\(L^ p\) and Lipschitz estimates for the \({\bar\partial}\)-equation and the \({\bar\partial}\)-Neumann problem
Optimal regularity results are proved for the canonical solution of the \({\bar \partial}\)-Neumann problem and the \({\bar \partial}\)-equation in the standard \(L^ p\) and Lipschitz spaces under the minimal assumption on the Levi form, condition Z(q). Pseudoconvexity of the domain is not required. These generalize the \(L^ 2\)-regularity results of Hörmander in case \(Z(q)\), the \(L^ p\) and Lipschitz results of the second author and Stein and of Chang for \((0,1)\)-forms on strictly pseudoconvex domains, and the Lipschitz results of Lieb and Range for \((p,q)\)-forms on strictly pseudoconvex domains. For example if u is a \((p,q)\) form orthogonal to the kernel of \({\bar \partial}_ q\) and \({\bar \partial}_ qu\) is in \(L^{\infty}\) (resp. \(L^ r\), \(1<r<\infty)\), then u is in \(\Lambda_{1/2}\) (resp. \(L^ r_{1/2})\). The proofsuse the method of the second author and Stein together with the full asymptotic pseudodifferential calculus which was developed by the authors in studying the boundary operator \(\square_ b\) and the heat equations for \(\square_ b\) and for the \({\bar \partial}\)- Neumann problem.
scientific article; zbMATH DE number 3964528
scientific article