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The envelope of holomorphy of a truncated tube. (English) Zbl 0727.32006

Let \(I=(-1,1)\subset {\mathbb{R}}\) and let \(h\in {\mathcal C}^{\infty}(I)\) be such that \(h(0)=0\), \(h'(0)=0\), and h attains both positive and negative values in any neighbourhood of 0. Let \(\Omega_ h\subset {\mathbb{C}}^ 2\) be the truncated tube domain which consists of all (z,w) such that Re \(z\in I\), Re \(w\in I\), Im \(z\in I\), Im \(w\in I\), and Re w\(>h(Re z)\). The author shows that the function h can be chosen so that the envelope of holomorphy of \(\Omega_ h\) does not contain a neighbourhood of 0.
Reviewer: M.Klimek (Dublin)

MSC:

32D10 Envelopes of holomorphy
32A07 Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube) (MSC2010)
32D15 Continuation of analytic objects in several complex variables
31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
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References:

[1] M. S. Baouendi and F. Trèves, A microlocal version of Bochner’s tube theorem, Indiana Univ. Math. J. 31 (1982), no. 6, 885 – 895. · Zbl 0505.32013 · doi:10.1512/iumj.1982.31.31060
[2] J.-M. Trépreau, Sur le prolongement holomorphe des fonctions C-R défines sur une hypersurface réelle de classe \?² dans \?\(^{n}\), Invent. Math. 83 (1986), no. 3, 583 – 592 (French). · Zbl 0586.32016 · doi:10.1007/BF01394424
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