Theorem: Descent by a finite delta correction

Let $Z=\{z_0\}$ and suppose $T^{\circ }$ has some extension $\tilde{T}\in {\mathcal{D}}^{\prime }(X)$. Let $\bar{B}=\bar{B}(z_0,{\varrho }_0)\subset X$ be a closed ball and $k$ an order of $\tilde{T}$ on $\bar{B}$. Then there is a distribution

such that $T=\tilde{T}-s$ extends $T^{\circ }$ and $\mathrm{Ann}(T)=\mathrm{Ann}(T^{\circ })$. Consequently $H_T=H$, ${d_0}^T=0$, and ${F_0}^T={F_0}^{\tilde{T}}-{\mathsf{Rec}}_s$ is a descending lift in the class of extension changes.

Proof. Let $B=B(z_0,{\varrho }_0)$ be the open ball and $A_B=\{g\in \mathrm{Ann}(T^{\circ }):\mathrm{supp}g\subset B\}$. Define $\lambda :A_B\to \mathbb{C}$, $\lambda (g)=⟨\tilde{T},g⟩$.

{Step 1: $\lambda$ factors through $k$-jets.} Let $g\in A_B$ have vanishing $k$-jet at $z_0$. Choose $\chi \in \mathcal{D}(B)$ with $\chi =1$ on $\mathrm{supp}g$. Apply Lemma lem:flat with $K=\mathrm{supp}\chi$, $\varrho$ so small that $K^{\prime }\subset \bar{B}$, and the order-$k$ bound of $\tilde{T}$ on $\bar{B}$: $⟨\tilde{T},g⟩=⟨g\tilde{T},\chi ⟩=0$. Hence $\lambda (g)$ depends only on ${j^k}_{z_0}g=({\partial }^{\mu }g(z_0){)}_{|\mu |\le k}$, i.e. $\lambda ={\Lambda }_0\circ {j^k}_{z_0}$ for a linear functional ${\Lambda }_0$ on the subspace ${j^k}_{z_0}(A_B)$ of the finite-dimensional jet space $J^k={\mathbb{C}}^{\#\{|\mu |\le k\}}$.

Step 2: the correction. Extend ${\Lambda }_0$ linearly to $\Lambda$ on $J^k$ and write $\Lambda (j)={\sum }_{|\mu |\le k}{c}_{\mu }{j}_{\mu }$. With $s$ as displayed, $⟨s,g⟩={\sum }_{\mu }{c}_{\mu }{\partial }^{\mu }g(z_0)=\Lambda ({j^k}_{z_0}g)$ for all $g\in \mathcal{D}(X)$. Set $T=\tilde{T}-s$. Since $\mathrm{supp}s=\{z_0\}\subset Z$, $T$ extends $T^{\circ }$. For $g\in A_B$, $⟨T,g⟩=\lambda (g)-\Lambda ({j^k}_{z_0}g)=0$.

{Step 3: $\mathrm{Ann}(T)=\mathrm{Ann}(T^{\circ })$.} The inclusion $\mathrm{Ann}(T)\subseteq \mathrm{Ann}(T^{\circ })$ holds for every extension. Let $g\in \mathrm{Ann}(T^{\circ })$ and $\varphi \in \mathcal{D}(X)$; fix $\chi \in \mathcal{D}(B)$ with $\chi =1$ near $z_0$. Then $(1-\chi )g\varphi \in \mathcal{D}(X^{\circ })$ and $⟨T,(1-\chi )g\varphi ⟩=⟨T^{\circ },(1-\chi )g\varphi ⟩=⟨g{|}_{X^{\circ }}{T}^{\circ },(1-\chi )\varphi ⟩=0$; and $\chi g\varphi \in A_B$, so $⟨T,\chi g\varphi ⟩=0$ by Step 2. Hence $⟨gT,\varphi ⟩=0$ for all $\varphi$, i.e. $g\in \mathrm{Ann}(T)$.

The consequences follow from Theorem thm:ann and linearity of $T\mapsto {\mathsf{Rec}}_T$.