Lemma: Uniqueness of normal-jet coefficients

If ${\sum }_{|\gamma |\le k}{\mathsf{E}}_{\gamma }{b}_{\gamma }=0$ in ${\mathcal{D}}^{\prime }(V_{+})$ with $b_{\gamma }\in {\mathcal{D}}^{\prime }(S^{+})$, then all $b_{\gamma }=0$. Hence the map $(b_{\gamma }{)}_{|\gamma |\le k}\mapsto {\sum }_{\gamma }{\mathsf{E}}_{\gamma }{b}_{\gamma }$ from ${\mathcal{D}}^{\prime }(S^{+}{)}^{\#\{|\gamma |\le k\}}$ onto ${\mathcal{J}}^k$ is a linear isomorphism.

Proof. For $\psi \in \mathcal{D}(S^{+})$ and $\chi \in \mathcal{D}({\mathbb{R}}^5)$ equal to one near $0$, set ${\mathsf{J}}_{{\gamma }^{\prime }}\psi (x^{\prime },n)=\frac{(-1{)}^{|{\gamma }^{\prime }|}}{{\gamma }^{\prime }!}{n}^{{\gamma }^{\prime }}\chi (n)\psi (x^{\prime })$. By (eq:insert), $⟨{\mathsf{E}}_{\gamma }b,{\mathsf{J}}_{{\gamma }^{\prime }}\psi ⟩=(-1{)}^{|\gamma |}⟨b,\psi ⟩\cdot \frac{(-1{)}^{|{\gamma }^{\prime }|}}{{\gamma }^{\prime }!}{{\partial }_n}^{\gamma }(n^{{\gamma }^{\prime }}\chi )(0)={\delta }_{\gamma {\gamma }^{\prime }}⟨b,\psi ⟩$. Pairing the vanishing sum with ${\mathsf{J}}_{{\gamma }^{\prime }}\psi$ gives $⟨b_{{\gamma }^{\prime }},\psi ⟩=0$ for all $\psi$.