Theorem: Weighted native Newton realization

Suppose $q>1$ and all required seminorms (eq:weighted) are finite. Then $\mathsf{N}b$ is absolutely defined, smooth, and tends to zero at spatial infinity. It solves

and is the unique smooth decaying solution.

Uniformly for $t\in J$, its undifferentiated output, and each fixed time derivative with the same source bound, obey

For a bounded cutoff profile ${\chi }_R=\chi (\cdot /R)$ equal to one on the radius-$R$ ball, put ${\mathsf{N}}_{R}b=\mathsf{N}({\chi }_{R}b)$. For every compact output region $K$ and fixed derivative order $m$,

for all sufficiently large $R$.

Proof. Near $\mathbf{y}=\mathbf{x}$, the dipole kernel has locally integrable size $C|\mathbf{x}-\mathbf{y}{|}^{-2}$. At infinity, for compact $\mathbf{x}$, its product with the source has radial majorant $C{r}^{-q}\mathrm{d}r$, integrable for $q>1$.

Local smoothness follows by writing $\mathbf{r}=\mathbf{x}-\mathbf{y}$, splitting with a smooth local cutoff in $\mathbf{r}$, and transferring output derivatives to the smooth source in the singular part. The nonsingular part may be differentiated directly. Weighted source derivatives justify all these operations.

For the decay estimate let $r=|\mathbf{x}|$ be large. In $|\mathbf{y}|\le r/2$, the kernel is bounded by $C{r}^{-2}$. The source integral is bounded by $C{r}^{3-q}$ for $q<3$, $C\log r$ for $q=3$, and $C$ for $q>3$.

In $r/2<|\mathbf{y}|<2r$, bound the source by $C{r}^{-q}$ and the dipole integral over a ball of radius $3r$ by $Cr$. In $|\mathbf{y}|\ge 2r$, use the radial integral $C{{\int }_{2r}}^{\infty }{\rho }^{-q}\mathrm{d}\rho$. These give (eq:weighteddecay). Time derivatives are treated by differentiating only the source.

For compact output and sufficiently large $R$, the cutoff difference is supported away from the kernel singularity. Each output spatial derivative of $P_i$ improves its tail; time derivatives act on $b$. Bounding all such terms by the undifferentiated radial estimate gives (eq:cutoffestimate).

Compact cutoff sources satisfy the distributional Poisson equation. The local smooth convergence, also for two output derivatives, passes it to the limit. A difference of smooth decaying solutions is entire harmonic and vanishes by the maximum principle.