Translation estimates and optional germ certificates

The order-four internal estimate gives

Thus the relative component functionals in (eq:Omega) have order at most five and compact support. If $C^{\chi }$ has order at most $h$, tensor evaluation gives

where $D$ is a fixed compact neighborhood of the difference support. Moreover,

The pushforward uses a cutoff equal to one near the compact tensor support, so it is independent of the choice of that cutoff. No unsmoothed same-variable product is defined.