Definition: Scale-dependent extension

For $u\in {\mathcal{H}}_k\subset {\mathcal{V}}_{\mathrm{adm}}$ and $\rho =k-4$, let ${\mathfrak{R}}_{\mathrm{L},\mu }(u)\in {\mathcal{D}}^{\prime }({\mathbb{R}}^4)$ be the Brunetti–Fredenhagen extension of scaling degree $k$ restricting to $u$ on ${\mathbb{R}}^4\setminus \{0\}$, built with the projection ${W^{(\mu )}}_{\rho }$ of (eq:L-Wrhomu) and the zero moments

Extend ${\mathfrak{R}}_{\mathrm{L},\mu }$ to ${\mathcal{V}}_{\mathrm{adm}}$ by linearity. Equivalently, on each graded piece,

(the right side uses the unique continuous extension of $u$ to test functions vanishing to order $\rho$ at $0$).