Inner Functions in Lipschitz, Besov, and Sobolev Spaces
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 part...
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| Format: | Article |
| Language: | English |
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Wiley
2011-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2011/626254 |
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| author | Daniel Girela Cristóbal González Miroljub Jevtić |
| author_facet | Daniel Girela Cristóbal González Miroljub Jevtić |
| author_sort | Daniel Girela |
| collection | DOAJ |
| description | 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 Hαp with 1/p≤α<∞ or any of the Besov spaces
Bαp, q with
0<p,q≤∞ and
α≥1/p, except when
p=∞,
α=0, and 2<q≤∞ or when
0<p<∞,
q=∞, and α=1/p are finite Blaschke products. Our assertion for the spaces
B0∞,q,
0<q≤2, follows from the fact that they are included in the space
VMOA. We prove also that for 2<q<∞, VMOA is not contained in B0∞,q 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
{I-1(a)},
|a|<1. In addition, we include a section devoted to Blaschke products with zeros in a Stolz angle. |
| format | Article |
| id | doaj-art-b85295d8f8b04936a31a9dd18e291c93 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2011-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-b85295d8f8b04936a31a9dd18e291c932025-02-03T07:24:38ZengWileyAbstract and Applied Analysis1085-33751687-04092011-01-01201110.1155/2011/626254626254Inner Functions in Lipschitz, Besov, and Sobolev SpacesDaniel Girela0Cristóbal González1Miroljub Jevtić2Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, 29071 Málaga, SpainDepartamento de Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, 29071 Málaga, SpainMatematički Fakultet, University of Belgrade, PP. 550, 11000 Belgrade, SerbiaWe 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 Hαp with 1/p≤α<∞ or any of the Besov spaces Bαp, q with 0<p,q≤∞ and α≥1/p, except when p=∞, α=0, and 2<q≤∞ or when 0<p<∞, q=∞, and α=1/p are finite Blaschke products. Our assertion for the spaces B0∞,q, 0<q≤2, follows from the fact that they are included in the space VMOA. We prove also that for 2<q<∞, VMOA is not contained in B0∞,q 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 {I-1(a)}, |a|<1. In addition, we include a section devoted to Blaschke products with zeros in a Stolz angle.http://dx.doi.org/10.1155/2011/626254 |
| spellingShingle | Daniel Girela Cristóbal González Miroljub Jevtić Inner Functions in Lipschitz, Besov, and Sobolev Spaces Abstract and Applied Analysis |
| title | Inner Functions in Lipschitz, Besov, and Sobolev Spaces |
| title_full | Inner Functions in Lipschitz, Besov, and Sobolev Spaces |
| title_fullStr | Inner Functions in Lipschitz, Besov, and Sobolev Spaces |
| title_full_unstemmed | Inner Functions in Lipschitz, Besov, and Sobolev Spaces |
| title_short | Inner Functions in Lipschitz, Besov, and Sobolev Spaces |
| title_sort | inner functions in lipschitz besov and sobolev spaces |
| url | http://dx.doi.org/10.1155/2011/626254 |
| work_keys_str_mv | AT danielgirela innerfunctionsinlipschitzbesovandsobolevspaces AT cristobalgonzalez innerfunctionsinlipschitzbesovandsobolevspaces AT miroljubjevtic innerfunctionsinlipschitzbesovandsobolevspaces |