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Georgian Mathematical Journal

Editor-in-Chief: Kiguradze, Ivan / Buchukuri, T.

Editorial Board: Kvinikadze, M. / Bantsuri, R. / Baues, Hans-Joachim / Besov, O.V. / Bojarski, B. / Duduchava, R. / Engelbert, Hans-Jürgen / Gamkrelidze, R. / Gubeladze, J. / Hirzebruch, F. / Inassaridze, Hvedri / Jibladze, M. / Kadeishvili, T. / Kegel, Otto H. / Kharazishvili, Alexander / Kharibegashvili, S. / Khmaladze, E. / Kiguradze, Tariel / Kokilashvili, V. / Krushkal, S. I. / Kurzweil, J. / Kwapien, S. / Lerche, Hans Rudolf / Mawhin, Jean / Ricci, P.E. / Tarieladze, V. / Triebel, Hans / Vakhania, N. / Zanolin, Fabio

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IMPACT FACTOR 2017: 0.482
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Online
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1572-9176
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Volume 23, Issue 4

Issues

The Sobolev space of half-differentiable functions and quasisymmetric homeomorphisms

Armen Sergeev
Published Online: 2016-10-18 | DOI: https://doi.org/10.1515/gmj-2016-0047

Abstract

In this paper, we give an interpretation of some classical objects of function theory in terms of Banach algebras of linear operators in a Hilbert space. We are especially interested in quasisymmetric homeomorphisms of the circle. They are boundary values of quasiconformal homeomorphisms of the disk and form a group QS(S1) with respect to composition. This group acts on the Sobolev space H01/2(S1,) of half-differentiable functions on the circle by reparameterization. We give an interpretation of the group QS(S1) and the space H01/2(S1,) in terms of noncommutative geometry.

Keywords: Connes quantization; half-differentiable functions; quasiconformal maps

MSC 2010: 81S10

Dedicated to the memory of Academician N. Muskhelishvili on the occasion of his 125th birthday anniversary

References

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    Ahlfors L. V., Lectures on Quasiconformal Mappings, Van Nostrand Math. Stud. 10, D. Van Nostrand, Princeton, 1966. Google Scholar

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    Connes A., Noncommutative Geometry, Academic Press, San Diego, 1994. Google Scholar

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    Nag S. and Sullivan D., Teichmüller theory and the universal period mapping via quantum calculus and the H1/2 space on the circle, Osaka J. Math. 32 (1995), no. 1, 1–34. Google Scholar

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    Sergeev A., Lectures on Universal Teichmüller Space, EMS Ser. Lect. Math., European Mathematical Society, Zürich, 2014. Google Scholar

About the article

Received: 2016-07-31

Accepted: 2016-09-19

Published Online: 2016-10-18

Published in Print: 2016-12-01


Funding Source: Russian Foundation for Basic Research

Award identifier / Grant number: 16-01-00117

Award identifier / Grant number: 16-52-12012

While preparing this paper the author was partially supported by the RFBR grants 16-01-00117, 16-52-12012, Program of Supporting of Leading Scientific Schools NSh-9110.2016.1 and Scientific Program of Presidium of RAS “Nonlinear Dynamics”.


Citation Information: Georgian Mathematical Journal, Volume 23, Issue 4, Pages 615–622, ISSN (Online) 1572-9176, ISSN (Print) 1072-947X, DOI: https://doi.org/10.1515/gmj-2016-0047.

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