Lemma: Combined source/vertex placement and sign

Let $z=y-w$ and let $u\in {\mathcal{D}}^{\prime }({\mathbb{R}}^4)$. On $y\ne w$,

In particular, for the edge kernel $S$,

where

The source-end derivatives carry no sign; the vertex-end sign is $(-1{)}^{|p|+|q|}$, exactly as in XVI-K, eq:sign, for the $w$-derivatives alone.

Proof. XVI-K, Lemma lem:sign (imported): ${{\partial }_w}^p[u(y-w)]=(-1{)}^{|p|}({\partial }^{p}u)(y-w)$ and ${{\partial }_y}^a[u(y-w)]=({\partial }^{a}u)(y-w)$. These commute on $y\ne w$ (sequential continuity of pullback by the linear submersion $\Phi (y,w)=y-w$, Hörmander, Theorem 8.2.4), and ${{\partial }_y}^a{{\partial }_w}^p={{\partial }_w}^p{{\partial }_y}^a$ gives (eq:L-yw). The product identity follows as in XVI-K, Lemma lem:sign: Hörmander products on $y\ne w$ are sequentially continuous, so the two signs multiply.