Proposition: An explicit nonextendable coefficient

For any $a>0$, the smooth function on the punctured interval

defines a distribution on $(-1,1)\setminus \{0\}$ with no extension in ${\mathcal{D}}^{\prime }((-1,1))$.

Proof. Suppose an extension $B$ exists. On a fixed compact neighborhood of zero, there are $C>0$ and $M\in \mathbb{N}$ such that

for tests supported in that neighborhood.

Choose a nonnegative nonzero $\varphi \in \mathcal{D}((0,1))$ and set

Its support lies in $(\eta ,\eta +{\eta }^3)$ and

For small $\eta$, the support lies in the fixed compact neighborhood and satisfies $x\le 2\eta$. Since $B$ agrees there with ${b_a}^{\circ }$,

The lower bound exceeds every fixed power of ${\eta }^{-1}$ as $\eta \downarrow 0$, contradicting the distributional estimate.