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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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