Proposition: Response and propagated constraint

Equation (eq:NW) defines a smooth response ${\mathcal{R}}^{\mathrm{NW}}Q=j$, continuous from suitable source Schwartz seminorms to compact smooth seminorms. It satisfies

These equations are built into the prescribed response; they are not derived gravitational dynamics.

Proof. The Newton multiplier $-|p{|}^{-2}$ is locally integrable in three dimensions. Its differentiated multipliers and the wave propagators give smoothness and compact seminorm bounds. The Newton data can have Coulomb tails; square integrability of the metric itself is not needed.

The wave formula gives $\square \overline{j}=-2Q$. Set $d_{\nu }=-{\dot{\overline{j}}}_{0\nu }+{\partial }_i{\overline{j}}_{i\nu }$. The prescribed data give $d(0)=0$. Since $\Delta a=-2\rho$ and $\Delta V_i=2J_i$, the wave equation also gives $\dot{d}(0)=0$. Conservation gives $\square d=0$; uniqueness implies $d=0$. Substitution into the linearized Einstein operator gives $G^{(1)}(j)=-\frac{1}{2}\square \overline{j}=Q$.