Theorem: Structure and dimension of the puncture-supported ambiguity

1. Let $z_0\in X$ and $s\in {\mathcal{D}}^{\prime }(X)$ with $\mathrm{supp}s\subseteq \{z_0\}$ satisfy $|⟨s,\psi ⟩|\le C{p}_k(\psi )$ for all $\psi \in \mathcal{D}(X)$ supported in a compact neighborhood ${K^{\prime }}_0$ of $z_0$. Then there are unique $c_{\mu }\in \mathbb{C}$, $|\mu |\le k$, with $s={\sum }_{|\mu |\le k}{c}_{\mu }(-1{)}^{|\mu |}{\partial }^{\mu }{\delta }_{z_0}$, i.e. $⟨s,\psi ⟩={\sum }_{|\mu |\le k}{c}_{\mu }{\partial }^{\mu }\psi (z_0)$.

2. The ${\mathsf{E}}_{\gamma }{\partial }^{\mu }{\delta }_{z_0}$, $|\gamma |\le r$, $\mu \in {\mathbb{N}}^d$, are linearly independent in ${\mathcal{D}}^{\prime }(U)$.

3. If $Z\ne \varnothing$, ${{\mathcal{V}}^{\mathrm{jet}}}_Z$ is infinite-dimensional.

4. If $Z=\{z_0\}$, the subspace ${{\mathcal{V}}^{\mathrm{jet}}}_{z_0,k}$ of elements all of whose coefficients have order at most $k$ near $z_0$ has basis $\{{\mathsf{E}}_{\gamma }{\partial }^{\mu }{\delta }_{z_0}:|\gamma |\le r,|\mu |\le k\}$ and dimension $M_r{N}_k$.

5. $s\mapsto {\mathsf{Rec}}_s$ is injective on $\{s:\mathrm{supp}s\subseteq Z\}$; hence ${\mathcal{A}}_{\mathrm{ext}}$ is infinite-dimensional whenever $Z\ne \varnothing$, and for $Z=\{z_0\}$ its order-$\le k$ subspace has dimension $N_k$.

Proof. (i) Let $Z_0=\{z_0\}$ and ${T^{\circ }}_0=0\in {\mathcal{D}}^{\prime }(X\setminus Z_0)$; $s$ extends ${T^{\circ }}_0$ and $\mathrm{Ann}({T^{\circ }}_0)=C^{\infty }(X)$. Choose a closed ball $K$ centered at $z_0$ and $\varrho >0$ with closed $\varrho$-neighborhood $K^{\prime }\subset {K^{\prime }}_0$. Let $\psi \in \mathcal{D}(X)$ have vanishing $k$-jet at $z_0$ and $\chi \in \mathcal{D}(\mathrm{int}K)$ with $\chi =1$ near $z_0$. Then $⟨s,(1-\chi )\psi ⟩=0$ since $(1-\chi )\psi$ vanishes near $z_0$, and (R3) applied to $T=s$, $g=\psi$, $\varphi =\chi$ gives $⟨s,\chi \psi ⟩=⟨\psi s,\chi ⟩=0$. So $⟨s,\psi ⟩=0$: the functional factors through the $k$-jet map ${j^k}_{z_0}:\mathcal{D}(X)\to {\mathbb{C}}^{N_k}$, which is surjective (use $(x-z_0{)}^{\mu }\chi$). A linear form on ${\mathbb{C}}^{N_k}$ is $j\mapsto \sum c_{\mu }{j}_{\mu }$; uniqueness follows from surjectivity. (This is the classical structure theorem Horm (Thm. 2.3.4), rederived from (R3) to keep the order bookkeeping explicit.)

(ii) If $\sum c_{\gamma \mu }{\mathsf{E}}_{\gamma }{\partial }^{\mu }{\delta }_{z_0}=0$ (finite sum), pair with ${\mathsf{J}}_{{\gamma }^{\prime }}\psi$ using (eq:extract): ${\sum }_{\mu }{c}_{{\gamma }^{\prime }\mu }(-1{)}^{|\mu |}{\partial }^{\mu }\psi (z_0)=0$ for all $\psi$; with $\psi =(x-z_0{)}^{\nu }\chi$ this is $c_{{\gamma }^{\prime }\nu }(-1{)}^{|\nu |}\nu !=0$.

(iii) Pick $z_0\in Z$; the ${\mathsf{E}}_0{\partial }^{\mu }{\delta }_{z_0}$ lie in ${{\mathcal{V}}^{\mathrm{jet}}}_Z$ and are independent by (ii).

(iv) By (i) the displayed family spans; by (ii) it is independent; its cardinality is $M_r{N}_k$.

(v) For the presentation $e_0$ with $(e_0{)}_0=1$ and all other components zero, ${\mathsf{Rec}}_s(e_0)={\mathsf{E}}_{0}s$, which vanishes iff $s=0$ by (R1). The dimension statements follow from (i)–(iii).