Lemma: Products and derivative signs

For all multi-indices $p,q$,

exists, and ${q^{\circ }}_{pq}={q^{\circ }}_{qp}$.

For $r_2(y,w)=y-w$ and any distribution $u$ on the relative space,

Consequently, on $y\ne w$,

Proof. Derivatives do not enlarge wavefront sets. At a nonzero null point, two nonzero covectors in the relevant wavefront fibers have the same time orientation, and hence cannot sum to zero. At nonsingular points there is no obstruction. Hörmander's product criterion therefore defines (eq:qpq); see Hormander (Theorem 8.2.10) and BDH (Theorem 13). That product is commutative.

The relative map is a linear submersion. In coordinates $(y,z)$, $z=y-w$, its pullback is a distribution constant in the extra $y$-variable. Differentiation at fixed $y$ therefore gives ${\partial }_w=-{\partial }_z$, proving the distributional identities. Pullback by a submersion is compatible with the products when their wavefront criterion holds, yielding the final assertion.