Cross-ratio distortion and Douady-Earle extension. II: Quasiconformality and asymptotic conformality are local (Q351336)

The authors investigate local properties of the Douady-Earle extension. Namely, let \(f\) be an orientation-preserving homeomorphism of the unit circle \(T\), and let \(\Phi\) be its Douady-Earle extension in the unit disc \(D\). Then the authors prove that if \(f\) is quasisymmetric in a neighborhood of a point \(p\in T\), then \(\Phi\) is quasiconformal in a neighborhood of \(p\) intersected with \(D\). Moreover, if \(f\) is symmetric in a neighborhood of \(p\), then \(\Phi\) is asymptotically conformal near \(p\). For Part I, see [the authors, J. Lond. Math. Soc., II. Ser. 86, No. 2, 387--406 (2012; Zbl 1294.30048)].