Composition operators induced by smooth self-maps of the real or complex unit balls

Hyungwoon Koo; Maofa Wang

doi:10.1016/j.jfa.2008.11.002

Composition operators induced by smooth self-maps of the real or complex unit balls

Hyungwoon Koo, Maofa Wang

Department of Mathematics

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

7 Citations (Scopus)

Abstract

In this paper, we study the composition operator C_Φ with a smooth but not necessarily holomorphic symbol Φ. A necessary and sufficient condition on Φ for C_Φ to be bounded on holomorphic (respectively harmonic) weighted Bergman spaces of the unit ball in Cⁿ (respectively Rⁿ) is given. The condition is a real version of Wogen's condition for the holomorphic spaces, and a non-vanishing boundary Jacobian condition for the harmonic spaces. We also show certain jump phenomena on the weights for the target spaces for both the holomorphic and harmonic spaces.

Original language	English
Pages (from-to)	2747-2767
Number of pages	21
Journal	Journal of Functional Analysis
Volume	256
Issue number	9
DOIs	https://doi.org/10.1016/j.jfa.2008.11.002
Publication status	Published - 2009 May 1

Keywords

Bergman space
Boundedness
Carleson measure
Composition operator
Smooth map
Unit ball

ASJC Scopus subject areas

Analysis

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10.1016/j.jfa.2008.11.002

Cite this

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title = "Composition operators induced by smooth self-maps of the real or complex unit balls",

abstract = "In this paper, we study the composition operator CΦ with a smooth but not necessarily holomorphic symbol Φ. A necessary and sufficient condition on Φ for CΦ to be bounded on holomorphic (respectively harmonic) weighted Bergman spaces of the unit ball in Cn (respectively Rn) is given. The condition is a real version of Wogen's condition for the holomorphic spaces, and a non-vanishing boundary Jacobian condition for the harmonic spaces. We also show certain jump phenomena on the weights for the target spaces for both the holomorphic and harmonic spaces.",

keywords = "Bergman space, Boundedness, Carleson measure, Composition operator, Smooth map, Unit ball",

author = "Hyungwoon Koo and Maofa Wang",

note = "Funding Information: ✩ Koo is partially supported by the KRF-2008-314-C00012 and Wang is partially supported by the NSF-10671147 and NSF-10571044 of China. * Corresponding author. E-mail addresses: koohw@korea.ac.kr (H. Koo), mfwang.math@whu.edu.cn (M. Wang).",

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AU - Koo, Hyungwoon

AU - Wang, Maofa

N1 - Funding Information: ✩ Koo is partially supported by the KRF-2008-314-C00012 and Wang is partially supported by the NSF-10671147 and NSF-10571044 of China. * Corresponding author. E-mail addresses: koohw@korea.ac.kr (H. Koo), mfwang.math@whu.edu.cn (M. Wang).

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N2 - In this paper, we study the composition operator CΦ with a smooth but not necessarily holomorphic symbol Φ. A necessary and sufficient condition on Φ for CΦ to be bounded on holomorphic (respectively harmonic) weighted Bergman spaces of the unit ball in Cn (respectively Rn) is given. The condition is a real version of Wogen's condition for the holomorphic spaces, and a non-vanishing boundary Jacobian condition for the harmonic spaces. We also show certain jump phenomena on the weights for the target spaces for both the holomorphic and harmonic spaces.

AB - In this paper, we study the composition operator CΦ with a smooth but not necessarily holomorphic symbol Φ. A necessary and sufficient condition on Φ for CΦ to be bounded on holomorphic (respectively harmonic) weighted Bergman spaces of the unit ball in Cn (respectively Rn) is given. The condition is a real version of Wogen's condition for the holomorphic spaces, and a non-vanishing boundary Jacobian condition for the harmonic spaces. We also show certain jump phenomena on the weights for the target spaces for both the holomorphic and harmonic spaces.

KW - Bergman space

KW - Boundedness

KW - Carleson measure

KW - Composition operator

KW - Smooth map

KW - Unit ball

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U2 - 10.1016/j.jfa.2008.11.002

DO - 10.1016/j.jfa.2008.11.002

M3 - Article

AN - SCOPUS:62049083098

SN - 0022-1236

VL - 256

SP - 2747

EP - 2767

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

IS - 9

ER -

Composition operators induced by smooth self-maps of the real or complex unit balls

Abstract

Keywords

ASJC Scopus subject areas

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