Proposition: Moment, scaling degree, and uniqueness

The formula defines an extension of $q^{\circ }$ with scaling degree four, satisfying

It is the unique scaling-degree-four extension with this moment. Every other extension of scaling degree at most four has the form

Proof. Continuity follows from Lemma lem:T0 and continuity of $W_{\varpi }$. If a test is supported away from zero, its value at zero vanishes and (eq:extension) returns the original punctured distribution. Also $W_{\varpi }\varpi =0$, giving the stated moment.

The radial estimate gives a logarithmic scaling correction: under shrinking of the test, the subtracted action is bounded by a constant times ${\lambda }^{-4}(1+|\log \lambda |)$ in fixed test seminorms. Thus its scaling degree is at most four. Its nonzero homogeneous restriction already has degree four, so the degree is exactly four.

The difference of any two such extensions is supported at zero. A point-supported distribution is a sum of derivatives of $\delta$, and $\mathrm{sd}({\partial }^{\alpha }\delta )=4+|\alpha |$. Only a multiple of $\delta$ is permitted. The moment fixes that multiple because $\varpi (0)=1$. This also agrees with the point-extension theorem BF (Theorem 5.3).