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The closed range property for \(\overline\partial\) on domains with pseudoconcave boundary. (English) Zbl 1204.32028

Ebenfelt, Peter (ed.) et al., Complex analysis. Several complex variables and connections with PDE theory and geometry. Proceedings of the conference in honour of Linda Rothschild, Fribourg, Switzerland, July 7–11, 2008. Basel: Birkhäuser (ISBN 978-3-0346-0008-8/hbk). Trends in Mathematics, 307-320 (2010).
The present paper gives a nice review about the closed range property and regularity of the \(\overline{\partial}\)-operator in the \(L^2\)-sense on pseudoconcave domains. Moreover, it contains some new results about infinite-dimensionality of the space of harmonic \(L^2\)-forms in the critical case of forms of degree \((p,n-1)\) on pseudoconcave domains in \(\mathbb{C}^n\). The main tool of the paper are regularity properties of the \(\overline{\partial}\)-Neumann operator.
The essential content of the paper can be summarized as follows (see Theorem 3.5): Let \(\Omega \subset\subset \mathbb{C}^n\) be the annulus \(\Omega = \Omega_1 \setminus \overline{\Omega_2}\) between two pseudoconvex domains \(\Omega_1\) and \(\Omega_2\) with smooth boundary and \(\Omega_2\subset\subset \Omega_1\). Then the \(\overline{\partial}\)-Neumann operator \(N_{(p,q)}\) exists on \(L^2_{(p,q)}(\Omega)\) for \(0\leq p\leq n\) and \(1\leq q\leq n-1\). For any \(f\in L^2_{(p,q)}(\Omega)\), there is the representation \[ \begin{aligned} f &= \overline{\partial} \overline{\partial}^* N_{(p,q)} f + \overline{\partial}^* \overline{\partial} N_{(p,q)} f, \quad 1\leq q\leq n-2,\\ f &= \overline{\partial} \overline{\partial}^* N_{(p,n-1)} f + \overline{\partial}^* \overline{\partial} N_{(p,n-1)} f + H_{(p,n-1)} f, \quad q=n-1,\end{aligned} \] where \(H_{(p,n-1)}\) is the orthogonal projection onto the harmonic space \(\mathcal{H}_{(p,n-1)}(\Omega)\) which is infinite-dimensional.
In the last section of the paper, the author gives a short survey about known existence and regularity results for \(\overline{\partial}\) on pseudoconcave domains with Lipschitz boundary in \(\mathbb{CP}^n\) when \(n\geq 3\). The closed range property for \(\overline{\partial}\) for \((0,1)\)-forms with \(L^2\)-coefficients on pseudoconcave domains in \(\mathbb{CP}^n\) is still an open problem. This question is of particular importance in the case of \(\mathbb{CP}^2\) for the closed range property would imply non-existence of Levi-flat hypersurfaces also in this dimension.
For the entire collection see [Zbl 1188.32003].

MSC:

32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
35N15 \(\overline\partial\)-Neumann problems and formal complexes in context of PDEs
58J32 Boundary value problems on manifolds
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