BMO in the Bergman metric on bounded symmetric domains

D. Békollé; C. A. Berger; L. A. Coburn; K. H. Zhu

doi:10.1016/0022-1236(90)90131-4

BMO in the Bergman metric on bounded symmetric domains

D. Békollé
, C. A. Berger
, L. A. Coburn
, K. H. Zhu

University at Buffalo, University at Albany

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

129 Scopus citations

Abstract

For bounded symmetric domains Ω in Cⁿ, a notion of "bounded mean oscillation" in terms of the Bergman metric is introduced. It is shown that for f{hook} in L²(Ω, dv), f{hook} is in BMO(Ω) if and only if the densely-defined operator [M_f{hook}, P] ≡ M_f{hook}P - PM_f{hook} on L²(Ω, dv) is bounded (here, M_f{hook} is "multiplication by f{hook}" and P is the Bergman projection with range the Bergman subspace H²(Ω, dv) = L_a²(Ω, dv) of holomorphic functions in L²(Ω, dv)). An analogous characterization of compactness for [M_f{hook}, P] is provided by functions of "vanishing mean oscillation at the boundary of Ω".

Original language	English
Pages (from-to)	310-350
Number of pages	41
Journal	Journal of Functional Analysis
Volume	93
Issue number	2
DOIs	https://doi.org/10.1016/0022-1236(90)90131-4
State	Published - Oct 15 1990

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@article{504941cd0247492698503279ef988846,

title = "BMO in the Bergman metric on bounded symmetric domains",

abstract = "For bounded symmetric domains Ω in Cn, a notion of {"}bounded mean oscillation{"} in terms of the Bergman metric is introduced. It is shown that for f\{hook\} in L2(Ω, dv), f\{hook\} is in BMO(Ω) if and only if the densely-defined operator [Mf\{hook\}, P] ≡ Mf\{hook\}P - PMf\{hook\} on L2(Ω, dv) is bounded (here, Mf\{hook\} is {"}multiplication by f\{hook\}{"} and P is the Bergman projection with range the Bergman subspace H2(Ω, dv) = La2(Ω, dv) of holomorphic functions in L2(Ω, dv)). An analogous characterization of compactness for [Mf\{hook\}, P] is provided by functions of {"}vanishing mean oscillation at the boundary of Ω{"}.",

author = "D. B{\'e}koll{\'e} and Berger, \{C. A.\} and Coburn, \{L. A.\} and Zhu, \{K. H.\}",

year = "1990",

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day = "15",

doi = "10.1016/0022-1236(90)90131-4",

language = "English",

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TY - JOUR

T1 - BMO in the Bergman metric on bounded symmetric domains

AU - Békollé, D.

AU - Berger, C. A.

AU - Coburn, L. A.

AU - Zhu, K. H.

PY - 1990/10/15

Y1 - 1990/10/15

N2 - For bounded symmetric domains Ω in Cn, a notion of "bounded mean oscillation" in terms of the Bergman metric is introduced. It is shown that for f{hook} in L2(Ω, dv), f{hook} is in BMO(Ω) if and only if the densely-defined operator [Mf{hook}, P] ≡ Mf{hook}P - PMf{hook} on L2(Ω, dv) is bounded (here, Mf{hook} is "multiplication by f{hook}" and P is the Bergman projection with range the Bergman subspace H2(Ω, dv) = La2(Ω, dv) of holomorphic functions in L2(Ω, dv)). An analogous characterization of compactness for [Mf{hook}, P] is provided by functions of "vanishing mean oscillation at the boundary of Ω".

AB - For bounded symmetric domains Ω in Cn, a notion of "bounded mean oscillation" in terms of the Bergman metric is introduced. It is shown that for f{hook} in L2(Ω, dv), f{hook} is in BMO(Ω) if and only if the densely-defined operator [Mf{hook}, P] ≡ Mf{hook}P - PMf{hook} on L2(Ω, dv) is bounded (here, Mf{hook} is "multiplication by f{hook}" and P is the Bergman projection with range the Bergman subspace H2(Ω, dv) = La2(Ω, dv) of holomorphic functions in L2(Ω, dv)). An analogous characterization of compactness for [Mf{hook}, P] is provided by functions of "vanishing mean oscillation at the boundary of Ω".

UR - https://www.scopus.com/pages/publications/38249016721

U2 - 10.1016/0022-1236(90)90131-4

DO - 10.1016/0022-1236(90)90131-4

M3 - Article

SN - 0022-1236

VL - 93

SP - 310

EP - 350

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

IS - 2

ER -

BMO in the Bergman metric on bounded symmetric domains

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