Theorem: Existence, profile independence, and strong convergence

The limit

exists strongly in ${\mathcal{D}}^{\prime }(\overline{M})$, is independent of $\rho$, and restricts to $r^{-2}$ on $M$. It is rotation invariant, translation invariant in $t,z$, and real.

For every compact test support there is a constant $C$ such that

where $p_2$ is a fixed-support smooth seminorm. In particular, all fixed distributional derivatives converge strongly, with the corresponding extra derivatives in the seminorm.

Proof. Let

Taylor's theorem and angular averaging give

uniformly on the relevant compact set.

For the sharp cutoff, the expression

has a limit: subtract $f(y,0)$ in a fixed small radial neighbourhood and integrate the remainder with measure ${\sigma }_{\alpha }\mathrm{d}r/r$. The omitted remainder on $0<r<\epsilon$ is $O({\epsilon }^2)$.

The smooth cutoff differs from the sharp cutoff only on $\epsilon <r<2\epsilon$. Its constant Taylor term contributes

which is exactly removed in (eq:Uepsilon). Its remaining contribution is $O({\epsilon }^2{p}_2(f))$. This proves existence, profile independence, and (eq:Urate).

The estimates are uniform on bounded sets of tests with fixed compact support, and hence give strong distributional convergence. The symmetries and reality follow directly from the construction. Derivative convergence follows by transposition.