Theorem: Continuous admission of test families

The maps

and

are continuous linear maps.

For fixed compact supports of finitely many observation and coordinate tests, every finite pointwise word in these generators depends jointly continuously on those tests in their fixed-support smooth topologies. No joint-continuity statement for unrestricted support variation is included.

Proof. For coordinate functionals, the source-to-advanced-solution map and its trace at $t=0$ are continuous on each fixed compact source stage. This follows from smooth finite-time Cauchy estimates and finite propagation. Formula (eq:qa) then gives continuous dependence of the compact distribution kernel in its strong topology.

For the sharp mixed kernel, use

on a compact truncated causal region. The product with the step function exists: its wavefront covectors have zero $x$-component, whereas nonzero wavefront pairs of the wave kernel have nonzero $x$-component. Move $C$ to the kernel and the scalar derivative to the $x$-kernel. Multiplication by $f(x)$ and finite differentiation give the kernel of $B_f$.

On a fixed source support, these operations against bounded smooth joint tests are bounded by finitely many smooth seminorms of $f$. Thus $f\mapsto B_f$ is continuous into the strong coefficient space.

The nonzero boundary coordinate with no directions is

With one direction, only the scalar direction contributes to the first metric derivative, and the same formula holds with that direction replacing $\varphi$. Higher such coordinates vanish. Multiplication by a fixed smooth scalar and advanced trace are continuous on each compact test stage. This proves continuity of all boundary coordinates. The test-function LF topology is final with respect to its fixed compact support stages, so the linear maps are continuous on the full test spaces.

For a finite word with fixed compact test supports, coefficient kernels are exterior tensor products of the fixed distributional kernels multiplied by smooth test coefficients. Finite-order distribution estimates on the common compact region give joint continuity. Boundary coordinates follow from the finite Leibniz formula and continuous evaluations and traces.