Lemma: Normal convergence of the same kernels

The mixed kernel $b_f$ satisfies $\mathrm{WF}(b_f)\subset {\Gamma }_B$. On a common compact support stage containing a neighborhood of $\mathrm{supp}{b}_f$,

Proof. The retarded wave kernel has diagonal conormals and direct null propagation covectors $(\xi ,-\xi )$. Multiplication by $\theta (z^0)$ adds boundary conormals in the $z$-slot, giving

At the boundary, the observation variable has strictly positive time, so the bulk diagonal does not meet that boundary branch. Differentiation and multiplication by the smooth observation test do not enlarge the wavefront bound.

Strong distribution convergence of compact convolution approximants is immediate from convergence of the corresponding smooth test functions on bounded sets.

For the Fourier seminorms, compactness and the translation-invariant angular bound give arbitrary rapid decrease in every closed cone separated from the angular cone in (eq:Gamma). Convolution multiplies the Fourier transform by bounded smooth multipliers converging to one on fixed frequency balls. Split into a fixed ball and its complement; uniform convergence handles the ball and arbitrary rapid decrease handles the tail.

Localized seminorms are treated by convolution with the rapidly decreasing Fourier transform of the localization. Angular separation controls the far-frequency part, while a slightly larger separated cone controls the near-frequency part. This is the usual compact approximate-identity argument for the normal seminorms.