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 μ is M-quasisymmetric (http://planetmath.org/QuasisymmetricMapping). Furthermore there exists an extension of μ to a quasiconformal mapping of the upper half planes such that the maximal dilatation of the extension depends only on M and not on μ.
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 ϕ:H→H (where H is the upper half plane), then the mapping μ:ℝ→ℝ, such that μ(x)=ϕ(x) where x∈ℝ, is the boundary correspondence of ϕ.
To prove the sufficiency of the above theorem Beurling and Ahlfors [2] define ϕ as follows. Given a μ that is a quasisymmetric mapping of the real line onto itself and fixes ∞, we define a map ϕ(x,y)=u(x,y)+iu(x,y) where
u(x,y) | =12y∫y-yμ(x+t)𝑑t, | ||
v(x,y) | =12y∫y0(μ(x+t)-μ(x-t))𝑑t. |
Intuitively ϕ is a function which “smoothes” out any kinks in the function μ as we get further and further away from the real line. It therefore intuitively follows that ϕ 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 |