# Article

 Title: The $\bar\partial$-Neumann operator and commutators of the Bergman projection and multiplication operators (English) Author: Haslinger, Friedrich Language: English Journal: Czechoslovak Mathematical Journal ISSN: 0011-4642 (print) ISSN: 1572-9141 (online) Volume: 58 Issue: 4 Year: 2008 Pages: 1247-1256 Summary lang: English . Category: math . Summary: We prove that compactness of the canonical solution operator to $\bar \partial$ restricted to $(0,1)$-forms with holomorphic coefficients is equivalent to compactness of the commutator $[\mathcal P,\bar M]$ defined on the whole $L^2_{(0,1)}(\Omega ),$ where $\bar M$ is the multiplication by $\bar z$ and $\mathcal P$ is the orthogonal projection of $L^2_{(0,1)}(\Omega )$ to the subspace of $(0,1)$ forms with holomorphic coefficients. Further we derive a formula for the $\bar \partial$-Neumann operator restricted to $(0,1)$ forms with holomorphic coefficients expressed by commutators of the Bergman projection and the multiplications operators by $z$ and $\bar z$. (English) Keyword: $\bar{\partial}$-equation Keyword: $\bar{\partial}$-Neumann operator Keyword: compactness MSC: 32A36 MSC: 32W05 idZBL: Zbl 1174.32015 idMR: MR2471181 . Date available: 2010-07-21T08:18:02Z Last updated: 2016-04-07 Stable URL: http://hdl.handle.net/10338.dmlcz/140455 . Reference: [1] Boas, H. P., Straube, E. J.: Global regularity of the $\bar{\partial}$-Neumann problem: a survey of the $L^2$-Sobolev theory.Several Complex Variables (M. Schneider and Y.-T. Siu, eds.) MSRI Publications, vol. 37, Cambridge University Press (1999), 79-111. MR 1748601 Reference: [2] Catlin, D.: Global regularity of the $\bar{\partial}$-Neumann problem.Proc. Sympos. Pure Math. 41 39-49; A.M.S. Providence, Rhode Island, 1984. 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