Proposition: Fixed-support kernel continuity

On ${C^{\infty }}_{K_f}(M)$, the maps $f\mapsto b_f$ and $f\mapsto {{\mathcal{R}}_z}^{\mathsf{T}}{b}_f$ are continuous into their stages with cone $Z_2$. The bilinear map $(f,g)\mapsto t_{f,g}$ is jointly continuous from ${C^{\infty }}_{K_f}\times {C^{\infty }}_{K_g}$ to the stage ${{\mathcal{E}}^{\prime }}_{K_T,Z_3}$.

Proof. For $b_f$, multiply the fixed $\Lambda$ by $f\otimes 1$ and differentiate. Each operation is continuous. For $t_{f,g}$, first use the smooth tensor map. Its seminorm estimate is directly

Then multiply the fixed $\Xi$, push forward over $y$, differentiate, and symmetrize. The projection is proper on $M\times K_g\times M^2$. The support bounds are (eq:compositesupports).