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 ( Furthermore there exists an extensionPlanetmathPlanetmath 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 ( if and only if the boundary correspondence is M-quasisymmetric ( where K depends purely on M.

Supposing that we have the mapping ϕ:HH (where H is the upper half plane), then the mapping μ:, such that μ(x)=ϕ(x) where x, 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-yyμ(x+t)𝑑t,
v(x,y) =12y0y(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.


  • 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
Title Beurling-Ahlfors quasiconformal extension
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