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  "body": "\n## Statement\n\nLet $A \\colon \\R \\to \\R$ be a function that satisfies a [[Lipschitz condition|0AW1E6YTJY]], and let $\\Gamma$ be its graph $\\set{x + iA(x) : x \\in \\R}$ in $\\C$.\nFor $\\varepsilon \u003e 0$ and $f$ continuous with compact support on $\\R$, let\n$$C_\\varepsilon f(x) = \\frac{1}{\\pi i} \\int_{\\abs{x - y} \u003e \\varepsilon} \\frac{f(y) \\, (1 + iA'(y))}{(y - x) + i(A(y) - A(x))} \\, dy, \\quad x \\in \\R.$$\nThen there is a constant $C$, depending only on $\\norm{A'}_{L^\\infty}$, such that\n$$\\norm{C_\\varepsilon f}_{L^2(\\R)} \\le C \\norm{f}_{L^2(\\R)}$$\nfor all $\\varepsilon \u003e 0$ and all such $f$.\nSo the Cauchy integral on $\\Gamma$ extends to a bounded operator on the [[Lp space|SASP14MA1Z]] $L^2(\\R)$.\n\n## Remarks\n\nThe operator $C_\\varepsilon f$ is the truncated Cauchy integral $\\frac{1}{\\pi i}\\int \\frac{F(\\zeta)}{\\zeta - z} \\, d\\zeta$ over $\\Gamma$, written in the parameter $x$, with $F(y + iA(y)) = f(y)$ and $z = x + iA(x)$.\nA Lipschitz function is differentiable almost everywhere, with $A'$ bounded, so the formula makes sense.\nLet $Hf(x) = \\frac{1}{\\pi} \\lim_{\\varepsilon \\to 0} \\int_{\\abs{x - y} \u003e \\varepsilon} \\frac{f(y)}{x - y} \\, dy$ be the [[Hilbert transform|NSCEBXM7ZM]].\nFor $A = 0$ the curve is the real line and the operator is $iH$.\nSo the theorem generalizes the $L^2$ bound of the Hilbert transform.\n\nThe case of small $\\norm{A'}_{L^\\infty}$ is due to Calderón.\nExpanding the kernel in powers of $A(y) - A(x)$ writes the operator as a series of Calderón commutators, and the series converges when $\\norm{A'}_{L^\\infty}$ is small.\nThe general case, for every Lipschitz function $A$, was proved by Coifman, McIntosh and Meyer.\nSince the kernel is a Calderón–Zygmund kernel, the $L^2$ bound also gives boundedness on $L^p(\\R)$ for $1 \u003c p \u003c \\infty$.\n\n## Explanations\n\n### L1 human\n\nThe Cauchy integral is the basic way to build holomorphic functions from values on a curve.\nOn a smooth curve it behaves like the Hilbert transform on the line.\nThe theorem says that the same good behaviour holds on much rougher curves.\nThey may have corners, as long as they are graphs of Lipschitz functions, such as the graph of $\\abs{x}$.\n\n### L2 human\n\nThe theorem is a central example of a [[singular integral operator|HTMCEARGDD]] whose kernel is not of convolution type.\nIt is the starting point for the method of layer potentials on Lipschitz domains, which solves the Dirichlet problem for the Laplacian on domains with rough boundary.\n",
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  "msc": [
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  "name": "Cauchy integral on Lipschitz curves",
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