On \(L^ p\) and Hölder estimates for the \({\bar \partial}\)-Neumann problem on strongly pseudo-convex domains
Let \(\Omega\) be an open, relatively compact subset in \({\mathbb{C}}^{n+1}\), and assume that the boundary of \(\Omega\), \(\partial \Omega\), is smooth and strongly pseudoconvex. Suppose \(\Omega\) has an arbitrary Hermitian metric. Consider the \({\bar \partial}\)-Neumann problem: Given a (0,1)- form f, find a (0,1)-form u such that \(\square u=({\bar \partial}{\bar \partial}^ *+{\bar \partial}^ *{\bar \partial})u=f\) with two boundary conditions: \(u \in domain ({\bar \partial}^ *),\) \({\bar \partial}u \in domain ({\bar \partial}^ *).\) In this paper, we can write down a parametrix \({\mathfrak N}\) for this problem as a product of some well-known operators plus a remainder type operator and then get the optimal estimates.
scientific article; zbMATH DE number 4025828
scientific article