Composition operators on the polydisc induced by smooth symbols

Hyungwoon Koo; Michael Stessin; Kehe Zhu

doi:10.1016/j.jfa.2008.03.003

Composition operators on the polydisc induced by smooth symbols

Hyungwoon Koo, Michael Stessin, Kehe Zhu

Department of Mathematics

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

10 Citations (Scopus)

Abstract

We study composition operators C_Φ on the Hardy spaces H^p and weighted Bergman spaces A_α^p of the polydisc Dⁿ in Cⁿ. When Φ is of class C² on over(Dⁿ, -), we show that C_Φ is bounded on H^p or A_α^p if and only if the Jacobian of Φ does not vanish on those points ζ on the distinguished boundary Tⁿ such that Φ (ζ) ∈ Tⁿ. Moreover, we show that if ε > 0 and if C_Φ : A_α^p → A_{α + frac(1, 2 n) - ε}^p, then C_Φ is bounded on A_α^p.

Original language	English
Pages (from-to)	2911-2925
Number of pages	15
Journal	Journal of Functional Analysis
Volume	254
Issue number	11
DOIs	https://doi.org/10.1016/j.jfa.2008.03.003
Publication status	Published - 2008 Jun 1

Keywords

Composition operator
Jacobian
Polydisc

ASJC Scopus subject areas

Analysis

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10.1016/j.jfa.2008.03.003

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title = "Composition operators on the polydisc induced by smooth symbols",

abstract = "We study composition operators CΦ on the Hardy spaces Hp and weighted Bergman spaces Aαp of the polydisc Dn in Cn. When Φ is of class C2 on over(Dn, -), we show that CΦ is bounded on Hp or Aαp if and only if the Jacobian of Φ does not vanish on those points ζ on the distinguished boundary Tn such that Φ (ζ) ∈ Tn. Moreover, we show that if ε > 0 and if CΦ : Aαp → Aα + frac(1, 2 n) - εp, then CΦ is bounded on Aαp.",

keywords = "Composition operator, Jacobian, Polydisc",

author = "Hyungwoon Koo and Michael Stessin and Kehe Zhu",

note = "Funding Information: ✩ H. Koo is partially supported by SBS and K. Zhu is partially supported by NSF. * Corresponding author. E-mail addresses: koohw@korea.ac.kr (H. Koo), stessin@math.albany.edu (M. Stessin), kzhu@math.albany.edu (K. Zhu).",

year = "2008",

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doi = "10.1016/j.jfa.2008.03.003",

language = "English",

volume = "254",

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journal = "Journal of Functional Analysis",

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T1 - Composition operators on the polydisc induced by smooth symbols

AU - Koo, Hyungwoon

AU - Stessin, Michael

AU - Zhu, Kehe

N1 - Funding Information: ✩ H. Koo is partially supported by SBS and K. Zhu is partially supported by NSF. * Corresponding author. E-mail addresses: koohw@korea.ac.kr (H. Koo), stessin@math.albany.edu (M. Stessin), kzhu@math.albany.edu (K. Zhu).

PY - 2008/6/1

Y1 - 2008/6/1

N2 - We study composition operators CΦ on the Hardy spaces Hp and weighted Bergman spaces Aαp of the polydisc Dn in Cn. When Φ is of class C2 on over(Dn, -), we show that CΦ is bounded on Hp or Aαp if and only if the Jacobian of Φ does not vanish on those points ζ on the distinguished boundary Tn such that Φ (ζ) ∈ Tn. Moreover, we show that if ε > 0 and if CΦ : Aαp → Aα + frac(1, 2 n) - εp, then CΦ is bounded on Aαp.

AB - We study composition operators CΦ on the Hardy spaces Hp and weighted Bergman spaces Aαp of the polydisc Dn in Cn. When Φ is of class C2 on over(Dn, -), we show that CΦ is bounded on Hp or Aαp if and only if the Jacobian of Φ does not vanish on those points ζ on the distinguished boundary Tn such that Φ (ζ) ∈ Tn. Moreover, we show that if ε > 0 and if CΦ : Aαp → Aα + frac(1, 2 n) - εp, then CΦ is bounded on Aαp.

KW - Composition operator

KW - Jacobian

KW - Polydisc

UR - http://www.scopus.com/inward/record.url?scp=41949106537&partnerID=8YFLogxK

U2 - 10.1016/j.jfa.2008.03.003

DO - 10.1016/j.jfa.2008.03.003

M3 - Article

AN - SCOPUS:41949106537

SN - 0022-1236

VL - 254

SP - 2911

EP - 2925

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

IS - 11

ER -

Composition operators on the polydisc induced by smooth symbols

Abstract

Keywords

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