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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

##### MSC:
 32A38 Algebras of holomorphic functions of several complex variables 32B05 Analytic algebras and generalizations, preparation theorems
##### Keywords:
ideals of holomorphic functions; division problems
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