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Integral representations and estimates in the theory of the \({\bar\partial}\)-Neumann problem. (English) Zbl 0589.32034
The main purpose of this paper is to give an essentially explicit construction of the operator \({\bar \partial}^*N\) on a smoothly bounded strongly pseudoconvex domain in \({\mathbb{C}}^ n\) (or more generally in a complex manifold). Here N is the \({\bar \partial}\)-Neumann operator and \({\bar \partial}^*\) is the adjoint of the \({\bar \partial}\)-operator calculated in a suitable Levi metric. The construction of a semi-explicit kernel for \(\partial^*N\) allows one to calculate optimal estimates, in a variety of norms, for the canonical solution to the \({\bar \partial}\)- problem. As we become more familiar with the construction in this paper, it is bounded to become a standard tool in complex analysis of several variables.
The paper is very carefully written and includes a useful history of the subject. The authors are careful to describe other methods of constructing N and \({\bar \partial}^*N\). The authors claim that their methods should also give a means of constructing the Neumann operator N; it will be a significant achievement if they can do this as well.
Reviewer: S.Krantz

MSC:
32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
35N15 \(\overline\partial\)-Neumann problems and formal complexes in context of PDEs
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