Action on tests vanishing at the origin

Choose an auxiliary smooth radial cutoff $\zeta \in {C_c}^{\infty }({\mathbb{R}}^4)$ equal to one near zero. For $\psi \in {C_c}^{\infty }({\mathbb{R}}^4)$ with $\psi (0)=0$, define

For each positive $\delta$, the test vanishes near the origin, so this pairing is already defined on the punctured domain. This cutoff limit defines the action on a larger test subspace; it is not a change of the preparation width.