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  "body": "\n## Summary\n\nThe Beurling–Ahlfors transform is a singular integral operator on functions of the complex plane that turns the derivative $\\partial f / \\partial \\bar{z}$ into the derivative $\\partial f / \\partial z$.\nIt is the complex analogue of the [[Hilbert transform|NSCEBXM7ZM]], and it is an isometry of $L^2(\\C)$ and a bounded operator on $L^p(\\C)$ for $1 \u003c p \u003c \\infty$.\nIt is the main analytic tool for solving the [[Beltrami equation|9J7GFJ1GPF]], and hence for the existence theory of [[quasiconformal maps|SB8KG46NHP]].\n\n## Explanations\n\n### L1 human\n\nA function of a complex variable has two Wirtinger derivatives, one with respect to $z$ and one with respect to $\\bar{z}$.\nFor a compactly supported smooth function, the first is determined by the second, and the Beurling–Ahlfors transform is the operator that computes it.\nIt works like a Fourier multiplier that only rotates the phase, by the factor $\\bar{\\xi}/\\xi$, so it preserves the $L^2$ norm.\nFor example, it sends $z e^{-\\abs{z}^2}$ to $\\bar{z} e^{-\\abs{z}^2}$.\n",
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