### Abstract

A holomorphic self-map φ of the unit disk is constructed such that the composition operator induced by φ is bounded on the Hardy-Sobolev space H
^{1}/
^{2} of order 2 as well as on the ordinary holomorphic Lipschitz space Lip
_{1} but unbounded on the Zygmund class Λ
_{1}. Among these three function spaces we have embedding relations H
^{1}/
^{2} ⊂ Lip
_{1} ⊂ Λ
_{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.

Original language | English |
---|---|

Pages (from-to) | 4369-4375 |

Number of pages | 7 |

Journal | Proceedings of the American Mathematical Society |

Volume | 139 |

Issue number | 12 |

DOIs | |

Publication status | Published - 2011 Dec 1 |

### Fingerprint

### Keywords

- Composition operator
- Holomorphic sobolev spaces
- Zygmund class

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

**A note on composition operators acting on holomorphic sobolev spaces.** / Choe, Boo Rim; Koo, Hyung Woon; Smith, Wayne.

Research output: Contribution to journal › Article

*Proceedings of the American Mathematical Society*, vol. 139, no. 12, pp. 4369-4375. https://doi.org/10.1090/S0002-9939-2011-10944-4

}

TY - JOUR

T1 - A note on composition operators acting on holomorphic sobolev spaces

AU - Choe, Boo Rim

AU - Koo, Hyung Woon

AU - Smith, Wayne

PY - 2011/12/1

Y1 - 2011/12/1

N2 - A holomorphic self-map φ of the unit disk is constructed such that the composition operator induced by φ is bounded on the Hardy-Sobolev space H 1/ 2 of order 2 as well as on the ordinary holomorphic Lipschitz space Lip 1 but unbounded on the Zygmund class Λ 1. Among these three function spaces we have embedding relations H 1/ 2 ⊂ Lip 1 ⊂ Λ 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.

AB - A holomorphic self-map φ of the unit disk is constructed such that the composition operator induced by φ is bounded on the Hardy-Sobolev space H 1/ 2 of order 2 as well as on the ordinary holomorphic Lipschitz space Lip 1 but unbounded on the Zygmund class Λ 1. Among these three function spaces we have embedding relations H 1/ 2 ⊂ Lip 1 ⊂ Λ 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.

KW - Composition operator

KW - Holomorphic sobolev spaces

KW - Zygmund class

UR - http://www.scopus.com/inward/record.url?scp=80052004000&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=80052004000&partnerID=8YFLogxK

U2 - 10.1090/S0002-9939-2011-10944-4

DO - 10.1090/S0002-9939-2011-10944-4

M3 - Article

AN - SCOPUS:80052004000

VL - 139

SP - 4369

EP - 4375

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 12

ER -