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On the Membership in Bergman Spaces of the Derivative of a Blaschke Product With Zeros in a Stolz Domain

Published online by Cambridge University Press:  20 November 2018

Daniel Girela  and
José Ángel Peláez
Daniel Girela
Affiliation:
Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, 29071 Málaga, Spain e-mail: girela@uma.es e-mail: pelaez@anamat.cie.uma.es
José Ángel Peláez
Affiliation:
Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, 29071 Málaga, Spain e-mail: girela@uma.es e-mail: pelaez@anamat.cie.uma.es
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Abstract

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It is known that the derivative of a Blaschke product whose zero sequence lies in a Stolz angle belongs to all the Bergman spaces ${{A}^{P}}$ with $0<p<3/2$. The question of whether this result is best possible remained open. In this paper, for a large class of Blaschke products $B$ with zeros in a Stolz angle, we obtain a number of conditions which are equivalent to the membership of ${B}'$ in the space ${{A}^{p}}\left( p>1 \right)$. As a consequence, we prove that there exists a Blaschke product $B$ with zeros on a radius such that ${B}'\,\notin \,{{A}^{3/2}}$.

Keywords

30D5030D5532A36Blaschke productsHardy spacesBergman spaces
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 49 , Issue 3 , 01 September 2006 , pp. 381 - 388
Copyright
Copyright © Canadian Mathematical Society 2006

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