Lemma: The cross-line pullback

Let $D\in {\mathcal{D}}^{\prime }({{\mathbb{R}}^4}_u\times {{\mathbb{R}}^4}_v)$ be any distribution (in particular the difference in (eq:cross)) and $\pi (a,b,z,y)=(y-a,y-b)$ on ${\mathbb{R}}^{16}$.

1. $\pi$ is a surjective linear map, hence a submersion; ${\pi }^{\ast }D\in {\mathcal{D}}^{\prime }({\mathbb{R}}^{16})$ exists, depends continuously on $D$, and $\mathrm{supp}{\pi }^{\ast }D={\pi }^{-1}(\mathrm{supp}D)$.

2. ${\pi }^{\ast }D$ is independent of $z$ and invariant under simultaneous translation of $(a,b,z,y)$.

3. $\mathrm{WF}({\pi }^{\ast }D)\subseteq \{(a,b,z,y;-\xi ,-\eta ,0,\xi +\eta ):(y-a,y-b;\xi ,\eta )\in \mathrm{WF}(D)\}$. In particular every covector of ${\mathcal{C}}_{\mathrm{cross}}$ has zero $z$-component and $y$-component equal to minus the sum of its $a$- and $b$-components.

4. If $D=D_1\otimes {H^{\prime }}_2+H_1\otimes D_2$ with $D_1={H^{\prime }}_1-H_1$, $D_2={H^{\prime }}_2-H_2$ smooth (the smooth-difference clause of XII-A's admissibility XIIAhad, if XIV-J's blocks satisfy it), then $\mathrm{WF}(D)\subseteq \{\xi =0\}\cup \{\eta =0\}$ and $\mathrm{WF}({\pi }^{\ast }D)\subseteq \{(0,-\eta ,0,\eta )\}\cup \{(-\xi ,0,0,\xi )\}$.

Proof. (i) The Jacobian of $\pi$ has rank $8$. Existence, continuity, and the support statement are Horm (Thm. 8.2.4). (ii) $\pi$ is invariant under $z\mapsto z+c$ and under $(a,b,z,y)\mapsto (a+c,b+c,z+c,y+c)$. (iii) is the wavefront bound of Horm (Thm. 8.2.4) with ${}^{t}d\pi (\xi ,\eta )=(-\xi ,-\eta ,0,\xi +\eta )$. (iv) For $D_1$ smooth, $\mathrm{WF}(D_1\otimes {H^{\prime }}_2)\subseteq \{(u,v;0,\eta )\}$ Horm (Thm. 8.2.9); similarly for the second term; insert into (iii).