Lemma: Canonical action on D ρ

The limit (eq:operational) exists, is independent of the auxiliary cutoff, and is continuous in the induced test topology. It is linear in $u$. It equals the action on ${\mathcal{D}}_{\rho }$ of any extension of $u$ with scaling degree at most $k$.

Proof. Fix a compact annulus separated from zero. The restriction of $u$ has a finite-order seminorm bound there. For $\psi \in {\mathcal{D}}_{\rho }$, Taylor's theorem and the chain rule give, on this fixed annulus,

for every fixed finite $m$, with bounds in finitely many seminorms of $\psi$. After scaling an annular pairing to this annulus, homogeneity supplies the factor $a^{4-k}$. The pairing is therefore

A dyadic decomposition near zero has summable bounds ${\sum }_{j\ge 0}O(2^{-j})$. This proves convergence and continuity. Differences of two scaled cutoff choices have annular support at scale $a$, so their action is also $O(a)$, proving independence.

The point-extension theorem provides an extension with scaling degree at most $k$ BF (Theorems 5.2–5.3). Any two such extensions differ by derivatives of $\delta$ of order at most $k-4=\rho$; these annihilate ${\mathcal{D}}_{\rho }$. Equivalently, for $|\beta |=\rho +1$, the distribution $z^{\beta }u$ has scaling degree at most $3<4$ and has a unique extension. Its action reconstructs the action on jet-vanishing tests, as in the proof of BF (Theorem 5.3). The cutoff construction realizes that same action.