Theorem: Uniform local finite-order criterion

Let $X\subset {\mathbb{R}}^d$ be open, let $Z\subset X$ be relatively closed, and let $b^{\circ }\in {\mathcal{D}}^{\prime }(X\setminus Z)$. Then $b^{\circ }$ extends to a distribution in ${\mathcal{D}}^{\prime }(X)$ if and only if, for every $K⋐X$, there exist $C_K>0$ and $M_K\in \mathbb{N}$ such that

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

The constants and order in (eq:closedestimate) are uniform as these supports approach $Z$. The criterion imposes no wavefront or prescribed scaling-degree bound.

Proof. Necessity follows from the finite-order estimate of any extension on the fixed compact set $K$.

For sufficiency, choose a locally finite cover of $X$ by relatively compact open sets $U_i$ whose closures $K_i$ are contained in $X$. On the subspace $\mathcal{D}(U_i\setminus Z)\subset \mathcal{D}(U_i)$, the functional $b^{\circ }$ is bounded by

where tests are extended by zero when needed. The complex Hahn–Banach theorem extends it to a linear functional $B_i$ on $\mathcal{D}(U_i)$ with the same bound. This bound proves that $B_i$ is a distribution on $U_i$.

Choose a smooth locally finite partition of unity $({\eta }_i)$ subordinate to the cover and define

Only finitely many terms occur for any fixed compact support, and their estimates show distributional continuity. If $\varphi$ is supported in $X\setminus Z$, every term agrees with the original distribution, giving

Thus $b$ is an extension.