Lemma: Smooth crossed-coefficient bounds

For every integer $r\ge 0$,

The constants depend on the fixed compact array and finitely many derivatives of $\rho$.

Proof. Use

If a compact-support cutoff is needed for the pairing, take it equal to one near $\mathrm{supp}(\chi C)$. Each derivative in the $q$-order distribution estimate costs at most ${\delta }^{-1}$, and the kernel contributes ${\delta }^{-8-|\alpha |}$. The cutoff derivatives contribute only lower powers, absorbed for $\delta \le 1$.