Theorem: Scalar double-cone limit and rate

The family converges strongly to

where

On compact test supports,

More precisely,

The constants are uniform on compact positive parameter sets. If both profiles are even, then

Proof. Rescale $v=\delta s$, $\xi =\sqrt{\delta }X$:

The Jacobian cancels the ${\delta }^{-2}$ factor. Polar integration gives ${\mathrm{d}}^{2}X=\pi \mathrm{d}Q$. The absolute determinant of $(s,Q)\mapsto (U,V)$ is $4\beta$, and $Q\ge 0$ means $hU+\alpha V\le 0$. Hence ${\mathfrak{m}}_0=c_0$.

Reflection averaging in $X$ bounds the test difference from $\psi (0,0)$ by

This gives the first rate and strong convergence. The refined expansion follows from Lemma lem:scalar-moments.

For the scale derivative, differentiate (eq:scalar-rescale):

Subtract ${\nabla }_{\xi }\psi (\delta s,0)$ in the second term; its integral against $X{F}_0$ is zero. The remaining term is $O(\delta \Vert \psi {\Vert }_2)$.

When both profiles are even, simultaneous reflection $(U,V)\mapsto (-U,-V)$ exchanges the two half-planes. The boundary has measure zero, so each receives half the mass.