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Extremal Problems for Quasiconformal Maps of Punctured Plane Domains

Marković, Vladimir (2001) Extremal Problems for Quasiconformal Maps of Punctured Plane Domains. Transactions of the American Mathematical Society, 354 (4). pp. 1631-1650. ISSN 0002-9947.

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The main goal of this paper is to give an affirmative answer to the long-standing conjecture which asserts that the affine map is a uniquely extremal quasiconformal map in the Teichmüller space of the complex plane punctured at the integer lattice points. In addition we derive a corollary related to the geometry of the corresponding Teichmüller space. Besides that we consider the classical dual extremal problem which naturally arises in the tangent space of the Teichmüller space. In particular we prove the uniqueness of Hahn-Banach extension of the associated linear functional given on the Bergman space of the integer lattice domain. Several useful estimates related to the local and global properties of integrable meromorphic functions and the delta functional (see the definition below) are also obtained. These estimates are intended to study the behavior of integrable functions near singularities and they are valid in general settings.

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Additional Information:© 2001 American Mathematical Society. Received by the editors March 23, 2000. I am thankful to V. Bozin and E.Reich for reading the manuscript and for their valuable suggestions.
Issue or Number:4
Classification Code:1991 Mathematics Subject Classification. Primary 41A65, 30C75.
Record Number:CaltechAUTHORS:20170505-133323269
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Official Citation:Extremal Problems for Quasiconformal Maps of Punctured Plane Domains Vladimir Marković Transactions of the American Mathematical Society Vol. 354, No. 4 (Apr., 2002), pp. 1631-1650 Published by: American Mathematical Society Stable URL:
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:77231
Deposited By: Ruth Sustaita
Deposited On:05 May 2017 21:09
Last Modified:03 Oct 2019 17:55

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