In this article, we examine stretching and rotation of planar quasiconformal mappings on a line. We show that for almost every point on the line, the set of complex stretching exponents (describing stretching and rotation jointly) is contained in the disk $ \overline{B}(1/(1-k^4),k^2/(1-k^4))$. This yields a quadratic improvement over the known optimal estimate for general sets of Hausdorff dimension $1$. Our proof is based on holomorphic motions and estimates for dimensions of quasicircles. We also give a lower bound for the dimension of the image of a $1$-dimensional subset of a line under a quasiconformal mapping.
If $Ω$ is a simply connected domain in $\overline{\mathbb C}$ then, according to the Ahlfors-Gehring theorem, $Ω$ is a quasidisk if and only if there exists a sufficient condition for the univalence of holomorphic functions in $Ω$ in relation to the growth of their Schwarzian derivative. We extend this theorem to harmonic mappings by proving a univalence criterion on quasidisks. We also show that the mappings satisfying this criterion admit a homeomorphic extension to $\overline{\mathbb C}$ and, under the additional assumption of quasiconformality in $Ω$, they admit a quasiconformal extension …
Sharp Examples for Planar Quasiconformal Distortion of Hausdorff Measures and Removability
在他著名的论文面积失真的拟共形映射，Astala显示了最佳面积失真界和维数失真估计平面拟共形映射。他问（问题4.4）是否更精细的结果举行，即绝对连续性的豪斯多夫措施下推进拟共形映射。这在Astala、Clop、Mateu、Orobitg和作者的联合工作（“Distortion of Hausdorff measures and improved Painlev 'e removability for bounded quasirregular mappings”，Duke Math J.，出现[ACMOU]）（定理1.1），其他情况保持开放。一个相关的问题，我们离开开放的[ACMOU]（问题4.2）（这是问Astala的作者在[ACMOU]之前，在一个等价的形式在个人通信）是是否BMO可移动的K-拟正则映射和（$L^{\infty}$）可移动的K-拟正则映射确实是不同的问题。本文给出了[ACMOU]中正问题4.2的一系列例子，同时证明了[ACMOU]中定理1.1在两种不同意义上的清晰性，并给出了在拟共形映射推进下Hausdorff测度的绝对连续性在这两种不同意义上的清晰性的例子.
We consider quasiconformal mappings of the unit disk that have a planar extension which have $p$-integrable distortion. In this paper, we establish a bound for the modulus of continuity for the inverse mapping and show sharpness of this bound. Furthermore, we also obtain bounds for the compression multifractal spectra of such mappings.
We study quasiconformal mappings of the unit disk that have planar extension with controlled distortion. For these mappings we prove a bound for the modulus of continuity of the inverse map, which somewhat surprisingly is almost as good as for global quasiconformal maps. Furthermore, we give examples which improve the known bounds for the three point property of generalized quasidisks. Finally, we establish optimal regularity of such maps when the image of the unit disk has cusp type singularities.
In this article we study fine regularity properties for mappings of finite distortion. Our main theorems yield strongly localized regularity results in the borderline case in the class of maps of exponentially integrable distortion. Analogues of such results were known earlier in the case of quasiconformal mappings. Moreover, we study regularity for maps whose distortion has higher exponential integrability.
Given any $ε>0$ and any planar region $Ω$ bounded by a simple n-gon $P$ we construct a ($1 + ε)$-quasiconformal map between $Ω$ and the unit disk in time $C(ε)n$. One can take $ C(ε) = C + C \log (1/ε) \log \log (1/ε)$.
