Proposition: Scalar-only representation

The function $q_{f,h}$ is real, smooth, and compactly supported strictly to the future of ${\Sigma }_0$. For every smooth free solution,

More precisely,

For fixed $h,\chi$ and fixed compact observation support $K$, the map $f\mapsto q_{f,h}$ is continuous from ${C_K}^{\infty }$ into ${C_{K\cup \mathrm{supp}\chi }}^{\infty }$.

Proof. Transpose the retarded Green operator and then the density operator:

The integrations by parts have no boundary contribution because $\chi h$ is compactly supported away from ${\Sigma }_0$. For the coordinate term,

This proves (eq:representation) with the sign in (eq:q).

Smoothness follows from smooth Green evolution and (eq:w). The first term has support in $\mathrm{supp}\chi \cap J_{\eta }^{-}(K_f)$; the second has support in $K_f$. Both sets are compact and separated from $t=0$.

For continuity, on the finite time interval containing the source and observation supports, (eq:advanced) gives continuous dependence of every derivative of $u_f$ on finitely many derivatives of $f$. This can be seen by commuting spatial derivatives through the sine propagator and applying Sobolev estimates on an enlarged compact spatial set; time derivatives follow from $P_m{u}_f=f$. Multiplication by the fixed smooth coefficients and application of the displayed finite differential operators then give the asserted fixed-support continuity.