Theorem: First scale theorem

For every $u\in {\mathcal{V}}_{\mathrm{adm}}$ of declared order $N$ (so $u\in {\mathcal{H}}_{4+N}$ up to lower graded pieces) and every $\mu >0$,

with coefficients $c_{\mu ,{\mu }_0}(u,\gamma )\in \mathbb{C}$. In particular the difference is supported at $\{0\}$ and has contact order at most the declared order of $u$.

Proof. Both ${\mathfrak{R}}_{\mathrm{L},\mu }(u)$ and ${\mathfrak{R}}_{\mathrm{L},{\mu }_0}(u)$ extend $u$ on ${\mathbb{R}}^4\setminus \{0\}$ with scaling degree $\le 4+N$ (Definition def:RLmu; Proposition prop:RL(2) at ${\mu }_0$). Their difference is therefore supported at $0$ and of scaling degree $\le 4+N$, hence of the form (eq:L-scale-diff) (Brunetti–Fredenhagen, Theorem 5.3, after the statement).