Theorem: Conditional supported extension interface

Let $A\subset U$ be relatively closed and let $t^{\circ }\in {\mathcal{D}}^{\prime }(U\setminus A)$ have supported normal order at most $r$ along $S\setminus A$. Write its unique coefficients as

on that open set.

The following are equivalent:

1. There exists $t\in {\mathcal{D}}^{\prime }(U)$ supported in $S$, of normal order at most $r$, such that

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

2. Each ${b_{\gamma }}^{\circ }$ extends to a distribution $b_{\gamma }\in {\mathcal{D}}^{\prime }(S_0)$.

For any such coefficient extensions,

is the corresponding supported realization. The equivalence does not assert that the coefficient extensions exist or obey additional normalization, growth, or wavefront requirements.

Proof. If (i) holds, Proposition prop:normal extracts its unique coefficients. On the open part of $S$ outside $A$, uniqueness of normal-jet coefficients shows that their restrictions are ${b_{\gamma }}^{\circ }$. This argument is local on that open part and hence applies even if $A$ is not a product set.

If (ii) holds, the finite insertion sum is defined on $U$, is supported in $S$, and has normal order at most $r$. Its restriction has the prescribed coefficients and therefore equals $t^{\circ }$ by Proposition prop:normal. No step constructs the coefficient extensions or supplies further bounds on them.