Abstract
Let \(\Omega \) be a \(C^2\)-smooth bounded pseudoconvex domain in \(\mathbb {C}^n\) for \(n\ge 2\) and let \(\varphi \) be a holomorphic function on \(\Omega \) that is \(C^2\)-smooth on the closure of \(\Omega \). We prove that if \(H_{\overline{\varphi }}\) is in Schatten p-class for \(p\le 2n\) then \(\varphi \) is a constant function. As a corollary, we show that the \(\overline{\partial }\)-Neumann operator on \(\Omega \) is not Hilbert–Schmidt.
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Acknowledgements
Part of this work was done while the second author was visiting Sabancı University. He thanks this institution for its hospitality and good working conditions. He also thanks Trieu Le for fruitful discussions. We are thankful to the anonymous referee for constructive comments that improved the presentation of the paper.
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Communicated by A. Kilian.
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Göğüş, N.G., Şahutoğlu, S. Schatten class Hankel and \(\overline{\partial }\)-Neumann operators on pseudoconvex domains in \(\mathbb {C}^{n}\) . Monatsh Math 187, 237–245 (2018). https://doi.org/10.1007/s00605-017-1099-x
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DOI: https://doi.org/10.1007/s00605-017-1099-x
Keywords
- Hankel operators
- \(\overline{\partial }\)-Neumann problem
- Hilbert–Schmidt
- Schatten p-class
- Pseudoconvex domains