\(L^ 2\) estimates and existence theorems for the tangential Cauchy-Riemann complex
The author proves that if \(\alpha\) is a \({\bar \partial}\)-closed (p,q) form on the boundary \(\partial \Omega\) of a smoothly bounded pseudoconvex domain \(\Omega \subseteq {\mathbb{C}}^ n\), \(0<q<n-2\), then there is a (p,q- 1) form u on \(\partial \Omega\) such that \({\bar \partial}_ bu=\alpha\) and u satisfies Sobolev estimates. The starting point for this work is an observation of Rosay that this result with \(C^{\infty}\) estimates follows from earlier work of Kohn and Kohn/Rossi. The author uses these ideas, together with a two-sided extension theorem for forms \(\alpha\), to obtain the more precise Sobolev estimates. These results and techniques should prove useful in later work.
scientific article; zbMATH DE number 3929567
scientific article