De Rham cohomology of manifolds containing the points of infinite type, and Sobolev estimates for the \(\bar \partial\)-Neumann problem
Main result of the paper is the following theorem: Let \(\Omega\) be a smooth pseudoconvex domain in \(\mathbb{C}^ n\). Suppose there is a smooth real submanifold \(M\) (with or without boundary of \(b\Omega\) that contains all the points of infinite type of \(b \Omega\) and whose real tangent space at each point is contained in the null space of the Levi form at the point (under the usual identification of \(\mathbb{R}^ n\) with \(\mathbb{C}^ n)\). If the \(H^ 1(M)\) de Rham cohomology class \([\alpha |_ M]\) is zero, then the \(\overline \partial\)-Neumann operators \(N_ q\) and the Bergman projections \(P_ q\) are continuous on the Sobolev space \(W^ s_{(0,q)} (\Omega)\) when \(0 \leq q \leq n\) and \(s \geq 0\). Here \(N_ q\) denotes the inverse of the complex Laplacian \(\overline \partial^*\overline \partial+\overline {\partial \partial}^*\) on \((0,q)\)-forms, and \(P_ q\) is the Bergman orthogonal projection from the space of square-integrable \((0,q)\)-forms onto the subspace of \(\overline \partial\)-closed \((0,q)\)-forms.
scientific article; zbMATH DE number 425631
scientific article