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 $\varphi :H\to H$ (where $H$ is the upper half plane), then the mapping $\mu :\mathbb{R}\to \mathbb{R}$, such that $\mu (x)=\varphi (x)$ where $x\in \mathbb{R}$, is the boundary correspondence of $\varphi $.
To prove the sufficiency of the above theorem Beurling and Ahlfors [2] define $\varphi $ as follows. Given a $\mu $ that is a quasisymmetric mapping of the real line onto itself and fixes $\mathrm{\infty}$, we define a map $\varphi (x,y)=u(x,y)+iu(x,y)$ where
$u(x,y)$ | $={\displaystyle \frac{1}{2y}}{\displaystyle {\int}_{-y}^{y}}\mu (x+t)\mathit{d}t,$ | ||
$v(x,y)$ | $={\displaystyle \frac{1}{2y}}{\displaystyle {\int}_{0}^{y}}(\mu (x+t)-\mu (x-t))\mathit{d}t.$ |
Intuitively $\varphi $ 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 $\varphi $ 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 |
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Canonical name | BeurlingAhlforsQuasiconformalExtension |
Date of creation | 2013-03-22 14:06:49 |
Last modified on | 2013-03-22 14:06:49 |
Owner | jirka (4157) |
Last modified by | jirka (4157) |
Numerical id | 12 |
Author | jirka (4157) |
Entry type | Theorem |
Classification | msc 30C62 |
Synonym | Beurling-Ahlfors theorem |
Related topic | QuasiconformalMapping |
Related topic | QuasisymmetricMapping |