Weighted Composition Operators from Hardy Spaces into Logarithmic Bloch Spaces

The logarithmic Bloch space B log ⁡ is the Banach space of analytic functions on the open unit disk whose elements f satisfy the condition ∥ f ∥ = sup ⁡ z ∈ ( 1 - | z | 2 ) log ⁡    (2 / ( 1 - | z | 2 )) | f ' ( z ) | < ∞ . In this work we characterize the bounded and the compact weighted co...

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Bibliographic Details
Published in:Journal of function spaces 2012-01, Vol.2012, p.1-20
Main Authors: Colonna, Flavia, Li, Songxiao
Format: Article
Language:English
Subjects:
Banach spaces
Power plants
Probability
Studies
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Summary:The logarithmic Bloch space B log ⁡ is the Banach space of analytic functions on the open unit disk whose elements f satisfy the condition ∥ f ∥ = sup ⁡ z ∈ ( 1 - | z | 2 ) log ⁡    (2 / ( 1 - | z | 2 )) | f ' ( z ) | < ∞ . In this work we characterize the bounded and the compact weighted composition operators from the Hardy space H p (with 1 ≤ p ≤ ∞ ) into the logarithmic Bloch space. We also provide boundedness and compactness criteria for the weighted composition operator mapping H p into the little logarithmic Bloch space defined as the subspace of B log ⁡ consisting of the functions f such that lim ⁡ | z | → 1 ( 1 - | z | 2 ) log ⁡    (2 / ( 1 - | z | 2 )) | f ' ( z ) | = 0 .
ISSN:0972-6802
2314-8896
1758-4965
2314-8888
DOI:10.1155/2012/454820