Theorem: Exact supported realization test

Under these hypotheses, the following are equivalent:

1. There exists a supported realization $t\in {\mathcal{D}}^{\prime }(U)$ of normal order at most $r$ with

   $$t{|}_{U\setminus A}=t^{\circ }.$$

2. For every compact $K\subset X$ and every $|\gamma |\le r$, there are constants $C_{K,\gamma }$ and integers $k_{K,\gamma }$ such that

   $$|⟨{b_{\gamma }}^{\circ },\psi ⟩|\le C_{K,\gamma }{p}_{k_{K,\gamma }}(\psi )$$

   whenever $\psi \in \mathcal{D}(X\setminus Z)$ and $\mathrm{supp}\psi \subset K$.

When these conditions hold, every such realization has a unique representation

with $b_{\gamma }$ extending ${b_{\gamma }}^{\circ }$. The unrestricted supported ambiguity is

Proof. A distribution of the stated supported normal order has a unique finite insertion representation. For completeness, define its coefficients by

For a test $\varphi$, the difference between $\varphi$ and the finite sum of its lifted signed normal jets has zero normal jets through order $r$. The defining annihilation condition gives (eq:realization). Lemma lem:extraction proves uniqueness.

If (i) holds, near each point of $S$ outside $A$ choose a product neighborhood disjoint from $A$. The realization and the prescribed reference agree there. Extracting coefficients shows that each $b_{\gamma }$ restricts to ${b_{\gamma }}^{\circ }$. Theorem thm:extension gives (eq:jetbound).

Conversely, (ii) and Theorem thm:extension extend each ${b_{\gamma }}^{\circ }$ to a distribution on $X$. The finite insertion sum is defined on $U$, supported in $S$, and has normal order at most $r$. On each product neighborhood outside $A$ meeting $S$, its coefficients agree with those in (eq:puncturedform); hence the distributions agree by coefficient uniqueness. Off $S$, both vanish. These local equalities prove the required restriction.

The extension differences of the coefficients are exactly the distributions supported in $Z$, by Theorem thm:extension. Uniqueness of the insertion representation gives (eq:ambiguity).