Lemma: Split calculus

The following operations preserve split type:

1. translation kernels $u(x-z)$ and diagonal delta kernels;

2. differentiation, smooth multiplication, and adjoining unused variables;

3. multiplication of two split distributions whose variable sets have exactly one common slot;

4. integration of internal variables when the projection is proper on the relevant support.

The product in (iii) is defined by the ordinary wavefront product criterion.

Proof. For (i), pullback by the difference map gives endpoint covectors $(p,-p)$; the delta cases are analogous. The operations in (ii) do not enlarge wavefront sets, and unused variables have zero covectors.

For (iii), lift both distributions to the union of their variables. An opposing pair would have zero covectors in every unshared slot. Split type would then force the covector in the single shared slot to be zero for each factor as well, contradicting a nonzero wavefront covector. The product criterion therefore holds. Its wavefront bound consists of sums of factor covectors, allowing a zero factor contribution. Those sums remain split.

For (iv), the pushforward bound retains only covectors with zero internal components. The surviving external components must still sum to zero. These are the standard pullback, product, and proper-pushforward rules Hormander (Chapter 8).