Inner Functions in Lipschitz, Besov, and Sobolev Spaces

Author:

Girela Daniel1,González Cristóbal1,Jevtić Miroljub2

Affiliation:

1. Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, 29071 Málaga, Spain

2. Matematički Fakultet, University of Belgrade, PP. 550, 11000 Belgrade, Serbia

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 spacesHαpwith1/pα<or any of the Besov spacesBαp,  qwith0<p,qandα1/p, except whenp=,α=0, and2<qor when0<p<,q=, andα=1/pare finite Blaschke products. Our assertion for the spacesB0,q,0<q2, follows from the fact that they are included in the spaceVMOA. We prove also that for2<q<,VMOAis not contained inB0,qand that this space contains infinite Blaschke products. Furthermore, we obtain distinct results for other values ofαrelating the membership of an inner functionIin 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.

Funder

Ministerio de Ciencia e Innovación

Publisher

Hindawi Limited

Subject

Applied Mathematics,Analysis

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