Proposition: Supports of the supported sector

Under the hypotheses of Theorem thm:exist,

so $\mathrm{supp}{K}_{\mathrm{supp}}\subseteq S_{J^{+}}\subseteq \{a=z\}\cap \{b^0\ge a^0\}$, and $S_{\partial }\subseteq \{(b-a{)}^2=0,a=z\}$.

Proof. $K_6$ is a smooth multiple of ${\partial }_b$ of ${\Phi }^{\ast }[({\partial }_{\alpha }R)\otimes \delta ]$, supported in ${\Phi }^{-1}(\partial J^{+}\times \{0\})=S_{\partial }$. For $K_9$: by (eq:Qpair), $⟨{\mathcal{T}}_0,\varphi ⟩$ is a finite sum of integrals over $v$ of derivatives ${{\partial }_a}^{{\beta }^{\prime }}{{\partial }_z}^{{\alpha }^{\prime }}{{\Phi }^R}_{\varphi }(v,v,v)=⟨R(u),({{\partial }_a}^{{\beta }^{\prime }}{{\partial }_z}^{{\alpha }^{\prime }}\varphi )(v,v+u,v)⟩$, which involve derivatives of $\varphi$ only at points $(v,v+u,v)$ with $u\in \mathrm{supp}R=\partial J^{+}$, i.e.\ on $S_{\partial }$. If $\varphi$ vanishes on a neighborhood of $S_{\partial }$, all its derivatives vanish there and $⟨{\mathcal{T}}_0,\varphi ⟩=0$. Multiplication by smooth functions and differentiation in $b$ preserve the statement. The same argument with $\mathcal{B}$, $\mathrm{supp}\mathcal{B}=J^{+}$, gives $\mathrm{supp}{\mathcal{T}}_{\mathcal{B}}\subseteq S_{J^{+}}$ and hence $\mathrm{supp}{K}_8\subseteq S_{J^{+}}$.