Inner functions in Lipschitz, Besov, and Sobolev spaces
type: Item
instance of: scholarly article
title: Inner functions in Lipschitz, Besov, and Sobolev spaces
zbMATH Open document ID: 1234.30041
DOI: 10.1155/2011/626254
author: Daniel Girela
author: Cristóbal González
author: Miroljub Jevtić
published in: Abstract and Applied Analysis
publication date: 2011-07-22T00:00:00Z
review text: Summary: 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 \(H_{\alpha}^p\) with \(1/p \leq \alpha < \infty\) or any of the Besov spaces \(B_{\alpha}^{p,q}\), with \(0 < p, q \leq \infty\) and \(\alpha \geq 1/p\), except when \(p = \infty\), \(\alpha = 0\), and \(2 < q \leq \infty\) or when \(0 < p < \infty\), \(q = \infty\), and \(\alpha = 1/p\) are finite Blaschke products. Our assertion for the spaces \(B_0^{\infty,q}\), \(0 < q \leq 2\), follows from the fact that they are included in the space VMOA. We prove also that for \(2 < q < \infty\), VMOA is not contained in \(B_0^{\infty,q}\) and that this space contains infinite Blaschke products. Furthermore, we obtain distinct results for other values of \(\alpha\) relating the membership of an inner function \(I\) in the spaces under consideration with the distribution of the sequences of preimages \(\{ I^{-1}(a)\}\), \(|a| < 1\). In addition, we include a section devoted to Blaschke products with zeros in a Stolz angle.
Mathematics Subject Classification ID: 30J05
Mathematics Subject Classification ID: 30H25
Mathematics Subject Classification ID: 30H10
Mathematics Subject Classification ID: 30J10
Mathematics Subject Classification ID: 30J15
zbMATH DE Number: 5930764
zbMATH Keywords: inner functions
zbMATH Keywords: Blaschke product
zbMATH Keywords: finite Blaschke product
zbMATH Keywords: Hardy-Sobolev space
zbMATH Keywords: Besov space
zbMATH Keywords: VMOA
Wikidata QID: Q58654357
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2011/626254
OpenAlex ID: W2006550336
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description: scientific article; zbMATH DE number 5930764