A note on composition operators acting on holomorphic sobolev spaces

Boo Rim Choe; Hyungwoon Koo; Wayne Smith

doi:10.1090/S0002-9939-2011-10944-4

A note on composition operators acting on holomorphic sobolev spaces

Boo Rim Choe, Hyungwoon Koo, Wayne Smith

Department of Mathematics

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

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 H¹/² of order 2 as well as on the ordinary holomorphic Lipschitz space Lip₁ but unbounded on the Zygmund class Λ₁. Among these three function spaces we have embedding relations H¹/² ⊂ Lip₁ ⊂ Λ₁. 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	https://doi.org/10.1090/S0002-9939-2011-10944-4
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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10.1090/S0002-9939-2011-10944-4

Cite this

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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 H1/2 of order 2 as well as on the ordinary holomorphic Lipschitz space Lip1 but unbounded on the Zygmund class Λ1. Among these three function spaces we have embedding relations H1/2 ⊂ Lip1 ⊂ Λ1. 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.

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KW - Zygmund class

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A note on composition operators acting on holomorphic sobolev spaces

Abstract

Keywords

ASJC Scopus subject areas

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