Lemma: Split calculus

The following operations preserve split type:

1. Translation kernels $u(x-z)$ are split, as are diagonal delta kernels.

2. Smooth multiplication, differentiation, and adjoining unused variables preserve split type.

3. If split distributions depend on variable sets $I,J$ with one shared slot, their product exists and is split.

4. Proper integration of internal variables preserves split type.

Proof. A translation kernel has endpoint covectors $(p,-p)$. A diagonal delta has covectors whose sum is zero. Smooth multiplication and differentiation do not enlarge the wavefront set, while unused variables carry zero covectors.

For (iii), lift both factors to the union of their variable sets. A violation of the distributional product criterion would require opposite wavefront covectors whose sum vanishes in every slot. All unshared slots would then be zero. Split type of either factor forces its shared covector to be zero as well, contradicting a nonzero wavefront covector. The wavefront bound for the product consists of sums of factor covectors, allowing a zero contribution from one factor; these sums remain split.

For (iv), the pushforward wavefront bound retains only covectors with zero internal components. The surviving external components therefore still sum to zero. Properness is required on the support over each compact external set. These are the usual product and pushforward rules Hormander (Chapter 8).