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The Gleason property for Reinhardt domains. (English) Zbl 0869.32001
We prove that bounded pseudoconvex Reinhardt domains in \(\mathbb{C}^2\) with \(C^2\)-boundary have the Gleason \(A\)-property. The main difficulties occur when the base point \(p\) is close to the boundary, but at the same time far away, in a sense, from the set of strictly pseudoconvex points.
As a first auxiliary result we solve a \(\overline\partial\)-equation in pseudoconvex Reinhardt domains in \(\mathbb{C}^2\) with \(C^2\)-boundary. The crucial step in the proof of the main theorem is a lemma which says that one can always find analytic polynomials with certain properties that enable us to construct a finite open covering of the closure of the domain. The covering is constructed so that the pairwise intersections of its open sets intersect the boundary of the domain only in strictly pseudoconvex points and is in turn used to formulate an additive Cousin problem. We solve this problem using the above-mentioned \(\overline \partial\)-result.
Reviewer: U.Backlund (Umeå)

MSC:
32A07 Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube) (MSC2010)
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
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