---
tag: TX32GDQ330
kind: concept
name: Beurling–Ahlfors transform
msc: [42B20, 30C62, 47G10]
sources:
  - url: https://en.wikipedia.org/wiki/Singular_integral_operators_of_convolution_type
---

## 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|NSCEBXM7ZM]], 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|9J7GFJ1GPF]], and hence for the existence theory of [[quasiconformal maps|SB8KG46NHP]].

## Explanations

### L1 human

A function of a complex variable has two Wirtinger derivatives, one with respect to $z$ and one with respect to $\bar{z}$.
For a compactly supported smooth function, the first is determined by the second, and the Beurling–Ahlfors transform is the operator that computes it.
It works like a Fourier multiplier that only rotates the phase, by the factor $\bar{\xi}/\xi$, so it preserves the $L^2$ norm.
For example, it sends $z e^{-\abs{z}^2}$ to $\bar{z} e^{-\abs{z}^2}$.