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  "body": "\n## Summary\n\nA weakly singular kernel is a [[kernel|GSMBXT7FPK]] that may become infinite on the diagonal $x = s$.\nIt may grow there only like a power $\\abs{x - s}^{-\\alpha}$ with exponent $\\alpha$ below the dimension.\nSuch a singularity is integrable, so the integral operator still makes sense on continuous functions and is compact there.\nSo the Fredholm theory for continuous kernels carries over.\nKernels of this type arise in potential theory and in Abel's integral equation.\n\n## Explanations\n\n### L1 human\n\nThe kernel blows up where the two points meet, but slowly enough that the area under it stays finite.\nOn an interval this means a blow-up like $1/\\abs{x - s}^{\\alpha}$ with $\\alpha \u003c 1$.\nFor example, $1/\\sqrt{\\abs{x - s}}$ and $\\log \\abs{x - s}$ are weakly singular on $[0, 1]$.\nThe kernel $1/(x - s)$ is not, because $\\int_0^1 ds / \\abs{x - s}$ is infinite.\n",
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