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\(L^ 2\) estimates and existence theorems for the tangential Cauchy-Riemann complex. (English) Zbl 0581.35057
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.
Reviewer: S.G.Krantz

35N15 \(\overline\partial\)-Neumann problems and formal complexes in context of PDEs
32T99 Pseudoconvex domains
Full Text: DOI EuDML
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