Proposition: Constructed harmonic-coordinate coefficients

The unique decaying coefficients are

They are smooth. On compact time intervals, $Y_1=O(|\mathbf{x}{|}^{-2})$ and $Y_2=O(|\mathbf{x}{|}^{-1})$ at spatial infinity.

Proof. The harmonic equation at the two successive orders is

All coefficients of $P_1,P_2$ have spatially compact support. Hence both sources are smooth and compactly supported, and (eq:Coulomb) constructs the solutions.

Any difference is a smooth harmonic function tending to zero at infinity. The maximum principle on successively larger balls gives zero, proving uniqueness.

The second estimate is the usual direct $1/|\mathbf{x}|$ bound for the displayed compact-source integral. For the first, (eq:B) is a spatial divergence, so its integral vanishes. Subtract $|\mathbf{x}{|}^{-1}$ from the kernel in (eq:Coulomb) outside a ball containing the source. The remaining kernel is $O(|\mathbf{x}{|}^{-2})$. The same estimates are uniform on compact time intervals.