Strongly symmetric homeomorphisms on the real line with uniform continuity
Strongly symmetric homeomorphisms on the real line with uniform continuity
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DOI：
10.1512/iumj.2023.72.9323
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发表时间：
2022-07
影响因子：
1.1
通讯作者：
Huaying Wei;Katsuhiko Matsuzaki
中科院分区：
文献类型：
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作者：
Huaying Wei;Katsuhiko Matsuzaki
研究了拟共形Teichm-Uller理论调和分析中出现的实直线的强对称同胚.这类元素可以用一个性质来刻画，即它可以拟共形扩张到上半平面，使得它的复伸缩导致Carleson测度消失。然而，与单位圆上的情形不同，实直线上的强对称同胚在合成或逆变换下都不能保持。在本文中，我们介绍了这两种情况的区别和联系。特别地，我们证明了如果对实直线的强对称同胚假设一致连续，则它们被这些运算保持。我们还证明了一致连续同态的重心扩张会导致Carleson测度消失，上半平面上的拟共形同胚的合成和逆也是如此。
We investigate strongly symmetric homeomorphisms of the real line which appear in harmonic analysis aspects of quasiconformal Teichm\"uller theory. An element in this class can be characterized by a property that it can be extended quasiconformally to the upper half-plane so that its complex dilatation induces a vanishing Carleson measure. However, differently from the case on the unit circle, strongly symmetric homeomorphisms on the real line are not preserved under either the composition or the inversion. In this paper, we present the difference and the relation between these two cases. In particular, we show that if uniform continuity is assumed for strongly symmetric homeomorphisms of the real line, then they are preserved by those operations. We also show that the barycentric extension of uniformly continuous one induces a vanishing Carleson measure and so do the composition and the inverse of those quasiconformal homeomorphisms of the upper half-plane.