Affordable Access

Access to the full text

On the non-smoothness of the vector fields for the dynamically invariant Beltrami coefficients

Authors
  • Huo, Shengjin1
  • Guo, Hui2
  • 1 Tianjin Polytechnic University, Department of Mathematics, Tianjin, 300387, China , Tianjin (China)
  • 2 Shenzhen University, College of Mathematics and Statistics, Shenzhen, 518060, China , Shenzhen (China)
Type
Published Article
Journal
Archiv der Mathematik
Publisher
Springer International Publishing
Publication Date
Feb 08, 2018
Volume
110
Issue
4
Pages
377–389
Identifiers
DOI: 10.1007/s00013-018-1151-7
Source
Springer Nature
Keywords
License
Yellow

Abstract

For μ∈L∞(Δ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu \in L^{\infty }(\Delta )$$\end{document}, the vector fields on the unit circle determined by μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} play an important role in the theory of the universal Teichmüller space. The aim of this paper is to give some characterizations of the vector fields induced by dynamically invariant μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document}. We show that those vector fields are not contained in the Sobolev class H3/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^{3/2}$$\end{document}. At last, we give some results on dynamically invariant vectors to show that the vector fields, the quasi-symmetric homeomorphisms, and the quasi-circles are closely related.

Report this publication

Statistics

Seen <100 times