Proposition: Restriction and finite normal order leave ambiguity

Let $U=(-1,1)\times {\mathbb{R}}^d$, $d\ge 1$, with coordinates $(q,x)$, and let $S=\{q=0\}$. If a distribution $t^{\circ }$ on $U\setminus S$ has an extension $t$ to $U$, then

has the same restriction.

The space of the displayed ambiguities is infinite-dimensional, although every one has normal order zero.

Proof. The added distribution is supported on $S$, so its restriction to the complement vanishes. The map $f\mapsto f(x)\delta (q)$ is injective: pairing against $\chi (q)\psi (x)$, where $\chi (0)=1$, recovers $\int f\psi$. The domain ${C_c}^{\infty }({\mathbb{R}}^d)$ is infinite-dimensional, proving the last assertion.