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  "body": "\n## Summary\n\nA Calderón–Zygmund operator is a bounded linear operator on $L^2(\\R^n)$ that is given away from the diagonal by integration against a [[Calderón–Zygmund kernel|78N6AXP9VV]].\nThis class contains the Hilbert transform, the [[Riesz transforms|FRNDFJ9WQ5]], the operators with smooth multipliers of Hörmander–Mikhlin type, and many operators that are not translation invariant.\nEvery Calderón–Zygmund operator is bounded on $L^p(\\R^n)$ for $1 \u003c p \u003c \\infty$ and of weak type $(1, 1)$.\nThis makes it the standard class of [[singular integral operators|HTMCEARGDD]].\n\n## Explanations\n\n### L1 human\n\nA Calderón–Zygmund operator averages a function against a kernel that blows up near the diagonal at the borderline rate.\nThe definition asks for two things: the operator is already known to be bounded in the energy norm $L^2$, and its kernel has the right size and smoothness.\nThe theory then shows that the operator is bounded in every $L^p$ norm with $1 \u003c p \u003c \\infty$.\nThe Hilbert transform $f \\mapsto \\frac{1}{\\pi}\\,\\mathrm{p.v.}\\int \\frac{f(y)}{x - y} \\, dy$ is the basic example.\n",
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