Global analytic hypoellipticity of the \({\bar \partial}\)-Neumann problem on circular domains
We show that if \(D\subset {\mathbb{C}}^ n\), \(n\geq 2\), is a smooth bounded pseudoconvex circular domain with real analytic defining function r(z) such that \(\sum ^{n}_{k=1}z_ k\partial r/\partial z_ k\neq 0\) for all z near the boundary, then the solution u to the \({\bar \partial}\)- Neumann problem, \[ \square u=(\overline{\partial \partial}^ *+{\bar \partial}^ *{\bar \partial})u=f, \] is real analytic up to the boundary, if the given form f is real analytic up to the boundary. In particular, if \(D\subset {\mathbb{C}}^ n\), \(n\geq 2\), is a smooth bounded complete Reinhardt pseudoconvex domain with real analytic boundary. Then \(\square\) is analytic hypoelliptic.
scientific article; zbMATH DE number 4006838