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Compactness of the canonical solution operator of \(\bar\partial\) on bounded pseudoconvex domains

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Title: Compactness of the canonical solution operator of \(\bar\partial\) on bounded pseudoconvex domains
Authors: Knirsch, Wolfgang
Publisher Information: Wiley (Wiley-VCH), Weinheim
Subject Terms: Bergman spaces of functions in several complex variables, Toeplitz operators, Hankel operators, Wiener-Hopf operators, d-bar equation, Bergman-Toeplitz operators, \(\overline\partial\) and \(\overline\partial\)-Neumann operators, \(\overline\partial\)-Neumann operator, canonical solution operator, \((0,q)\)-forms with holomorphic coefficients
Description: The main result of this paper is an explicit formula for the canonical solution operator \(S_{q+1}\) to \(\overline{\partial} \) restricted to \((0,q)\)-forms with holomorphic coefficients. The restriction to forms with holomorphic coefficients is essential for the proof and it shows an interesting relationship of \(\overline{\partial} \) to the Bergman kernel and certain multiplication operators. In the second part the explicit formula for the solution operator to \(\overline{\partial} \) is applied to generalize a result of \textit{N. Salinas}, \textit{A. Sheu} and \textit{H. Upmeier} [Ann. Math. (2) 130, 531--565 (1989; Zbl 0708.47021)], concerning compactness of the solution operator, the \(\overline{\partial} \)- Neumann operator and certain Bergman-Toeplitz operators.
Document Type: Article
File Description: application/xml
DOI: 10.1002/1522-2616(200211)245:1<94::aid-mana94>3.0.co;2-h
Access URL: https://zbmath.org/1848362
Accession Number: edsair.c2b0b933574d..6d21c153c354d2d2581a48dc12bcc20e
Database: OpenAIRE
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  Data: Compactness of the canonical solution operator of \(\bar\partial\) on bounded pseudoconvex domains
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  Data: &lt;searchLink fieldCode=&quot;DE&quot; term=&quot;%22Bergman+spaces+of+functions+in+several+complex+variables%22&quot;&gt;Bergman spaces of functions in several complex variables&lt;/searchLink&gt;&lt;br /&gt;&lt;searchLink fieldCode=&quot;DE&quot; term=&quot;%22Toeplitz+operators%2C+Hankel+operators%2C+Wiener-Hopf+operators%22&quot;&gt;Toeplitz operators, Hankel operators, Wiener-Hopf operators&lt;/searchLink&gt;&lt;br /&gt;&lt;searchLink fieldCode=&quot;DE&quot; term=&quot;%22d-bar+equation%22&quot;&gt;d-bar equation&lt;/searchLink&gt;&lt;br /&gt;&lt;searchLink fieldCode=&quot;DE&quot; term=&quot;%22Bergman-Toeplitz+operators%22&quot;&gt;Bergman-Toeplitz operators&lt;/searchLink&gt;&lt;br /&gt;&lt;searchLink fieldCode=&quot;DE&quot; term=&quot;%22%5C%28%5Coverline%5Cpartial%5C%29+and+%5C%28%5Coverline%5Cpartial%5C%29-Neumann+operators%22&quot;&gt;\(\overline\partial\) and \(\overline\partial\)-Neumann operators&lt;/searchLink&gt;&lt;br /&gt;&lt;searchLink fieldCode=&quot;DE&quot; term=&quot;%22%5C%28%5Coverline%5Cpartial%5C%29-Neumann+operator%22&quot;&gt;\(\overline\partial\)-Neumann operator&lt;/searchLink&gt;&lt;br /&gt;&lt;searchLink fieldCode=&quot;DE&quot; term=&quot;%22canonical+solution+operator%22&quot;&gt;canonical solution operator&lt;/searchLink&gt;&lt;br /&gt;&lt;searchLink fieldCode=&quot;DE&quot; term=&quot;%22%5C%28%280%2Cq%29%5C%29-forms+with+holomorphic+coefficients%22&quot;&gt;\((0,q)\)-forms with holomorphic coefficients&lt;/searchLink&gt;
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  Data: The main result of this paper is an explicit formula for the canonical solution operator \(S_{q+1}\) to \(\overline{\partial} \) restricted to \((0,q)\)-forms with holomorphic coefficients. The restriction to forms with holomorphic coefficients is essential for the proof and it shows an interesting relationship of \(\overline{\partial} \) to the Bergman kernel and certain multiplication operators. In the second part the explicit formula for the solution operator to \(\overline{\partial} \) is applied to generalize a result of \textit{N. Salinas}, \textit{A. Sheu} and \textit{H. Upmeier} [Ann. Math. (2) 130, 531--565 (1989; Zbl 0708.47021)], concerning compactness of the solution operator, the \(\overline{\partial} \)- Neumann operator and certain Bergman-Toeplitz operators.
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        Value: 10.1002/1522-2616(200211)245:1<94::aid-mana94>3.0.co;2-h
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      – Text: Undetermined
    Subjects:
      – SubjectFull: Bergman spaces of functions in several complex variables
        Type: general
      – SubjectFull: Toeplitz operators, Hankel operators, Wiener-Hopf operators
        Type: general
      – SubjectFull: d-bar equation
        Type: general
      – SubjectFull: Bergman-Toeplitz operators
        Type: general
      – SubjectFull: \(\overline\partial\) and \(\overline\partial\)-Neumann operators
        Type: general
      – SubjectFull: \(\overline\partial\)-Neumann operator
        Type: general
      – SubjectFull: canonical solution operator
        Type: general
      – SubjectFull: \((0,q)\)-forms with holomorphic coefficients
        Type: general
    Titles:
      – TitleFull: Compactness of the canonical solution operator of \(\bar\partial\) on bounded pseudoconvex domains
        Type: main
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            NameFull: Knirsch, Wolfgang
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