Abstract
Let \({\Omega=\Omega_{1}\times\cdots\times\Omega_{n}\subset\mathbb{C}^{n}}\) , where \({\Omega_{j}\subset\mathbb{C}}\) is a bounded domain with smooth boundary. We study the solution operator to the \({\overline\partial}\) -Neumann problem for (0,1)-forms on Ω. In particular, we construct singular functions which describe the singular behavior of the solution. As a corollary our results carry over to the \({\overline\partial}\) -Neumann problem for (0,q)-forms. Despite the singularities, we show that the canonical solution to the \({\overline\partial}\) -equation, obtained from the Neumann operator, does not exhibit singularities when given smooth data.
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Partially supported by the Max-Planck-Institute for Mathematics in Bonn.
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Ehsani, D. The \({\bar{\partial}}\) -Neumann problem on product domains in \({\mathbb{C}^{n}}\) . Math. Ann. 337, 797–816 (2007). https://doi.org/10.1007/s00208-006-0056-0
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DOI: https://doi.org/10.1007/s00208-006-0056-0