Theorem: Exact extension criterion

Let $Z\subset X$ be relatively closed and $b^{\circ }\in {\mathcal{D}}^{\prime }(X\setminus Z)$. The following are equivalent:

1. There exists $b\in {\mathcal{D}}^{\prime }(X)$ with

   $$b{|}_{X\setminus Z}=b^{\circ }.$$

2. For every compact $K\subset X$, there are an integer $k_K\ge 0$ and a constant $C_K<\infty$ such that

   $$|⟨b^{\circ },\varphi ⟩|\le C_K{p}_{k_K}(\varphi )$$

   for every $\varphi \in \mathcal{D}(X\setminus Z)$ with $\mathrm{supp}\varphi \subset K$.

When extensions exist, their differences are exactly the distributions in ${\mathcal{D}}^{\prime }(X)$ supported in $Z$.

Proof. Suppose (i) holds. Distributional continuity on tests supported in a fixed compact $K$ gives a finite-order estimate

On the tests in (ii), the pairing with $b$ is the pairing with $b^{\circ }$. This proves necessity.

Assume (ii). Choose a relatively compact open set $V\subset X$ and a compact $K\subset X$ containing $\overline{V}$. On the linear subspace $\mathcal{D}(V\setminus Z)\subset \mathcal{D}(V)$, the functional defined by $b^{\circ }$ is bounded by $C_K{p}_{k_K}$.

The real or complex Hahn–Banach theorem extends this functional to a linear functional $B_V$ on $\mathcal{D}(V)$ satisfying the same bound. Since $p_{k_K}$ is continuous on every fixed-compact test space, $B_V$ is a distribution on $V$. It agrees with $b^{\circ }$ on $V\setminus Z$.

Take a locally finite cover $\{V_i\}$ of $X$ by such relatively compact open sets and a smooth partition of unity $\{{\chi }_i\}$ subordinate to it. Choose the extensions $B_{V_i}$ above and define

For each compactly supported test, only finitely many terms contribute. On each fixed compact test space this is a finite sum of continuous functionals, hence a distribution.

If $\varphi$ is supported in $X\setminus Z$, every term is the pairing of $b^{\circ }$ with ${\chi }_i\varphi$. The partition of unity gives

This proves sufficiency.

Finally, two extensions have zero difference after restriction to $X\setminus Z$, which is equivalent to support in $Z$. Conversely, adding a distribution supported in $Z$ changes no such restriction.