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  "body": "\n## Summary\n\nA singular integral operator is an integral operator $Tf(x) = \\int K(x, y)f(y) \\, dy$ whose kernel has a non-integrable singularity on the diagonal $x = y$, of size $\\abs{x - y}^{-n}$ in $\\R^n$.\nThe integral exists only as a principal value, thanks to cancellation in the kernel.\nThe Hilbert transform and the Riesz transforms are the basic examples.\nCalderón–Zygmund theory shows that such operators are bounded on $L^p$ for $1 \u003c p \u003c \\infty$, which is central in harmonic analysis and elliptic PDE.\n\n## Explanations\n\n### L1 human\n\nMany natural operators, such as the map from the Laplacian of a function to its second derivatives, are given by integrals against kernels that blow up like $1/\\abs{x}^n$.\nSuch an integral diverges if taken with absolute values, so it is not an ordinary integral.\nIt still makes sense because the kernel takes positive and negative values in balanced amounts near the singularity.\nFor example, the Hilbert transform on the line integrates $f(y)$ against $1/(\\pi(x - y))$, which is positive on one side of $x$ and negative on the other.\n",
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