Lemma: Numerical derivative transfer

For a fixed smooth multiplier $a$ and a differential operator $D$ of order $d$ on $X$, the parent finite part transfers to $aD{u}_{\mu }^{\rho }$, with residue $aDr$ and test order $64+d$. Conversion of $U_c$ becomes $caDr$, with the same $c$.

Proof. Transpose $aD$ onto the test in (eq:parentestimate). For a common compact support,

The constant is bounded by a finite polynomial in the $C^{64+d}$-seminorms of $a$ and the coefficients of $D$. The resulting bound tends to zero uniformly on bounded joint-test sets. The conversion assertion is linearity.