Theorem: The six internal expressions exist for every completion of the undefined data

Let $R$, $\mathcal{B}$ be as in (eq:R)–(eq:RR). Let the arrays ${\mathsf{A}}^{\alpha \beta ,I}$, ${\mathsf{A}}^{\alpha \mu }{}_L$, ${\mathsf{D}}^{\alpha \mu ,I,J}$, ${\mathsf{P}}^L{}_M$, ${\mathfrak{b}}_L$ be any constants, $m\in \mathbb{C}$, and $f,g,F,{\sigma }_{\epsilon },{q^{L,I,J}}_{\alpha \beta }\in C^{\infty }({\mathbb{R}}^4)$. Then (eq:L) defines $L\in {\mathcal{D}}^{\prime }({\mathbb{R}}^8)$ with $X^{\mu }[h]\in C^{\infty }({\mathbb{R}}^4)$ for $h\in \mathcal{D}$, and each of (eq:K5a), (eq:K5z), (eq:K47), (eq:K6), (eq:K8), (eq:K9) defines a distribution on ${{\mathbb{R}}^{12}}_{(a,b,z)}$. No microlocal hypothesis and no restriction to a patch is needed for existence.

Proof. {$L$.} $(s,a)\mapsto s-a$ is a surjective linear map; the pullback of ${\partial }_{\lambda }R$ exists Horm (Thm. 8.2.4); multiply by the smooth ${\sigma }_{\epsilon }(a)\mathsf{A}$. $X^{\mu }[h]={\mathsf{A}}^{\lambda \mu ,I}({\partial }_{\lambda }R)\ast ({\sigma }_{\epsilon }{h}_I)$ is a distribution convolved with a test function, hence smooth.

{${K_5}^{[a]}$, ${K_5}^{[z]}$.} The bracket of (eq:K5a) is a smooth function of $(a,b,z)$ times constants times $({\partial }_{\alpha }R)(b-a)\cdot ({\partial }_{\beta }{\partial }_{\lambda }R)(a-z)$, defined as ${\Phi }^{\ast }[({\partial }_{\alpha }R)\otimes ({\partial }_{\beta }{\partial }_{\lambda }R)]$ with $\Phi (a,b,z)=(b-a,a-z)$, a surjective linear map ${\mathbb{R}}^{12}\to {\mathbb{R}}^8$ (this is XIV-H's pullback existence Hkernel). Apply ${\partial }_{b^{\mu }}$. For (eq:K5z) use ${\Phi }^{\prime }(a,b,z)=(b-z,z-a)$.

{$K_{47}$.} Each product in (eq:K47) is a smooth function times a derivative tensor of $R\otimes R$ pulled back by $\Psi (a,b,z)=(b-z,b-a)$, surjective linear.

{$K_6$.} ${\Phi }^{\ast }[({\partial }_{\alpha }R)\otimes \delta ]$ times a smooth function, then ${\partial }_{b^{\mu }}$.

{$K_8$, $K_9$.} For $\varphi \in \mathcal{D}({{\mathbb{R}}^{12}}_{(a,b,z)})$ set

Both are $C^{\infty }$ ($R$ paired with a smooth family of test functions with supports in a fixed compact; $\mathcal{B}\in L^{\infty }$ with compact $u$-support of the integrand), and their derivatives in $(a,z)$ vanish when $a\notin {\mathrm{pr}}_a(\mathrm{supp}\varphi )$, so along the diagonal they are compactly supported in $v$. By Lemma lem:Q, $⟨{\mathcal{T}}_0,\varphi ⟩:=⟨Q,{{\Phi }^R}_{\varphi }⟩$ and $⟨{\mathcal{T}}_{\mathcal{B}},\varphi ⟩:=⟨Q,{{\Phi }^{\mathcal{B}}}_{\varphi }⟩$ are finite, linear, and bounded by $C{p}_N(\varphi )$ for $\varphi$ supported in a fixed compact; they define distributions on ${\mathbb{R}}^{12}$ agreeing with the displayed integrals when $Q$ is a test function. Then $T=\mathsf{P}{\mathcal{T}}_0$, $K_8=-{\partial }_{b^{\mu }}[f\mathsf{AP}{\partial }_{b^{\alpha }}{\mathcal{T}}_{\mathcal{B}}]$, and $K_9=-{\partial }_{b^{\nu }}[g{\partial }_{\mu }F\mathsf{A}T]-m^{2}gF\mathfrak{b}T$ are distributions on ${\mathbb{R}}^{12}$.