Theorem: Normal order of the supported sector

Under the hypotheses of (P1) with the order bound $|\alpha |+|\beta |\le 2$ of (eq:Q), the restrictions to $V_{+}$ satisfy

normal jets along $S^{+}$ with coefficients smooth on $S^{+}$ and independent of $y$. The supports of $K_6$, $K_8$, $K_9$ meet $W\setminus V_{+}$ only in $\{a=b=z\}$.

Proof. {$K_6$.} ${\partial }_{a^{\alpha }}R(b-a)=-({\partial }_{\alpha }R)(d)=-\frac{1}{\pi }{d}_{\alpha }{\delta }^{\prime }(q)$, and ${\delta }^{(4)}(a-z)=\delta (r)$; the product is $-\frac{1}{\pi }{d}_{\alpha }{\mathsf{E}}_{e_q}(1)\in {{\mathcal{J}}^1}_{\infty }$ by Lemma lem:taylor5(i). Multiplication by the smooth $f(b){\sigma }_{\epsilon }(a)\mathsf{D}$ keeps ${{\mathcal{J}}^1}_{\infty }$; the outer ${\partial }_{b^{\mu }}$ is first order (Lemma lem:vf), so $K_6\in {{\mathcal{J}}^2}_{\infty }$.

{$K_9$.} By Lemma lem:T0chart, ${\mathcal{T}}_0$ is a sum of at most two first-order operators (Lemma lem:vf) applied to elements of ${{\mathcal{J}}^0}_{\infty }$, hence ${\mathcal{T}}_0\in {{\mathcal{J}}^2}_{\infty }$ by Lemma lem:taylor5(iii); $T=\mathsf{P}{\mathcal{T}}_0\in {{\mathcal{J}}^2}_{\infty }$. The first term of (eq:K9) is ${\partial }_{b^{\nu }}$ of a smooth multiple of $T$: in ${{\mathcal{J}}^3}_{\infty }$. The second is a smooth multiple of $T$: in ${{\mathcal{J}}^2}_{\infty }$.

{$K_8$.} By (eq:TBchart), ${\partial }_{b^{\kappa }}{\mathcal{T}}_{\mathcal{B}}\in {{\mathcal{J}}^2}_{\infty }$; the smooth multiple and the outer ${\partial }_{b^{\mu }}$ give $K_8\in {{\mathcal{J}}^3}_{\infty }$.

$y$-independence: no entry depends on $y$. The support statement: $\mathrm{supp}{K}_{\mathrm{supp}}\subseteq S_{\partial }$ by (P3) and Corollary cor:K8supp, and $S_{\partial }\setminus V_{+}=\{a=z,b-a\in \partial J^{+},(b-a{)}^0\le 0\}=\{a=b=z\}$ since $\partial J^{+}\cap \{t\le 0\}=\{0\}$.