Theorem: Existence, support, continuity, and normalization

The two maps in (eq:second-map)–(eq:iteration) are defined on every $F_c\in {\mathcal{L}}_{\mathrm{off}}$. Their output tests are smooth and compact, and

They are continuous in the compact-test topology and the smooth-background topology.

For every smooth scalar $u$,

Consequently

No singular product, diagonal extension, or subtraction is used.

Proof. The function ${\mathsf{E}}_{\mathrm{A}}c$ is smooth and supported in $J_{\eta }^{-}(\mathrm{supp}c)$. The coefficients of each $L_{j,g}$ are smooth and supported in the fixed compact set $S=\mathrm{supp}g$. This proves the assertion for $s_c$ and for the intermediate test $a_c$.

The function ${\mathsf{E}}_{\mathrm{A}}{a}_c$ is smooth and has support in $J_{\eta }^{-}(\mathrm{supp}{a}_c)$. Applying $L_{1,g}[h]$ gives

by transitivity of the completion causal relation.

For continuity, the maps

are continuous into their respective smooth spaces; the last two have range in the fixed test space ${\mathcal{D}}_S$. The claim follows by composition, first on each compact-support test stage and then on the LF test space. The dependence on $h,k$ is polynomial.

For normalization, formal self-adjointness and ${{\mathsf{E}}_{\mathrm{R}}}^{\mathrm{T}}={\mathsf{E}}_{\mathrm{A}}$ give

Transposing the two Green operators in the iteration successively gives

At every step, the differential source factor is compactly supported. All propagations are applied to smooth tests or smooth compact sources.