Compactness of the ∂¯-Neumann operator and commutators of the Bergman projection with continuous functions

Let Ω be a bounded pseudoconvex domain in Cn,n≥2,0≤p≤n, and 1≤q≤n−1. We show that compactness of the ∂¯-Neumann operator, Np,q+1, on square integrable (p,q+1)-forms is equivalent to compactness of the commutators [Pp,q,z¯j] on square integrable ∂¯-closed (p,q)-forms for 1≤j≤n where Pp,q is the Bergm...

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Bibliographic Details
Published inJournal of mathematical analysis and applications Vol. 409; no. 1; pp. 393 - 398
Main Authors Çeli̇k, Mehmet, Şahutoğlu, Sönmez
Format Journal Article
LanguageEnglish
Published Elsevier Inc 01.01.2014
Subjects
[formula omitted]-Neumann operator
Bergman projection
Hankel operators
Pseudoconvex domain
Pseudoconvex domain
Hankel operators
Bergman projection
Neumann operator
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Summary:Let Ω be a bounded pseudoconvex domain in Cn,n≥2,0≤p≤n, and 1≤q≤n−1. We show that compactness of the ∂¯-Neumann operator, Np,q+1, on square integrable (p,q+1)-forms is equivalent to compactness of the commutators [Pp,q,z¯j] on square integrable ∂¯-closed (p,q)-forms for 1≤j≤n where Pp,q is the Bergman projection on (p,q)-forms. We also show that compactness of the commutator of the Bergman projection with bounded functions percolates up in the ∂¯-complex on ∂¯-closed forms and square integrable holomorphic forms.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2013.07.015