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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