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Ideals of holomorphic functions with \(C^{\infty}\) boundary values on a pseudoconvex domain. (English) Zbl 0631.32015
The authors give some natural sufficient conditions for the solution of several division problems in the space \({\mathcal A}^{\infty}(\Omega)\) of holomorphic functions with \({\mathcal C}^{\infty}\) boundary value on a pseudo-convex domain \(\Omega\) in \({\mathbb{C}}^ n\) with smooth boundary. We formulate here one of the results.
Every matrix A with entries in the \({\mathcal O}({\bar \Omega})\)-algebra of germs of holomorphic functions in a neighborhood of \({\bar \Omega}\) defines a continuous homomorphism \(A: {\mathcal A}^{\infty}(\Omega)^ q\to {\mathcal A}(\Omega)^ p\). The question is under what condition is \(A\cdot {\mathcal A}^{\infty}(\Omega)^ q\) a closed submodule of \({\mathcal A}^{\infty}(\Omega)^ p?\)
Each matrix in a natural way defines a filtration of some neighborhood of \({\bar \Omega}\subset U=\Sigma_ 0\supset \Sigma_ 1\supset..\). by a locally finite sequence of analytic sets. Define \((A\cdot {\mathcal A}^{\infty}(\Omega)^ q)^{\wedge}= \{f\in {\mathcal A}^{\infty}(\Omega)^ p:\hat f{}_ a\in \hat A_ a\cdot {\mathcal O}^ q_ a\) for all \(a\in {\bar \Omega}\}\). Here the symbol \(^{\wedge}\) denotes a completion in the Krull topology. Clearly \((A\cdot {\mathcal A}^{\infty}(\Omega)^ q)^{\wedge}\) is a closed submodule of \({\mathcal A}^{\infty}(\Omega)^ q.\)
Theorem 1.2. Suppose that \(\Sigma_ k\) and \({\bar \Omega}\) are regularly situated for all \(k=0,1,...\), then \(A\cdot {\mathcal A}^{\infty}(\Omega)^ q=(A\cdot {\mathcal A}^{\infty}(\Omega))^{\wedge}\); in particular, \(A\cdot {\mathcal A}^{\infty}(\Omega)^ q\) is a closed submodule of \({\mathcal A}^{\infty}(\Omega)^ p\). Recall that two closed subsets X and Y of U are regularly situated if every point of U admits of neighborhood V and constants \(e,r>0\) such that \(d(x,X)+d(x,Y)\geq cd(x,X\cap Y)^ r\) for all \(x\in V\).
Reviewer: S.M.Ivashkovich

32A38 Algebras of holomorphic functions of several complex variables
32B05 Analytic algebras and generalizations, preparation theorems
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