrm{Ann}(T)=\mathrm{Ann}(T^{\circ })$.

{XVE, Lemma lem:flat} Let $T$ extend $T^{\circ }$, $K\subset X$ compact, $\varrho >0$ with the closed $\varrho$-neighborhood $K^{\prime }$ of $K$ contained in $X$, and $|⟨T,\psi ⟩|\le C_T{p}_k(\psi )$ for all $\psi \in \mathcal{D}(X)$ with $\mathrm{supp}\psi \subset K^{\prime }$. If $g\in \mathrm{Ann}(T^{\circ })$ has vanishing $k$-jet at every point of $Z$, then $⟨gT,\varphi ⟩=0$ for every $\varphi \in \mathcal{D}(X)$ with $\mathrm{supp}\varphi \subset K$.

{XVE, Theorem thm:point} If $Z$ is finite and $T^{\circ }$ has an extension, it has an extension $T$ with $\mathrm{Ann}(T)=\mathrm{Ann}(T^{\circ })$.

{XVD, Proposition prop:example} Let $d=2$, $X={\mathbb{R}}^2$, $Z=\{0\}$, $m=1$, $r=0$, $T^{\circ }=\delta (x_1){|}_{X^{\circ }}$, $T_1=\delta (x_1)$, $T_2=\delta (x_1)+{\partial }_1{\delta }_0$, $a(x,q)=x_1$. Then $\mathrm{Ann}(T^{\circ })=\{g:g(0,x_2)=0\forall x_2\}$, $a\in H$, ${d_0}^{T_1}(a)=0$, ${d_0}^{T_2}(a)=-{\mathsf{E}}_0{\delta }_0\ne 0$.