Theorem: Elementary parallel-line extension

A primitive from Proposition prop:parallel-degree with finite degree $s$ extends from ${\mathbb{R}}^4\setminus \{0\}$ to ${\mathbb{R}}^4$ without increasing its scaling degree. If $s\ge 4$, its ambiguity is

For the temporal kinetic component the allowed derivative order is at most four. A relative-coordinate extension has a balanced two-endpoint wavefront bound.

Proof. Choose a compact smooth function $w$ equal to one near zero and put $N=\lfloor s-4\rfloor$. Subtract the normal Taylor jet:

The remainder vanishes to order $N+1$. On a dyadic annulus of radius $2^{-j}$, the scaling estimate and Taylor remainder give a summable bound with exponent $4+N+1-s-\delta >0$ for sufficiently small $\delta >0$. Thus the primitive has a continuous extension to tests with the specified vanishing jet. Assigning the omitted finite jet defines an extension on all tests.

Two such extensions differ by a distribution supported at zero. The scaling-degree bound restricts its derivative order to $\lfloor s-4\rfloor$. This is also the standard point-extension construction BF (Section 5.2).

An extension depending only on $r=y-v$ is pulled back by the relative-coordinate submersion. Its endpoint covectors are $(p,-p)$, including those of its contact terms. Hence it is balanced.