Lemma: Reduction to the bare tests

The operators ${\mathsf{L}}_{1,g}{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}$, ${\mathsf{L}}_{2,g}{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}$, and ${\mathsf{L}}_{1,g}{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}{\mathsf{L}}_{1,g}{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}$ are continuous from ${\mathcal{D}}_K(M)$ to ${\mathcal{D}}_C(M)$, where

If $c_{j,\epsilon }\to c_{j,0}$ in ${\mathcal{D}}_K(M)$, then $d_{j,\epsilon }\to d_{j,0}$ in ${\mathcal{D}}_C(M)$, with

For each output seminorm $p_N$, there are an integer $N^{\prime }$ and a constant $C_N$ such that

Proof. The advanced Green operator is continuous on compact smooth tests. Applying the smooth compact coefficients of the Euler operators gives the stated continuous maps and supports BGP (Chapter 3). For the iteration, the intermediate test is supported in $\mathrm{supp}g\cap J^{-}(K)$, and a second causal past remains in $J^{-}(K)$.

The differences are

Continuity on the fixed Fréchet test spaces gives the seminorm estimate. No unproved fixed derivative-loss count for the Green operator is needed.