Theorem: Conditional supported realization

Fix a vertex component $D$. The following two assertions are equivalent:

1. There exists $t_D\in {\mathcal{D}}^{\prime }(U)$ of normal order at most three such that

   $$t_D{|}_{U\setminus N_U}={t_D}^{\circ }.$$

2. There exist distributions $b_{D,\gamma }\in {\mathcal{D}}^{\prime }(S_0)$, $|\gamma |\le 3$, whose restrictions to $S_{\mathrm{resp}}\setminus N_U$ are the coefficients (eq:bcoeff).

Given (ii), the corresponding supported realization is

Its wavefront satisfies

No existence, degree, or characteristic normalization of the coefficient extensions is asserted by this equivalence.

Proof. If (i) holds, Proposition prop:characterization extracts its unique coefficient distributions. Tests ${\mathsf{J}}_{\gamma }\psi$ with $\psi$ supported away from $N_U$ compare the extracted coefficients with (eq:collected); uniqueness shows that their restrictions equal (eq:bcoeff).

If (ii) holds, the insertion maps are defined for arbitrary coefficient distributions. Their finite sum has normal order at most three and has the required restriction. The wavefront statement is Lemma lem:insertWF.