Lemma: Contact defects for

Let $u\in {\mathcal{H}}_k$ be homogeneous of degree $-k$ on ${\mathbb{R}}^4\setminus \{0\}$ with $4\le k\le 7$, and suppose ${\partial }_{\alpha }u\in {\mathcal{H}}_{k+1}\subseteq {\mathcal{V}}_{\mathrm{adm}}$ (as an identity in ${\mathcal{D}}^{\prime }({\mathbb{R}}^4\setminus \{0\})$), for some $\alpha \in \{0,1,2,3\}$. Here $k$ is the scaling degree of $u$ (equivalently the homogeneity degree with opposite sign), not the numerical order $N=k-4$. Put

Then $\mathrm{supp}{D}_{\alpha ,u}\subset \{0\}$, $\mathrm{sd}(D_{\alpha ,u})\le k+1$, and

where the moment order $k-3=(k+1)-4$ is that of the image, and the identity uses $⟨{\mathfrak{R}}_{\mathrm{L}}({\partial }_{\alpha }u),{w^{(k-3)}}_{\beta }⟩=0$. (Equivalently, for a general $u\in {\mathcal{V}}_{\mathrm{adm}}$ with ${\partial }_{\alpha }u\in {\mathcal{V}}_{\mathrm{adm}}$, the same support and scaling conclusions hold, and the contact coefficients are given by the full pairing formula keeping both terms $⟨{\mathfrak{R}}_{\mathrm{L}}({\partial }_{\alpha }u),w⟩$ and $⟨{\partial }_{\alpha }{\mathfrak{R}}_{\mathrm{L}}(u),w⟩$.) These defects are retained in every subsequent formula; they are not claimed to vanish.

Proof. Off $0$, ${\mathfrak{R}}_{\mathrm{L}}({\partial }_{\alpha }u)={\partial }_{\alpha }u$ and ${\partial }_{\alpha }{\mathfrak{R}}_{\mathrm{L}}(u)={\partial }_{\alpha }u$ (Proposition prop:RL(1); restriction commutes with ${\partial }_{\alpha }$), so $D_{\alpha ,u}$ is supported at $0$. Scaling: $\mathrm{sd}({\mathfrak{R}}_{\mathrm{L}}({\partial }_{\alpha }u))\le k+1$ and $\mathrm{sd}({\partial }_{\alpha }{\mathfrak{R}}_{\mathrm{L}}(u))\le \mathrm{sd}({\mathfrak{R}}_{\mathrm{L}}(u))+1\le k+1$, hence $\mathrm{sd}(D_{\alpha ,u})\le k+1$. A point-supported distribution of scaling degree $\le k+1$ in ${\mathbb{R}}^4$ is ${\sum }_{|\gamma |\le (k+1)-4}{c}_{\gamma }{\partial }^{\gamma }\delta$. Pair with ${w^{(k-3)}}_{\beta }$, $|\beta |\le k-3$: the left side of (eq:L-defect) yields $(-1{)}^{|\beta |}{c}_{\alpha ,u,\beta }$; the right side uses the moment vanishing for ${\mathfrak{R}}_{\mathrm{L}}({\partial }_{\alpha }u)$ and $⟨{\partial }_{\alpha }{\mathfrak{R}}_{\mathrm{L}}(u),w⟩=-⟨{\mathfrak{R}}_{\mathrm{L}}(u),{\partial }_{\alpha }w⟩$.