Randregularität des \({\bar\partial}\)-Problems für die Halbkugel in \({\mathbb{C}}^ n\). (Boundary regularity of the \({\bar\partial}\)-problem for the half ball in \({\mathbb{C}}^ n)\)
The author proves \(C^ k\) regularity estimates for the \({\bar \partial}\)-operator on the domain \(\{z\in {\mathbb{C}}^ n: Re z_ n>0,\quad | z| <1\}.\) As the author points out, such a domain is of interest because it is a mixture of the strongly pseudoconvex (ball) case and the Levi flat (polydisc) case, both of which are well understood. The integral kernels which arise involve interesting mixtures of singularities. Part of the work is to construct new integral formulas which give good regularity for \(C^ k\), \(k>1.\) It seems likely that the same proof will give results for a strongly convex domain intersected with a half space. With suitable modifications, the result ought to hold for the intersection of a general strongly pseudoconvex domain with a half space.
regularity estimates for the \({\bar \partial }\)-operator
scientific article; zbMATH DE number 3950980