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


32E35 Global boundary behavior of holomorphic functions of several complex variables
32A38 Algebras of holomorphic functions of several complex variables
Full Text: DOI Numdam EuDML


[1] AHERN (P.) and SCHNEIDER (R.) , Holomorphic Lipschitz functions in pseudo-convex domains , Amer. J. Math., vol. 101, n^\circ 3, 1979 , p. 543-565. MR 81f:32022 | Zbl 0455.32008 · Zbl 0455.32008
[2] HENKIN (G. M.) , Approximation of functions in pseudo-convex domains and the theorem of Z. L. Leibenson , Bull. Acad. Polon. Sc., Sér. Math. Astronom. Phys., vol. 19, 1971 , p. 37-42. MR 44 #4234 | Zbl 0214.33701 · Zbl 0214.33701
[3] KERZMAN (N.) and NAGEL (A.) , Finitely generated ideals in certain function algebras . J. Funct. Anal., vol., 7, 1971 , p. 212-215. MR 43 #929 | Zbl 0211.43902 · Zbl 0211.43902
[4] OVRELID (N.) , Generators of the maximal ideals of A (D) , Pacific J. Math., vol. 39, n^\circ 1, 1971 , p. 219-223. Article | MR 46 #9393 | Zbl 0231.46090 · Zbl 0231.46090
[5] RUDIN (W.) , Function theory in the unit ball of \Bbb C , Springer-Verlag, New York, Heidelberg, Berlin.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.