Theorem: Conditional supported extension

The following are equivalent:

1. there exists a distribution $T\in {\mathcal{D}}^{\prime }(U)$ of supported normal order at most $m$ satisfying $T{|}_{U^{\circ }}=T^{\circ }$;

2. each ${b^{\circ }}_{\gamma }$, $|\gamma |\le m$, has a distributional extension $b_{\gamma }\in {\mathcal{D}}^{\prime }(S_0)$.

Given (ii), the corresponding extension is

If an extension exists, all extensions in this normal-order class differ by

Proof. Assume (i). Lemma lem:normal-coefficients extracts the coefficients $b_{\gamma }$ of $T$. Testing with ${\mathsf{J}}_{\gamma }\psi$, where $\psi$ is supported in $S_0\setminus A$, proves

Thus (ii) holds.

Conversely, assume (ii) and form the displayed finite sum. It has supported normal order at most $m$. On $(S_0\setminus A)\times {\mathbb{R}}^k$, uniqueness of the normal coefficients gives equality with $T^{\circ }$. At points of $U^{\circ }$ with $z\ne 0$, both distributions are locally zero. These local statements cover $U^{\circ }$, proving (i).

The difference of two extensions has coefficients whose restrictions to $S_0\setminus A$ vanish. Those coefficients are precisely the distributions supported in $A$, proving (eq:jet-ambiguity).