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  "body": "\n## Definition\n\nLet $m \\ge 1$ and let $\\Omega \\subseteq \\R^m$ be a closed bounded set.\nWrite $\\abs{x - s}$ for the Euclidean distance of $x, s \\in \\Omega$.\nA *weakly singular kernel* on $\\Omega$ is a [[kernel|GSMBXT7FPK]] $K$ defined for $x, s \\in \\Omega$ with $x \\ne s$ such that:\n\n1. $K$ is a [[continuous function|0X3WGX95FE]] on $\\set{(x, s) \\in \\Omega \\times \\Omega : x \\ne s}$.\n2. There are constants $M \u003e 0$ and $\\alpha \\in [0, m)$ such that $\\abs{K(x, s)} \\le M \\abs{x - s}^{-\\alpha}$ for all $x \\ne s$.\n\n## Remarks\n\nExample: $K(x, s) = \\abs{x - s}^{-1/2}$ on $\\Omega = [0, 1]$, with $m = 1$, $M = 1$ and $\\alpha = 1/2 \u003c 1$.\nIt is continuous off the diagonal.\n\nNon-example: $K(x, s) = 1/(x - s)$ on $[0, 1]$.\nIt satisfies the bound only with $\\alpha = 1$, which is not less than $m = 1$; the integral $\\int_0^1 \\abs{x - s}^{-1} \\, ds$ diverges.\n\nSince $\\alpha \u003c m$, the function $s \\mapsto \\abs{x - s}^{-\\alpha}$ is integrable over $\\Omega$, so $\\int_\\Omega K(x, s) \\varphi(s) \\, ds$ exists for bounded continuous $\\varphi$.\nAssume also that $\\Omega$ is Jordan measurable, for example a closed interval or a closed ball.\nThen the integral operator of a weakly singular kernel maps continuous functions on $\\Omega$ to continuous functions and is compact for the maximum norm.\nThe kernel $\\log \\abs{x - s}$ is weakly singular on a bounded set: it satisfies the bound for every $\\alpha \\in (0, m)$.\n",
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