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  "body": "\n## Definition\n\nLet $f \\colon \\C \\to \\C$ be a smooth function with compact support.\nIts *Beurling–Ahlfors transform* is the function $\\mathcal{B} f \\colon \\C \\to \\C$ given by the [[principal value|0V3RDQV3ZX]] integral\n$$\\mathcal{B} f(z) = -\\frac{1}{\\pi} \\lim_{\\varepsilon \\to 0} \\int_{\\abs{w - z} \u003e \\varepsilon} \\frac{f(w)}{(z - w)^2} \\, dA(w),$$\nwhere $dA$ is the Lebesgue area measure on $\\C$.\nThe operator $\\mathcal{B}$ extends uniquely to a bounded linear operator on $L^p(\\C)$ for each $p$ with $1 \u003c p \u003c \\infty$, also called the Beurling–Ahlfors transform.\n\n## Remarks\n\nIt is a singular integral operator of Calderón–Zygmund type, with the kernel $-\\frac{1}{\\pi}(z - w)^{-2}$.\nThe principal value is needed: the kernel has absolute value $\\frac{1}{\\pi}\\abs{z - w}^{-2}$, which is not integrable near $w = z$ in the plane, but its integral over each circle around $z$ is $0$.\nWith the [[Wirtinger derivatives|WMXDVF36ZT]] $\\partial$ and $\\bar{\\partial}$, it satisfies $\\mathcal{B}(\\bar{\\partial}\\varphi) = \\partial\\varphi$ for every smooth compactly supported $\\varphi$.\nExample: for $\\varphi(z) = e^{-\\abs{z}^2}$, $\\bar{\\partial}\\varphi = -z e^{-\\abs{z}^2}$ and $\\partial\\varphi = -\\bar{z} e^{-\\abs{z}^2}$, so $\\mathcal{B}(z e^{-\\abs{z}^2}) = \\bar{z} e^{-\\abs{z}^2}$.\nOn the Fourier transform side, $\\mathcal{B}$ is multiplication by $\\bar{\\xi}/\\xi$, where $\\xi = \\xi_1 + i\\xi_2$ is the frequency variable; as $\\abs{\\bar{\\xi}/\\xi} = 1$, it is an isometry of $L^2(\\C)$ onto itself.\nThe norm of $\\mathcal{B}$ on $L^p(\\C)$ is conjectured by Iwaniec to be $\\max(p, p/(p-1)) - 1$; this is an open problem.\n",
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