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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

MSC:
 35N15 $$\overline\partial$$-Neumann problems and formal complexes in context of PDEs 32T99 Pseudoconvex domains
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References:
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