Lemma: Regulated compensation

For compact smooth $\xi$,

For the fixed witness,

as $\mu \downarrow 0$, $\Lambda \uparrow \infty$.

Proof. The multiplier identity $N_{\rho }\Delta ={\chi }_{\rho }(D)$ proves the first claim.

On a compact time interval containing $\mathrm{supp}f$, the spatial Fourier transforms of $\xi$ have a uniform Schwartz bound. Thus

tends to zero uniformly on that interval. One can use a fixed integrable Schwartz envelope and dominated convergence. Integration against the fixed compact test factors gives the result.