Authorization and normalization

Let $Z=U\setminus V_{\mathrm{ref}}$. Specify:

• a class $\mathcal{B}$ of authorized scale-dependent subtractions;

• a vector space $\mathcal{A}\subset {{\mathcal{D}}^{\prime }}_Z(U;E_{\mathrm{out}})$ of permitted scale-independent additions;

• the required tensor, symmetry, growth, overlap, and microlocal conditions;

• a linear normalization map $\mathcal{N}$, defined on the vector space containing $T+\mathcal{A}$, and a target $n_{\ast }$.

Here a “finite addition” is scale independent; it need not belong to a finite-dimensional vector space. A transverse-order bound limits the normal derivative types, not the dimension over $\mathbb{C}$ of the coefficient freedom. Already $a(\tau ){\delta }_S$, with arbitrary compact smooth $a$, gives infinite-dimensional freedom at normal order zero.