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  "body": "\n## Definition\n\nLet $n \\ge 3$.\nThe *Newtonian kernel* on $\\R^n$ is the function $k \\colon \\R^n \\times \\R^n \\to (0, +\\infty]$ given by\n$$k(x, y) = \\frac{1}{\\abs{x - y}^{n-2}} \\quad (x \\ne y), \\qquad k(x, x) = +\\infty.$$\n\n## Remarks\n\nExample: for $n = 3$, $k(x, y) = 1/\\abs{x - y}$; for $x = (1, 0, 0)$ and $y = 0$ it equals $1$.\nFor fixed $y$, the function $x \\mapsto k(x, y)$ is [[harmonic|0DEB0JMN0W]] 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 $r = \\abs{x - y}$.\nNon-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.\nThe kernel is symmetric, $k(x, y) = k(y, x)$, lower semicontinuous, and positive.\nWith $\\sigma_n$ 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|BSJ8X2XRN9]].\nSome authors include this constant in the kernel; the definition above is the usual one in potential theory.\nIn the plane its role is played by the logarithmic kernel $\\log(1/\\abs{x - y})$.\n",
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