Lemma: Existence and scaling of the reference extension

Equation (eq:finite-part) defines a real retarded distribution extending (eq:cone-response). Its scaling degree at the origin is four.

Proof. Near zero, $F(t,t\mathbf{\omega })-F(0)=O(t)$ uniformly in $\mathbf{\omega }$. Both terms are therefore integrable; away from zero all supports are bounded. On a fixed compact support, the formula is bounded by a finite smooth-test seminorm, so it is a distribution.

For a test supported away from zero, use radial variables $r=\text{abs}\mathbf{x}$ and $u=t^2-r^2$. The identity

shows agreement with the punctured kernel. Support is contained in the future null cone and the origin.

For $F_{\lambda }(z)={\lambda }^{-4}F(z/\lambda )$, a change of variables gives

The logarithm does not increase scaling degree above four. The nonzero homogeneous punctured restriction has degree $-4$, so the scaling degree is exactly four.