Proposition: Classical response and constraint propagation

Both prescriptions are linear maps on $\mathcal{Q}$, give smooth solutions, and satisfy

On every finite time interval,

and the response is continuous from finitely many source Schwartz seminorms to every prescribed compact $C^s$ seminorm. The metric has Coulomb-order spatial tails, not a general $L^2$ bound.

Proof. The Newton estimates give finite derivative energy of the position data and finite $L^2$ velocity data. The massless wave energy estimate, proved as in Lemma lem:wave, applies to (eq:response). Smoothness follows by differentiation or the Fourier formula; the potentially singular initial multipliers $|p{|}^{-2}$ and $|p{|}^{-1}$ are locally integrable in three dimensions.

Set

Both data rows have $d(t_0)=0$. For the first row,

For the second row,

Conservation gives $\square d=0$, so uniqueness gives $d=0$. Equation (eq:G1) then proves the sourced equation.

For continuity, finitely many source seminorms control arbitrary prescribed finite Fourier decay of the source and its time derivatives. The initial multipliers are locally integrable and have polynomial growth after differentiation. Fourier inversion on a compact time interval gives every stated compact $C^s$ bound. The Newton bounds, propagated by the finite-time Kirchhoff formula, give Coulomb-order tails. They do not make a nonzero $1/|x|$ term square integrable.