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Diviseurs de Leibenson et problème de Gleason pour \(H^{\infty}(\Omega)\) dans le cas convexe. (Leibenson divisors and the Gleason problem for \(H^{\infty}(\Omega)\) for convex open sets). (French) Zbl 0603.32012

Let \(\Omega\) be an open bounded and convex subset of \({\mathbb{C}}^ n\) (n\(\geq 2)\) such that \(0\in \Omega\). For \(\phi \in H^{\infty}(\Omega)\) put \(\Psi_ j(z):=\int^{1}_{0}\frac{\partial \phi}{\partial z_ j}(tz)dt\), \(z\in \Omega\), \(j=1,...\), n, and let \(L_{\phi}:=(\Psi_ 1,...\), \(\Psi_ n)\). The author proves that if \(\partial \Omega\) is of class \(C^{1+\epsilon}\) then there exists a constant \(\sigma\) (\(\Omega)\) such that: \[ (*)\quad \| L_{\phi}\|_{\infty}\leq \sigma (\Omega) \| \phi \|_{\infty},\quad \phi \in H^{\infty}(\Omega),\quad \phi (0)=0. \] Moreover, he presents an example of an \(\Omega\) with the boundary of class \(C^ 1\) for which there is no constant \(\sigma\) (\(\Omega)\) satisfying (*). In the case where \(\Omega ={\mathbb{B}}_ n\) is the unit ball in \({\mathbb{C}}^ n\), the author shows that the constant \(\sigma ({\mathbb{B}}_ n):=(1+\pi^ 2/4)^{1/2}\) is the best possible.
Reviewer: M.Jarnicki

MSC:

32E35 Global boundary behavior of holomorphic functions of several complex variables
32A38 Algebras of holomorphic functions of several complex variables
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References:

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