Theorem: Jet-detection is exactly ``some extension carries a defect''

Suppose $T^{\circ }$ has an extension. For $a\in H$ the following are equivalent: (i) $a$ is jet-detected; (ii) there is an extension $T$ with ${d_0}^T(a)\ne 0$. For a fixed extension a jet-detected relation may have zero defect: in (R5), $a=x_1$ is jet-detected, ${d_0}^{T_1}(a)=0$, ${d_0}^{T_2}(a)\ne 0$. If $Z$ is finite, every jet-detected $a$ admits extensions $T^{\prime }$, $T^{\prime \prime }$ with ${d_0}^{T^{\prime }}(a)=0$ and ${d_0}^{T^{\prime \prime }}(a)\ne 0$.

Proof. (ii)$\Rightarrow$(i). If $a$ is not jet-detected, every ${\beta }_{\gamma }(a)$ is flat on $Z$ and lies in $\mathrm{Ann}(T^{\circ })$ by (R2). Given $T$ and a compact $K$, choose $\varrho$, $K^{\prime }$ as in (R3) and let $k$ be an order of $T$ on $K^{\prime }$; (R3) gives $⟨{\beta }_{\gamma }(a)T,\varphi ⟩=0$ for $\mathrm{supp}\varphi \subset K$. As $K$ is arbitrary, ${\beta }_{\gamma }(a)T=0$ for all $\gamma$, so ${d_0}^T(a)=0$ by (R1).

(i)$\Rightarrow$(ii). Let $\tilde{T}$ extend $T^{\circ }$; choose $\gamma$ and $z_0\in Z$ with $g={\beta }_{\gamma }(a)$ not flat at $z_0$, and $\mu$ minimal with ${\partial }^{\mu }g(z_0)\ne 0$. For $c\in \mathbb{C}$ set $T_c=\tilde{T}+c{\partial }^{\mu }{\delta }_{z_0}$, an extension of $T^{\circ }$ since $\mathrm{supp}{\partial }^{\mu }{\delta }_{z_0}\subset Z$. By Lemma lem:deltamult, $g{T}_c=g\tilde{T}+c(-1{)}^{|\mu |}{\partial }^{\mu }g(z_0){\delta }_{z_0}$, an affine function of $c$ with nonzero direction; it vanishes for at most one $c$. For any other $c$, ${\beta }_{\gamma }(a)T_c\ne 0$, hence ${d_0}^{T_c}(a)\ne 0$ by (R1). The example is (R5); the last sentence is (R4) together with the construction of $T_c$.