Lemma: Localized relative smoothing before a fixed product

Let $f:\Omega \subset {\mathbb{R}}^n\to {\mathbb{R}}^d$ be a linear submersion, $u\in {\mathcal{D}}^{\prime }(\Omega )$, $v\in {\mathcal{D}}^{\prime }({\mathbb{R}}^d)$, and $D$ a constant-coefficient differential operator. Let $L⋐\Omega$, and let $\chi =1$ near $f(L)$. If on a neighborhood of $L$,

then

strongly in distributions locally near $L$. A finite list of component pairings can be treated simultaneously.

Proof. At a fixed point of $L$, compactness of normalized directions and closedness of wavefront sets give small base neighborhoods and closed angular cones containing the relevant wavefronts, with a strict nonopposition margin after pullback by $d{f}^T$.

Differentiate before smoothing:

Choose a relative cutoff $\sigma$ equal to one on a smaller neighborhood of the image and supported within the chosen relative neighborhood. Compact support of $\rho$ ensures that, for sufficiently small $\delta$, the smoothed factor on that smaller neighborhood equals ${\rho }_{\delta }\ast [\sigma D(\chi v)]$.

Lemma lem:mollification applies to this compact distribution in its constant angular cone. Submersion pullback continuity and fixed-factor product continuity then give the local convergence. A finite partition covers $L$.

For a finite list of pairings, make the neighborhood small enough for each pairing and apply the argument with its own cones. There is no need to take a union of incompatible component cones.