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Abstract and Applied Analysis
Volume 2011, Article ID 626254, 26 pages
http://dx.doi.org/10.1155/2011/626254
Research Article

Inner Functions in Lipschitz, Besov, and Sobolev Spaces

1Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, 29071 Málaga, Spain
2Matematički Fakultet, University of Belgrade, PP. 550, 11000 Belgrade, Serbia

Received 23 February 2011; Accepted 12 April 2011

Academic Editor: Wolfgang Ruess

Copyright © 2011 Daniel Girela et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study the membership of inner functions in Besov, Lipschitz, and Hardy-Sobolev spaces, finding conditions that enable an inner function to be in one of these spaces. Several results in this direction are given that complement or extend previous works on the subject from different authors. In particular, we prove that the only inner functions in either any of the Hardy-Sobolev spaces 𝐻𝑝𝛼 with 1/𝑝≀𝛼<∞ or any of the Besov spaces 𝐡𝛼𝑝,π‘ž with 0<𝑝,π‘žβ‰€βˆž and 𝛼β‰₯1/𝑝, except when 𝑝=∞, 𝛼=0, and 2<π‘žβ‰€βˆž or when 0<𝑝<∞, π‘ž=∞, and 𝛼=1/𝑝 are finite Blaschke products. Our assertion for the spaces 𝐡0∞,π‘ž, 0<π‘žβ‰€2, follows from the fact that they are included in the space VMOA. We prove also that for 2<π‘ž<∞, VMOA is not contained in 𝐡0∞,π‘ž and that this space contains infinite Blaschke products. Furthermore, we obtain distinct results for other values of 𝛼 relating the membership of an inner function 𝐼 in the spaces under consideration with the distribution of the sequences of preimages {πΌβˆ’1(π‘Ž)}, |π‘Ž|<1. In addition, we include a section devoted to Blaschke products with zeros in a Stolz angle.