A note on composition operators acting on holomorphic sobolev spaces

Research output: Contribution to journalArticle

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 languageEnglish
Pages (from-to)4369-4375
Number of pages7
JournalProceedings of the American Mathematical Society
Volume139
Issue number12
DOIs
Publication statusPublished - 2011 Dec 1

Fingerprint

Sobolev spaces
Composition Operator
Function Space
Sobolev Spaces
Mathematical operators
Lipschitz Spaces
Hardy Space
Chemical analysis
Unit Disk
Class
Standards

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.

In: Proceedings of the American Mathematical Society, Vol. 139, No. 12, 01.12.2011, p. 4369-4375.

Research output: Contribution to journalArticle

@article{4562551e2a9f47bf92d1e5d485ff5785,
title = "A note on composition operators acting on holomorphic sobolev spaces",
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.",
keywords = "Composition operator, Holomorphic sobolev spaces, Zygmund class",
author = "Choe, {Boo Rim} and Koo, {Hyung Woon} and Wayne Smith",
year = "2011",
month = "12",
day = "1",
doi = "10.1090/S0002-9939-2011-10944-4",
language = "English",
volume = "139",
pages = "4369--4375",
journal = "Proceedings of the American Mathematical Society",
issn = "0002-9939",
publisher = "American Mathematical Society",
number = "12",

}

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 -