Beurling–Ahlfors transform

Summary

The 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 $\partial f / \partial z$. It is the complex analogue of the Hilbert transform, and it is an isometry of $L^2(\C)$ and a bounded operator on $L^p(\C)$ for $1 < p < \infty$. It is the main analytic tool for solving the Beltrami equation, and hence for the existence theory of quasiconformal maps.