Newtonian kernel

Remarks

Example: for $n = 3$, $k(x, y) = 1/\abs{x - y}$; for $x = (1, 0, 0)$ and $y = 0$ it equals $1$. For fixed $y$, the function $x \mapsto k(x, y)$ is harmonic on $\R^n \setminus \set{y}$, since $\Delta r^{2-n} = (2-n)(1-n) r^{-n} + (n-1)(2-n) r^{-n} = 0$ for $r = \abs{x - y}$. Non-example: on $\R^4$, the function $1/\abs{x - y}$ is not the Newtonian kernel; its Laplacian in $x$ is $-\abs{x - y}^{-3} \ne 0$, so it is not harmonic off the pole. The kernel is symmetric, $k(x, y) = k(y, x)$, lower semicontinuous, and positive. With $\sigma_n$ the area of the unit sphere of $\R^n$, one has $k(x, y) = (n-2)\sigma_n \, \Phi(x - y)$, where $\Phi$ is the fundamental solution of the Laplacian. Some authors include this constant in the kernel; the definition above is the usual one in potential theory. In the plane its role is played by the logarithmic kernel $\log(1/\abs{x - y})$.