Lemma: Native Newton domain and tails

For $z\in \mathcal{X}$, $\text{calRz}$ is smooth and

uniformly on $J⋐I$. The map is continuous in the stated seminorms. It is the unique decaying solution of its rod equation. The same conclusions hold for uniformly compact smooth sources.

Proof. The source $\text{calCz}$ is spatial Schwartz, with smooth bounded time coefficients on $J$. For compact sources the derivative bounds follow directly from (eq:P) outside twice the source radius.

For Schwartz sources split the integral into $|\mathbf{y}|\le |\mathbf{x}|/2$ and its complement. In the first region the differentiated kernel has the claimed decay and the source has uniform finite moments. In the second region the source decreases to arbitrary order. Near $\mathbf{y}=\mathbf{x}$, use local integrability of $P$, transferring derivatives onto the smooth source. Away from that region differentiate the kernel directly. These estimates use finitely many Schwartz seminorms for each desired bound, proving continuity.

The equation follows from $\Delta {\mathsf{G}}_{\mathrm{sp}}=\mathrm{id}$. A difference of two smooth decaying solutions is an entire harmonic function tending to zero; the maximum principle makes it zero.