Definition: The extension map

For $b\in {\mathcal{B}}_k$, $\rho =k-4\in \{0,1,2,3,4\}$, let ${\mathfrak{R}}_{\mathrm{L}}(b)\in {\mathcal{D}}^{\prime }({\mathbb{R}}^4)$ be the extension of Brunetti–Fredenhagen, Theorem 5.3 (projection (38), construction (39)), of scaling degree $k$, restricting to $b$ on ${\mathbb{R}}^4\setminus \{0\}$, built with the projection $W_{\rho }$ of (eq:L-Wrho) and the declared moments

Extend ${\mathfrak{R}}_{\mathrm{L}}$ to ${\mathcal{V}}_{\mathrm{adm}}$ by linearity.