Existence of the Fourier-defined distribution

The kernel in $(x,y)$ is obtained by the invertible coordinate change to $(t,s,\mathbf{z},\mathbf{y})$, with constant dependence on the last variable. Consequently the compactly supported pairings in (eq:commutator) are meaningful for arbitrary smooth configuration factors.

All distribution identities below are first proved with ambient time variables in $\mathbb{R}$. Interior restrictions require no boundary choice. For inherited tests meeting $\Sigma$, smooth dependence on the time parameters supplies the corresponding half-space pairing. No new boundary contact is introduced into $K$.