Lemma: Differentiability in the scale

For each $u\in {\mathcal{V}}_{\mathrm{adm}}$ the map $\mu \mapsto {\mathfrak{R}}_{\mathrm{L},\mu }(u)$ is $C^{\infty }$ as a ${\mathcal{D}}^{\prime }({\mathbb{R}}^4)$-valued function of $\mu >0$. Writing

one has $\mathrm{supp}{A}_{\mu }(u)\subset \{0\}$, and if $u\in {\mathcal{H}}_{4+N}$ then

for coefficients $a_{\mu }(u,\gamma )\in \mathbb{C}$ depending smoothly on $\mu$. Moreover $A_{\mu }:{\mathcal{V}}_{\mathrm{adm}}\to {\mathcal{D}}^{\prime }({\mathbb{R}}^4)$ is linear, and every constant-coefficient relation ${\sum }_{\lambda }{r}_{\lambda }{q^{\circ }}_{\lambda }=0$ in ${\mathcal{D}}^{\prime }({\mathbb{R}}^4\setminus \{0\})$ implies ${\sum }_{\lambda }{r}_{\lambda }{A}_{\mu }({q^{\circ }}_{\lambda })=0$.

Proof. Smoothness of the projections. The map $\mu \mapsto {\varpi }_{\mu }$ is $C^{\infty }$ into ${C_c}^{\infty }({\mathbb{R}}^4)$ (chain rule; compact supports remain in a fixed large ball for $\mu$ in any compact subinterval of $(0,\infty )$). Hence $\mu \mapsto M^{(\rho ,\mu )}$ is $C^{\infty }$ into invertible matrices (unitriangular with smooth entries), $\mu \mapsto {w^{(\rho ,\mu )}}_{\alpha }$ is $C^{\infty }$ into ${C_c}^{\infty }$, and $\mu \mapsto {W^{(\mu )}}_{\rho }$ is $C^{\infty }$ as a map into continuous operators on $\mathcal{D}({\mathbb{R}}^4)$.

{Differentiability of ${\mathfrak{R}}_{\mathrm{L},\mu }$.} On ${\mathcal{H}}_{4+N}$, formula (eq:L-RLmu-formula) gives $⟨{\mathfrak{R}}_{\mathrm{L},\mu }(u),\phi ⟩=⟨u,{W^{(\mu )}}_{\rho }\phi ⟩$. The right side is $C^{\infty }$ in $\mu$ for each test $\phi$, so $\mu \mapsto {\mathfrak{R}}_{\mathrm{L},\mu }(u)$ is $C^{\infty }$ in the weak topology of ${\mathcal{D}}^{\prime }$; the distributional derivative $A_{\mu }(u)$ exists and is given by differentiating under the pairing. Linearity in $u$ is immediate from (eq:L-RLmu-formula).

Contact expansion. Differentiate (eq:L-scale-diff) in $\mu$ at fixed ${\mu }_0$, or compare ${\mathfrak{R}}_{\mathrm{L},\mu +\epsilon }(u)-{\mathfrak{R}}_{\mathrm{L},\mu }(u)$ by Theorem thm:Lscale with reference $\mu$: the difference quotient is a combination of ${\partial }^{\gamma }\delta$ with $|\gamma |\le N$, and the limit retains that form, yielding (eq:L-Amu-exp). Support at $0$ is immediate.

Coherence. If $\sum r_{\lambda }{q^{\circ }}_{\lambda }=0$ as an element of ${\mathcal{V}}_{\mathrm{adm}}$, then ${\mathfrak{R}}_{\mathrm{L},\mu }$ of that element vanishes for every $\mu$ (linearity), so its $\mu$-derivative $A_{\mu }$ vanishes on it.