L.4 Coherent extension preserving XVI-K

The extension acts only in the difference variable $z=y-w$. The coefficients ${a^{\mu \nu }}_{\lambda }(y,w;h)$ of (eq:QL2) depend on $y$ and $w$ separately; they are not extended and multiply the pulled-back extensions afterwards. This is the same order of operations as XVI-K, Section K.4.

{Projection, extended to declared orders $N\le 4$.} Fix the same function $\varpi \in {C_c}^{\infty }({\mathbb{R}}^4)$, $\varpi (0)=1$, of XVI-J, eq:moment, used in XVI-K, eq:Wrho. For $\rho \in \{0,1,2,3,4\}$ let

By Leibniz' rule ${M^{(\rho )}}_{\beta \alpha }=0$ unless $\alpha \le \beta$, and ${M^{(\rho )}}_{\alpha \alpha }=\varpi (0)=1$; so $M^{(\rho )}$ is unitriangular and invertible. Put

Then ${\partial }^{\beta }{w^{(\rho )}}_{\alpha }(0)={{\delta }^{\beta }}_{\alpha }$ for $|\alpha |,|\beta |\le \rho$, so $W_{\rho }$ projects onto the test functions vanishing to order $\rho$ at $0$. For $\rho \in \{0,1,2\}$ the matrices $M^{(\rho )}$, the functions ${w^{(\rho )}}_{\alpha }$, and the projections $W_{\rho }$ are identical to XVI-K, eq:Wrho (same $\varpi$); the cases $\rho =3,4$ are the declared-order extension required by Proposition prop:Lsd.

and basis. Write ${\mathcal{H}}_k$ for the homogeneous degree $-k$ piece of ${\mathcal{V}}_{\mathrm{adm}}$ (equivalently: elements with $\mathrm{sd}\le k$ that are homogeneous of degree $-k$ on ${\mathbb{R}}^4\setminus \{0\}$, in the sense of Proposition prop:Lsd), so ${\mathcal{V}}_{\mathrm{adm}}={{⨁}_{k=4}}^8{\mathcal{H}}_k$ as an explicit direct sum by distinct homogeneity degree on ${\mathbb{R}}^4\setminus \{0\}$ (as in XVI-K, Lemma lem:grading: every element of ${\mathcal{H}}_k$ is homogeneous of degree $-k$; vanishing kernels contribute nothing). For each $k$, choose a basis of ${\mathcal{V}}_{\mathrm{K}}\cap {\mathcal{H}}_k$, and extend it to a basis ${\mathcal{B}}_k$ of ${\mathcal{H}}_k\subseteq {\mathcal{V}}_{\mathrm{adm}}$. Set ${\mathcal{B}}_{\mathrm{adm}}={{\bigcup }_{k=4}}^8{\mathcal{B}}_k$. Which spanning elements enter each ${\mathcal{B}}_k$ is not specified beyond this extension requirement; the construction below is basis-independent on each homogeneous piece (Remark rem:Lbasisfree).