Beurling–Ahlfors transform

Remarks

It is a singular integral operator of Calderón–Zygmund type, with the kernel $-\frac{1}{\pi}(z - w)^{-2}$. The 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 $0$. With the Wirtinger derivatives $\partial$ and $\bar{\partial}$, it satisfies $\mathcal{B}(\bar{\partial}\varphi) = \partial\varphi$ for every smooth compactly supported $\varphi$. Example: 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 $\mathcal{B}(z e^{-\abs{z}^2}) = \bar{z} e^{-\abs{z}^2}$. On 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. The 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.