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A remark on semiglobal existence for \(\overline{\partial}\). (English) Zbl 0896.32009

Let \(\Omega\) be a domain in \(\mathbb{C}^n\) with \(C^2\)-boundary. Then it is known that \(\Omega\) is pseudoconvex if (and only if) \(\Omega\) can be exhausted by a sequence of relatively compact pseudoconvex subdomains \(\Omega_i \Subset \Omega\). Hörmander gave a proof of this result with the so called \(\overline \partial\) technique (reference [6] in this paper).
Here, the author provides a simpler proof of that result, by using the Hörmander technique. However, I have some trouble to follow his argument. Furthermore, the author lists 9 papers in the references; in fact actually, in the paper itself, only two were referred to.

MSC:

32T99 Pseudoconvex domains
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References:

[1] A. Andreotti - H. GRAUERT, Théorèmes de finitude pour la cohomologie des éspaces complexes , Bull. Soc. Math. France , 90 ( 1962 ), pp. 193 - 259 . Numdam | MR 150342 | Zbl 0106.05501 · Zbl 0106.05501
[2] A. Andreotti - C. D. HILL, E. E. Levi convexity and the Hans Lewy problem, Part II: Vanishing theorems , Ann. Sc. Norm. Sup. Pisa ( 1972 ), pp. 747 - 806 . Numdam | MR 477150 | Zbl 0283.32013 · Zbl 0283.32013
[3] H. Grauert , Kantenkohomologie , Compositio Math. , 44 ( 1981 ), pp. 79 - 101 . Numdam | MR 662457 | Zbl 0512.32011 · Zbl 0512.32011
[4] G.M. Henkin , H. Lewy’s equation and analysis on pseudoconvex manifolds (Russian) , I. Uspehi Mat. Nauk. , 32 ( 3 ) ( 1977 ), pp. 57 - 118 . MR 454067 | Zbl 0382.35038 · Zbl 0382.35038 · doi:10.1070/RM1977v032n03ABEH001628
[5] G.M. Henkin - J. Leiterer , Andreotti-Grauert theory by integral formulas , Birkhauser Progress in Math. , 74 ( 1988 ). MR 986248 | Zbl 0654.32002 · Zbl 0654.32002
[6] L. Hörmander , An Introduction to Complex Analysis in Several Complex Variables , Van Nostrand , Princeton N.J. ( 1966 ). MR 203075 | Zbl 0138.06203 · Zbl 0138.06203
[7] L. Hörmander , L2 estimates and existence theorems for the \partial operator , Acta Math. , 113 ( 1965 ), pp. 89 - 152 . Zbl 0158.11002 · Zbl 0158.11002 · doi:10.1007/BF02391775
[8] H. Komatsu , A local version of Bochner’s tube theorem , J. Fac. Sci. Univ. Tokyo , Sect. 1A, 19 ( 1972 ). MR 316749 | Zbl 0239.32012 · Zbl 0239.32012
[9] G. Zampieri , The Andreotti-Grauert vanishing theorem for dihedrons of Cn , J. Fac. Sci. Univ. Tokyo , to appear. Zbl 0859.32009 · Zbl 0859.32009
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