Insertion into the differentiated complete pairing

At positive $\delta$, all amplitudes ${H_{e,{\dot{c}}_{\delta }}}^{I,J}[\Phi ]$ are smooth, so the theorem applies to them. Define

Then

The ambient derivatives in these amplitudes precede their incidence evaluation.

If $C^{\chi }$ has distributional order at most $h$, the elementary convolution estimate and the derivative-placement ledger give

Indeed, one additional ambient derivative of each smooth amplitude adds at most one crossed derivative and one test derivative to the bounds in Appendix app:bounds.

These estimates do not provide an integrable complete $\delta$-remainder. Substituting $ϵ=ϵ(\delta )$ would also require control of the complementary region. The incidence expressions in (eq:transition-insertion) still contain the unidentified crossed coefficient and have not been evaluated as physical smoothing coefficients.