Extension of boundary homeomorphisms to mappings of finite distortion
Extension of boundary homeomorphisms to mappings of finite distortion
复制标题
边界同胚到有限畸变映射的扩展
DOI：
10.1112/plms.12462
复制
发表时间：
2022
影响因子：
1.8
通讯作者：
Ntalampekos, Dimitrios
中科院分区：
文献类型：
--
作者：
Karafyllia, Christina;Ntalampekos, Dimitrios
我们提供了充分条件，使得真实的直线或圆的同胚分别允许扩张到上半平面或圆盘中的有限畸变映射。此外，我们可以确保扩张的拟共形伸缩满足某些可积性条件，例如p$p$-可积或指数可积。满足后一个可积条件的映射也称为大卫同胚。我们的扩展操作是相同的Beurling和Ahlfors在他们的著名工作中使用的一个。本文证明了真实的直线的同胚的Beurling-Ahlfors扩张的拟共形扩张的最优界，并给出了它的对称畸变函数.更具体地说，拟共形伸缩的上界是对称失真函数的平均值，下界是对称失真函数本身。因此，真实的直线的同胚的Beurling-Ahlfors扩张的拟共形伸缩是（次）指数可积的，是p$p$-可积的，或具有BMO$BMO$优数当且仅当对称畸变分别是（次）指数可积的，是p$p$-可积的，或具有BMO$BMO$优数。这些定理都是新的，并调和了过去已经建立的几个充分扩张条件。
We provide sufficient conditions so that a homeomorphism of the real line or of the circle admits an extension to a mapping of finite distortion in the upper half‐plane or the disk, respectively. Moreover, we can ensure that the quasiconformal dilatation of the extension satisfies certain integrability conditions, such as p$p$‐integrability or exponential integrability. Mappings satisfying the latter integrability condition are also known as David homeomorphisms. Our extension operator is the same as the one used by Beurling and Ahlfors in their celebrated work. We prove an optimal bound for the quasiconformal dilatation of the Beurling–Ahlfors extension of a homeomorphism of the real line, in terms of its symmetric distortion function. More specifically, the quasiconformal dilatation is bounded above by an average of the symmetric distortion function and below by the symmetric distortion function itself. As a consequence, the quasiconformal dilatation of the Beurling–Ahlfors extension of a homeomorphism of the real line is (sub)exponentially integrable, is p$p$‐integrable, or has a BMO$BMO$ majorant if and only if the symmetric distortion is (sub)exponentially integrable, is p$p$‐integrable, or has a BMO$BMO$ majorant, respectively. These theorems are all new and reconcile several sufficient extension conditions that have been established in the past.
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DOI：
10.1090/ecgd/379
发表时间：
2019-12
期刊：
Conformal Geometry and Dynamics of the American Mathematical Society
影响因子：
--
作者：
Russell Lodge;M. Lyubich;S. Merenkov;S. Mukherjee
通讯作者：
Russell Lodge;M. Lyubich;S. Merenkov;S. Mukherjee
具有给定边界值的拟共形映射
影响因子：
2.5
作者：
T. J. Reed
通讯作者：
T. J. Reed
有限畸变的平面映射
影响因子：
2.1
作者：
P. Koskela
通讯作者：
P. Koskela
BMO-qc 自同构的边界行为
影响因子：
1
作者：
S. Sastry
通讯作者：
S. Sastry
$mu(z)$-同态下的边界对应。
影响因子：
0.9
作者：
Ji;Zhi;C. He
通讯作者：
C. He