Parametric Representations of Quasiconformal Mappings
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Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences, 2020
http://actams.wipm.ac.cn
PARAMETRIC REPRESENTATIONS OF QUASICONFORMAL MAPPINGS∗
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Zhenlian LIN (
School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China E-mail : [email protected]
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Qingtian SHI (
School of Mathematics and Computer Science, Quanzhou Normal University, Quanzhou 362000, China E-mail : [email protected] Abstract In this article, we first give two simple examples to illustrate that two types of parametric representation of the family of Σ0K have some gaps. Then we also find that the area derivative formula (1.6), which is used to estimate the area distortion of Σ0K , cannot be derived from [6], but that formula still holds for Σ0K through our amendatory parametric representation for the one obtained by Eremenko and Hamilton. Key words
quasiconformal mapping; Σ0K ; parametric representation; area distortion theorem; Cauchy transformation
2010 MR Subject Classification
1
30C62; 30C75; 31A10
Introduction
Let C be the complex plane and ∆ be the unit disk of C. Denote ΣK to be the set of all functions which admit a K-quasiconformal extension to ∆ and conformal in C\∆ with f (z) = z + o(1) as z → ∞, in particular, Σ0K = {f ∈ ΣK : f (0) = 0}. The definition of quasiconformal mapping carries over on the complex plane as follows: Definition 1.1 Let Ω be an open region in the complex plane C. A homeomorphism f : Ω → C is K-quasiconformal mapping if it satisfies that (1) f is ACL in Ω; (2) there is a constant k ∈ [0, 1) such that |fz | ≤ k|fz | almost everywhere holds in Ω, here 1+k K = 1−k . For h ∈ Lp , the Cauchy transformation P h and the Hilbert transformation T h of h are ∗ Received
May 20, 2019; revised July 31, 2020. This work is supported by National Natural Science Foundation of China (11971182), the Promotion Program for Young and Middle-aged Teacher in Science and Technology Research of Huaqiao University (ZQN-PY402), Research projects of Young and Middle-aged Teacher’s Education of Fujian Province (JAT190508) and Scientific research project of Quanzhou Normal University (H19009).
No.6
Z.L. Lin & Q.T. Shi: PARAMETRIC REPRESENTATIONS OF QUASICONFORMAL
1875
defined by ZZ 1 zh(ς) P h(z) = − dζdη, π ς(ς − z) C ZZ 1 h(ς) T h(z) = − lim dζdη π r→0 |ς−z|>r (ς − z)2
(1.1)
for ς = ζ + iη, respectively (see [2]). Letting µ ∈ L∞ (C) with kµk∞ < 1, there exists a quasiconformal mapping φ from C onto itself such that φ is the solution of the Beltrami equation ∂z φ = µ∂z φ. Moreover, φ is unique if φ is normalized by φ(0) = 0 and φ(z) = z+o(1) as z → ∞. Furthermore, if µ(λ, z) = λµ(z) for λ ∈ ∆, then ϕ(z, λ), which is the normalized solution of the Beltrami equation ∂z ϕ(z, λ) = µ(z, λ)∂z ϕ(z, λ), (1.2) depends holomorphically on the parametric λ. The above measurable Riemann mapping theorem plays an important role in the theories of quasiconformal mapping and Teichm¨ uller space, see [7, 11–15] for references. Applying the measurable Riemann mapping theorem, Eremenko and Hamilton obtai
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