Lemma: Root attachment for the complete-source kernels

Assume (R1)–(R3). For every numerical label $\lambda$, every pair of indices $\mu ,\nu$, and every smooth $h$:

1. ${\pi }^{\ast }{B}_{f,g,\mu \nu }$ and ${\Phi }_3^{\ast }{\mathfrak{R}}_{\mathrm{L}}({q^{\circ }}_{\lambda })$ are defined distributions on $M^3$, with

   $$\mathrm{WF}({\pi }^{\ast }{B}_{f,g,\mu \nu })\subset \{(y,x,w;\xi ,-\xi ,0):\xi \ne 0\},\mathrm{WF}({\Phi }_3^{\ast }{\mathfrak{R}}_{\mathrm{L}}({q^{\circ }}_{\lambda }))\subset \{(y,x,w;\kappa ,0,-\kappa ):\kappa \ne 0\}.$$

2. The product $B_{f,g,\mu \nu }(y,x){\mathfrak{R}}_{\mathrm{L}}({q^{\circ }}_{\lambda })(y-w)$ exists in ${\mathcal{D}}^{\prime }(M^3)$, including on $y=w$, and so does the product with the smooth coefficient ${a^{\mu \nu }}_{\lambda }(y,w;h)$.

3. $\mathrm{supp}{U^{{\mathfrak{R}}_{\mathrm{L}}}}_{\mathrm{L}}(\cdot ;h)\subset \{(y,x,w):(y,x)\in \mathrm{supp}{B}_{f,g},w\in \mathrm{supp}{\chi }_{\mathrm{I}}\}$, a compact set.

The statements use only that ${\mathfrak{R}}_{\mathrm{L}}({q^{\circ }}_{\lambda })$ is a distribution on ${\mathbb{R}}^4$; they hold for every other same-degree extension, and for every ${\mathfrak{R}}_{\mathrm{L},\mu }({q^{\circ }}_{\lambda })$.

Proof. Identical to XVI-K, Lemma lem:attachK, with ${\mathfrak{R}}_{\mathrm{K}}({q^{\circ }}_{\mathsf{k}})$ replaced by ${\mathfrak{R}}_{\mathrm{L}}({q^{\circ }}_{\lambda })$ and the finite sum extended over the complete-source labels of ${\mathcal{V}}_{\mathrm{adm}}$. (1) $\pi$ and ${\Phi }_3$ are linear submersions (Hörmander, Theorem 8.2.4); the covector shapes follow from (R3) and from ${}^t{\Phi }_3^{\prime }\kappa =(\kappa ,0,-\kappa )$, independently of which extension of ${q^{\circ }}_{\lambda }$ is pulled back. (2) A sum $(\xi ,-\xi ,0)+(\kappa ,0,-\kappa )$ vanishes only if $\xi =\kappa =0$; neither wavefront set contains a zero covector, so the product exists by Hörmander, Theorem 8.2.10. Smooth multiplication by ${a^{\mu \nu }}_{\lambda }$ (compactly supported in $w$ through ${\chi }_{\mathrm{I}}$) preserves existence. (3) Support of a product sits in the intersection of supports; (R2) and $\mathrm{supp}{\chi }_{\mathrm{I}}$ compact give compactness.