Percolation of estimates for \(\bar{\partial}\) by the method of alternating projections
On a bounded pseudoconvex domain in \(\mathbb{C}^{n}\), compactness of the \(\overline{\partial}\)-Neumann operator percolates up the Cauchy-Riemann complex. That is, if the property holds at form level \(q\), then it holds at form level \((q+1)\). In this paper, the authors generalize this fact to domains that are not assumed pseudoconvex nor bounded (but with Lipschitz boundary), but are only assumed to have the property that \(\overline{\partial}\) has closed range at the respective form levels. Their approach to these questions is new and very interesting. Rather than relating \textit{estimates} at different form levels, they study the relation(s) between various relevant \textit{operators} at (and between) these different levels. After studying in detail how to relate forms at consecutive levels, and analyzing relations between various naturally occuring operators, they establish the following key fact: the closed range property is equivalent to a geometric rate of convergence in von Neumann's alternating projection algorithm, applied to two basic projection operators. This then fact implies percolation of compactness. The point of view adopted in this paper is original and should prove useful in the future.
scientific article; zbMATH DE number 7388713
scientific article