# Beurling-Ahlfors quasiconformal extension

###### Theorem (Beurling-Ahlfors).

There exists a quasiconformal mapping of the upper half plane to itself if and only if the boundary correspondence mapping $\mu$ is $M$-quasisymmetric (http://planetmath.org/QuasisymmetricMapping). Furthermore there exists an extension  of $\mu$ to a quasiconformal mapping of the upper half planes such that the maximal dilatation of the extension depends only on $M$ and not on $\mu$.

That is, the extension is $K$-quasiconformal (http://planetmath.org/QuasiconformalMapping) if and only if the boundary correspondence is $M$-quasisymmetric (http://planetmath.org/QuasisymmetricMapping) where $K$ depends purely on $M$.

Supposing that we have the mapping $\phi:H\rightarrow H$ (where $H$ is the upper half plane), then the mapping $\mu:{\mathbb{R}}\rightarrow{\mathbb{R}}$, such that $\mu(x)=\phi(x)$ where $x\in{\mathbb{R}}$, is the boundary correspondence of $\phi$.

To prove the sufficiency of the above theorem Beurling and Ahlfors  define $\phi$ as follows. Given a $\mu$ that is a quasisymmetric mapping of the real line onto itself and fixes $\infty$, we define a map $\phi(x,y)=u(x,y)+iu(x,y)$ where

 $\displaystyle u(x,y)$ $\displaystyle=\frac{1}{2y}\int_{-y}^{y}\mu(x+t)dt,$ $\displaystyle v(x,y)$ $\displaystyle=\frac{1}{2y}\int_{0}^{y}(\mu(x+t)-\mu(x-t))dt.$

Intuitively $\phi$ is a function which “smoothes” out any kinks in the function $\mu$ as we get further and further away from the real line. It therefore intuitively follows that $\phi$ has the worst (highest) dilatation near the $x$ axis, which actually turns out to be true.

## References

• 1 L. V. Ahlfors. . Van Nostrand-Reinhold, Princeton, New Jersey, 1966
• 2 A. Beurling, L. V. Ahlfors. . Acta Math., 96:125-142, 1956.
• 3 J. Lebl. . . Also available at http://www.jirka.org/thesis.pdfhttp://www.jirka.org/thesis.pdf
Title Beurling-Ahlfors quasiconformal extension BeurlingAhlforsQuasiconformalExtension 2013-03-22 14:06:49 2013-03-22 14:06:49 jirka (4157) jirka (4157) 12 jirka (4157) Theorem msc 30C62 Beurling-Ahlfors theorem QuasiconformalMapping QuasisymmetricMapping