Lemma: Extraction and support

Let $b_{\gamma }\in {\mathcal{D}}^{\prime }(X)$ for $|\gamma |\le r$ and $t={\sum }_{|\gamma |\le r}{\mathsf{E}}_{\gamma }{b}_{\gamma }$. Then

1. $t=0$ if and only if $b_{\gamma }=0$ for all $|\gamma |\le r$;

2. $\mathrm{supp}t=\left({\bigcup }_{|\gamma |\le r}\mathrm{supp}{b}_{\gamma }\right)\times \{0\}$;

3. $t{|}_{U^{\circ }}={\sum }_{|\gamma |\le r}{{\mathsf{E}}^{\circ }}_{\gamma }(b_{\gamma }{|}_{X^{\circ }})$.

Proof. Choose $\chi \in \mathcal{D}(V_q)$ equal to one near $0$ and set ${\mathsf{J}}_{\gamma }\psi (x,q)=\frac{(-1{)}^{|\gamma |}}{\gamma !}{q}^{\gamma }\chi (q)\psi (x)$ for $\psi \in \mathcal{D}(X)$. Since $\chi$ is constant near $0$, $(-1{)}^{|\beta |}{{\partial }_q}^{\beta }{\mathsf{J}}_{\gamma }\psi (x,0)={\delta }_{\beta \gamma }\psi (x)$ for all $\beta$, so $⟨t,{\mathsf{J}}_{\gamma }\psi ⟩=⟨b_{\gamma },\psi ⟩$. This proves (i) and the inclusion “$\supseteq$” in (ii): if $t$ vanishes near $(x_0,0)$, choose $\chi$ supported in the normal neighborhood and $\psi$ supported near $x_0$ to see that every $b_{\gamma }$ vanishes near $x_0$. The inclusion “$\subseteq$” in (ii) is immediate from (eq:insert): a test vanishing near $(\bigcup \mathrm{supp}{b}_{\gamma })\times \{0\}$ has normal jets vanishing on $\bigcup \mathrm{supp}{b}_{\gamma }$. For (iii), a test $\varphi \in \mathcal{D}(U^{\circ })$ has ${{\partial }_q}^{\gamma }\varphi (\cdot ,0)\in \mathcal{D}(X^{\circ })$, so $⟨{\mathsf{E}}_{\gamma }{b}_{\gamma },\varphi ⟩=(-1{)}^{|\gamma |}⟨b_{\gamma }{|}_{X^{\circ }},{{\partial }_q}^{\gamma }\varphi (\cdot ,0)⟩$.