Global regularity of \(\overline\partial\) on an annulus between a \(Q\)-pseudoconvex and a \(P\)-pseudoconcave boundary
The authors consider the \(\overline{\partial}\)-Neumann problem on an annulus \(\Omega\subset\subset\mathbb{C}^n\) between an internal \(p\)-pseudoconcave and an external \(q\)-pseudoconvex hypersurface. They prove that the \(\overline{\partial}\)-cohomology of holomorphic \((0,k)\)-forms of degree \(k\) for \(q+1\leq k\leq p-1\) is finite-dimensional by showing that each cohomology class has a harmonic representative. In case of strong \(p\)-pseudoconcave and strong \(q\)-pseudoconvex boundaries, they obtain stronger regularity results implying hypoellipticity at the boundary for \((\overline{\partial},\overline{\partial}^*)\). The basic tool of the paper is Kohn's \(\overline{\partial}\)-Neumann method. The main results have been obtained previously by M.-C. Shaw in case of an annulus between pseudoconvex domains. The regularity results at strongly \(p\)-pseudoconcave and strongly \(q\)-pseudoconvex boundaries are classical and due to Kohn, only the presentation in this paper is a bit different and set in the context of the other results. The question under which circumstances the cohomology groups actually vanish (no harmonic forms) is not discussed.
\(\overline{\partial}\)-Neumann operator
\(q\)-pseudoconvex/\(q\)-pseudoconcave domains
scientific article; zbMATH DE number 5831507