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Composition operators acting on holomorphic Sobolev spaces

Author(s): Boo Rim Choe; Hyungwoon Koo; Wayne Smith
Journal: Trans. Amer. Math. Soc. 355 (2003), 2829-2855.
MSC (2000): Primary 47B33; Secondary 30D55, 46E15
Posted: March 14, 2003
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Abstract: We study the action of composition operators on Sobolev spaces of analytic functions having fractional derivatives in some weighted Bergman space or Hardy space on the unit disk. Criteria for when such operators are bounded or compact are given. In particular, we find the precise range of orders of fractional derivatives for which all composition operators are bounded on such spaces. Sharp results about boundedness and compactness of a composition operator are also given when the inducing map is polygonal.

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Additional Information:

Boo Rim Choe
Affiliation: Department of Mathematics, Korea University, Seoul 136--701, Korea
Email: choebr@math.korea.ac.kr

Hyungwoon Koo
Affiliation: Department of Mathematics, Korea University, Seoul 136--701, Korea
Email: koohw@math.korea.ac.kr

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

DOI: 10.1090/S0002-9947-03-03273-2
PII: S 0002-9947(03)03273-2
Keywords: Composition operator, fractional derivative, Bergman space
Received by editor(s): April 4, 2002
Received by editor(s) in revised form: August 6, 2002
Posted: March 14, 2003
Additional Notes: The second author's research was partially supported by KRF2001-041-D00012
Copyright of article: Copyright 2003, American Mathematical Society

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