Theorem: Derivative–spectator finite part

Under Assumption ass:VIID,

More precisely,

where $C_{A,a,D}$ can be chosen as the coefficient expression in (eq:leibniz). The estimate is uniform on families with those coefficient seminorms bounded. It includes compact sets crossing $\mathcal{R}\cap {\Delta }_{13}$.

Proof. Apply (eq:parentestimate) to $D^{\mathrm{t}}(a\Phi )$, whose support remains in $A$, and use Lemma lem:leibniz. A bounded subset of $\mathcal{D}(X)$ has a common compact support and bounded smooth seminorms of every fixed order. Thus the right side tends to zero uniformly on each bounded test set, which is the asserted strong convergence.

The parent estimate already holds at the reflected-diagonal intersection. Applying a support-preserving transpose does not remove that intersection or introduce a new exclusion. No new ultraviolet argument is required.