### 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 |
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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 |

### Keywords

- Composition operator
- Holomorphic sobolev spaces
- Zygmund class

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

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## Cite this

Choe, B. R., Koo, H., & Smith, W. (2011). A note on composition operators acting on holomorphic sobolev spaces.

*Proceedings of the American Mathematical Society*,*139*(12), 4369-4375. https://doi.org/10.1090/S0002-9939-2011-10944-4