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INNER FUNCTIONS, BLOCH SPACES AND SYMMETRIC MEASURES

Published online by Cambridge University Press:  01 September 1999

A. B. ALEKSANDROV ,
J. M. ANDERSON  and
A. NICOLAU
A. B. ALEKSANDROV
Affiliation:
St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, 191011 St Petersburg, Russia. [email protected]
J. M. ANDERSON
Affiliation:
Department of Mathematics, University College, Gower Street, London, WC1E 6BT. [email protected]
A. NICOLAU
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain. [email protected]
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Abstract

Schwarz's lemma asserts that analytic mappings from the unit disc into itself decrease hyperbolic distances. In this paper, inner functions which decrease hyperbolic distances as much as possible, when one approaches the unit circle, are constructed. Actually, it is shown that a quadratic condition governs the best decay of the hyperbolic derivative of an inner function. This is related to a result of L. Carleson on the existence of singular symmetric measures. As a consequence, some results on composition operators are obtained, bringing out the importance of the Bloch spaces in this connection. Another consequence is a uniform way of producing singular measures which are simultaneously symmetric and Kahane.

1991 Mathematics Subject Classification: primary 30D50; secondary 30D45, 26A30, 47B38.

Type
Research Article
Information
Proceedings of the London Mathematical Society , Volume 79 , Issue 2 , September 1999 , pp. 318 - 352
Copyright
1999 London Mathematical Society

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