A restricted flat limit

For the untwisted Friedrichs kernel, fix regular spatial points and a lifted angle that stays a positive distance from $\pm \pi$ as $\alpha \uparrow 1$. For $\alpha$ sufficiently close to one, only $k=0$ belongs to $\mathcal{K}$. Equation (eq:Carslaw) gives

On angular sets bounded away from the image boundaries, differentiation of the closed denominators gives, for $\alpha$ near one, a bound of the form

Consequently dominated convergence in (eq:master) gives the flat heat and positive resolvent kernels at the fixed pair.

For the retarded kernel tested against a fixed compact time test, only a bounded $u$-interval contributes. The same argument applied to the joint definition in Theorem thm:closed gives convergence as a time distribution at the fixed generic pair; on smooth kernel regions it also gives the ordinary pointwise limit. This is not uniform through a moving image boundary or a moving front.