Theorem: Fixed-support continuity and contracted tests

Fix common compact support stages for $f,g$, a common plateau reserve, and the positive width. Then:

1. the smooth coefficient arrays in (eq:t) depend continuously on the input tests in their fixed-support smooth Fréchet spaces;

2. the maps to $t\in {{\mathcal{E}}^{\prime }}_{Z_2}(C\times S)$ and $t_3\in {{\mathcal{E}}^{\prime }}_{Z_3}(C^3)$ are continuous on those stages;

3. for fixed smooth $u,v$, the partial contraction

   $${\ell }_{u,v}=⟨t_3,u\otimes v\otimes \cdot ⟩$$

   is an element of ${\mathcal{D}}_C$. It depends continuously on the data and on $u,v$ in their smooth topology near $C$;

4. the classes $[{\ell }_{u,v}]\in Q_m$ depend continuously on the same data.

The first derivative of the cubic normal symbol is represented by $3{\ell }_{u,u}$.

Proof. Green continuity makes $F={{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}f$ depend smoothly on $f$ on every compact coefficient reserve. The explicit expression (eq:t) consists of a finite number of fixed distributions $R,R_2$, smooth coefficient multiplications, and derivatives. The coefficient maps are continuous; multiplication by a varying smooth coefficient on a fixed distribution is continuous in the stated microlocal spaces. The fixed diagonal insertion and proper pushforwards are continuous in their normal topologies BDH. This proves (i)–(ii).

Smoothly testing two arguments of a distribution with wavefront in $Z_3$ leaves no possible nonzero covector in the remaining argument: that would require $(0,0,p)\in Z_3$, hence $p=0$. Thus the contraction is smooth. Its support is in $C$.

For continuity into every smooth seminorm, the same partial-pushforward wavefront argument applies with all other variables tested. Its output cone is empty, whose normal topology is the smooth topology. On a fixed compact stage, bounded smooth test families and their derivatives give the requisite estimates. This also follows by contracting the finite formula (eq:t) and moving derivatives to the smooth factors. Lemma lem:closed proves (iv).