The supported sector: a normal-jet presentation

jets along the null incidence. On $V_{+}$ let $S^{+}:=\{q=0,r=0\}$, so that $S^{+}=S_{\partial }\cap V_{+}$, with normal variables $n=(q,r)\in {\mathbb{R}}^5$ and base variables $x^{\prime }=(u,\zeta ,y)$, $\zeta \ne 0$; $S^{+}$ has codimension five and $S^{+}\cong {\mathbb{R}}^4\times ({\mathbb{R}}^3\setminus 0)\times {\mathbb{R}}^4$. For $b\in {\mathcal{D}}^{\prime }(S^{+})$ and $\gamma \in {\mathbb{N}}^5$ set

Let ${\mathcal{J}}^k$ be the space of finite sums ${\sum }_{|\gamma |\le k}{\mathsf{E}}_{\gamma }{b}_{\gamma }$, $b_{\gamma }\in {\mathcal{D}}^{\prime }(S^{+})$ ({normal order at most $k$}), and ${{\mathcal{J}}^k}_{\infty }\subseteq {\mathcal{J}}^k$ the subspace with $b_{\gamma }\in C^{\infty }(S^{+})$.