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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $\\Delta = \\set{(x, x) : x \\in \\R^n}$ be the diagonal of $\\R^n \\times \\R^n$.\nA *Calderón–Zygmund kernel*, or standard kernel, is a function $K \\colon (\\R^n \\times \\R^n) \\setminus \\Delta \\to \\C$ for which there are constants $A \u003e 0$ and $0 \u003c \\delta \\le 1$ such that:\n\n1. $\\abs{K(x, y)} \\le A \\abs{x - y}^{-n}$ for all $x \\ne y$;\n2. $\\abs{K(x, y) - K(x', y)} \\le A \\dfrac{\\abs{x - x'}^\\delta}{\\abs{x - y}^{n + \\delta}}$ whenever $\\abs{x - x'} \\le \\frac{1}{2}\\abs{x - y}$;\n3. $\\abs{K(x, y) - K(x, y')} \\le A \\dfrac{\\abs{y - y'}^\\delta}{\\abs{x - y}^{n + \\delta}}$ whenever $\\abs{y - y'} \\le \\frac{1}{2}\\abs{x - y}$.\n\n## Remarks\n\nExample: $n = 1$ and $K(x, y) = \\frac{1}{\\pi(x - y)}$, the kernel of the Hilbert transform.\nClause 1 holds with $A = 1/\\pi$.\nFor clause 2, if $\\abs{x - x'} \\le \\frac{1}{2}\\abs{x - y}$ then $\\abs{x' - y} \\ge \\frac{1}{2}\\abs{x - y}$, and\n$$\\abs{K(x, y) - K(x', y)} = \\frac{\\abs{x - x'}}{\\pi \\abs{x - y}\\abs{x' - y}} \\le \\frac{2}{\\pi}\\frac{\\abs{x - x'}}{\\abs{x - y}^2},$$\nso clause 2 holds with $\\delta = 1$; clause 3 is the same by symmetry.\n\nNon-example: $K(x, y) = \\abs{x - y}^{-n - 1/2}$ fails clause 1, because it is too large near the diagonal.\n\nA convolution kernel $K(x, y) = k(x - y)$ is a Calderón–Zygmund kernel when $\\abs{k(x)} \\le A\\abs{x}^{-n}$ and $\\abs{\\nabla k(x)} \\le A\\abs{x}^{-n-1}$ for $x \\ne 0$, by the mean value theorem.\nA weaker smoothness condition often used for convolution kernels is the Hörmander condition $\\int_{\\abs{x} \u003e 2\\abs{y}} \\abs{k(x - y) - k(x)} \\, dx \\le A$ for all $y \\ne 0$.\nThe size and smoothness conditions alone do not give an $L^2$ bound; one also needs cancellation, as for the [[singular integral operators|HTMCEARGDD]] with mean-zero angular part.\n",
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