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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A note on composition operators acting on holomorphic Sobolev spaces


Authors: Boo Rim Choe, Hyungwoon Koo and Wayne Smith
Journal: Proc. Amer. Math. Soc. 139 (2011), 4369-4375
MSC (2010): Primary 47B33; Secondary 30C35, 46E35
Published electronically: April 21, 2011
MathSciNet review: 2823082
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Abstract: A holomorphic self-map $ \phi$ of the unit disk is constructed such that the composition operator induced by $ \phi$ is bounded on the Hardy-Sobolev space $ H^1_2$ of order $ 2$ as well as on the ordinary holomorphic Lipschitz space $ {\rm Lip}_1$ but unbounded on the Zygmund class $ \Lambda_1$. Among these three function spaces we have embedding relations $ H^1_2\subset {\rm Lip}_1\subset \Lambda_1$. So, the main points here are that our construction provides a composition operator which is bounded on smaller spaces, but not on a larger space and that all the function spaces involved are standard ones.


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Additional Information

Boo Rim Choe
Affiliation: Department of Mathematics, Korea University, Seoul 136–713, Republic of Korea
Email: cbr@korea.ac.kr

Hyungwoon Koo
Affiliation: Department of Mathematics, Korea University, Seoul 136–713, Republic of Korea
Email: koohw@korea.ac.kr

Wayne Smith
Affiliation: Department of Mathematics, University of Hawaii, Honolulu, Hawaii 96822
Email: wayne@math.hawaii.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-10944-4
PII: S 0002-9939(2011)10944-4
Keywords: Composition operator, holomorphic Sobolev spaces, Zygmund class
Received by editor(s): October 18, 2010
Published electronically: April 21, 2011
Additional Notes: The first two authors were supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-314-C00012).
The third author wishes to acknowledge the hospitality of Korea University, where this research was carried out
Communicated by: Richard Rochberg
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.