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  "body": "\n## Definition\n\nIdentify $\\R^2$ with $\\C$.\nThe *logarithmic kernel* is the function $k \\colon \\C \\times \\C \\to (-\\infty, +\\infty]$ given by\n$$k(z, w) = \\log \\frac{1}{\\abs{z - w}} \\quad (z \\ne w), \\qquad k(z, z) = +\\infty.$$\n\n## Remarks\n\nExample: $k(2, 0) = -\\log 2 \u003c 0$ and $k(1/2, 0) = \\log 2 \u003e 0$.\nFor fixed $w$, the function $z \\mapsto k(z, w)$ is [[harmonic|0DEB0JMN0W]] on $\\C \\setminus \\set{w}$.\nIndeed, on every disc that does not contain $w$ it is the real part of a holomorphic branch of $-\\log(z - w)$.\nNon-example: in $\\R^3$ the function $\\log(1/\\abs{x - y})$ is not harmonic off the pole, since for $r = \\abs{x - y}$ one has $\\Delta \\log(1/r) = -1/r^2$; the three-dimensional analogue is the Newtonian kernel.\nThe kernel is symmetric and lower semicontinuous, and $k(z, w) = 2\\pi \\, \\Phi(z - w)$, where $\\Phi$ is the [[fundamental solution of the Laplacian|BSJ8X2XRN9]] in the plane.\nSo $\\Delta_z k(z, w) = -2\\pi \\delta_w$ in the sense of distributions.\n",
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