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  "body": "\n## Summary\n\nA Calderón–Zygmund kernel is a function $K(x, y)$ on $\\R^n \\times \\R^n$ off the diagonal.\nIt is at most of size $\\abs{x - y}^{-n}$, and it is smooth in a quantitative way away from the diagonal.\nThese are the kernels of the [[singular integral operators|HTMCEARGDD]] of Calderón–Zygmund theory, such as the Hilbert and [[Riesz transforms|FRNDFJ9WQ5]].\nThe size and smoothness conditions are what turn an $L^2$ bound into $L^p$ bounds for $1 \u003c p \u003c \\infty$.\n\n## Explanations\n\n### L1 human\n\nMany important operators act by integrating a function against a kernel $K(x, y)$ that blows up when $y$ comes close to $x$.\nThe blow-up is of the borderline size $\\abs{x - y}^{-n}$, just too big to integrate.\nA Calderón–Zygmund kernel has this size and, in addition, changes slowly when $x$ or $y$ moves by a small amount compared to their distance.\nThe model example on the line is $K(x, y) = \\frac{1}{\\pi(x - y)}$, the kernel of the Hilbert transform.\n",
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