Composition operators on holomorphic Sobolev spaces in Bn

Hyungwoon Koo; Inyoung Park

doi:10.1016/j.jmaa.2010.02.056

Composition operators on holomorphic Sobolev spaces in B_n

Hyungwoon Koo, Inyoung Park

Research output: Contribution to journal › Article › peer-review

4 Citations (Scopus)

Abstract

We study the composition operator C_Φ on holomorphic Sobolev spaces induced by an analytic self-map Φ of B_n in Cⁿ that extends to be smooth on B_n. We characterize the boundedness and the compactness of C_Φ on A_α,s^p, and prove the jump phenomenon of C_Φ on A_α,s^p. Moreover, we show an interesting result that the boundedness of C_Φ on A_α,s^p is equivalent to the compactness of C_Φ:A_α,s^p→A_β,t^q for appropriate A_β,t^q, for example A_β,t^q=A_α+1/4,s^p. We provide examples to show that our results are sharp.

Original language	English
Pages (from-to)	232-244
Number of pages	13
Journal	Journal of Mathematical Analysis and Applications
Volume	369
Issue number	1
DOIs	https://doi.org/10.1016/j.jmaa.2010.02.056
Publication status	Published - 2010 Sept

Bibliographical note

Funding Information:
✩ This work is partially supported by Korea Research Foundation Grant (KRF-2008-314-C00012).

Keywords

Boundedness
Compactness
Composition operator
Holomorphic Sobolev space
Unit ball

ASJC Scopus subject areas

Analysis
Applied Mathematics

Access to Document

10.1016/j.jmaa.2010.02.056

Cite this

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title = "Composition operators on holomorphic Sobolev spaces in Bn",

abstract = "We study the composition operator CΦ on holomorphic Sobolev spaces induced by an analytic self-map Φ of Bn in Cn that extends to be smooth on Bn. We characterize the boundedness and the compactness of CΦ on Aα,sp, and prove the jump phenomenon of CΦ on Aα,sp. Moreover, we show an interesting result that the boundedness of CΦ on Aα,sp is equivalent to the compactness of CΦ:Aα,sp→Aβ,tq for appropriate Aβ,tq, for example Aβ,tq=Aα+1/4,sp. We provide examples to show that our results are sharp.",

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AU - Park, Inyoung

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AB - We study the composition operator CΦ on holomorphic Sobolev spaces induced by an analytic self-map Φ of Bn in Cn that extends to be smooth on Bn. We characterize the boundedness and the compactness of CΦ on Aα,sp, and prove the jump phenomenon of CΦ on Aα,sp. Moreover, we show an interesting result that the boundedness of CΦ on Aα,sp is equivalent to the compactness of CΦ:Aα,sp→Aβ,tq for appropriate Aβ,tq, for example Aβ,tq=Aα+1/4,sp. We provide examples to show that our results are sharp.

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