×

Local holomorphic extendability and non-extendability of CR-functions on smooth boundaries. (English) Zbl 0587.32035

The authors are interested in local extendability of CR-functions on the smooth boundary of an open set \(\Omega\) in \({\mathbb{C}}^ n\). For their purpose they may assume the complex line \(C=\{(z_ 1,0,0,...,0)\}\) is tangent to \(\partial \Omega\) at 0 to order exactly k. They say 0 satisfies the rays condition if there are real rays \(\ell_ 1,...,\ell_{\nu}\) from 0 in C, all in \({\bar \Omega}\) with at least one in \(\Omega\) and their angles satisfying \(\ell_ j\ell^{\wedge}_{j+1}<\pi /k\) while \(\ell_ 1\ell^{\wedge}_{\nu}>\pi /k.\) If 0 satisfies the rays condition, the authors prove that CR-functions on \(\partial \Omega\) are holomorphically extendable (near 0) to the complement of \({\bar \Omega}\). In case \(n=2\), they also provide a sufficient condition for 0 to be a local peak point for \(\Omega\). The authors provide a number of examples and also discuss the exceptional cases to which their results do not apply.
Reviewer: G.Harris

MSC:

32V40 Real submanifolds in complex manifolds
32D10 Envelopes of holomorphy
PDFBibTeX XMLCite
Full Text: Numdam EuDML

References:

[1] M.S. Baouendi - F. Treves , About the holomorphic extension fo CR-functions on real hypersurfaces in complex space , Duke Math. J. , 51 ( 1984 ), pp. 77 - 107 . Article | MR 744289 | Zbl 0564.32011 · Zbl 0564.32011 · doi:10.1215/S0012-7094-84-05105-6
[2] E. Bedford , Local and global envelopes of holomorphy of domains in C2 . (Preprint). · Zbl 0599.32010
[3] E. Bedford - J.E. Fornæss , Local extension of CR-functions from weakly pseudoconvex boundaries , Michigan Math. J. , 25 ( 1978 ) pp. 259 - 262 . Article | MR 512898 | Zbl 0401.32007 · Zbl 0401.32007 · doi:10.1307/mmj/1029002109
[4] E. Bedford and J.E. Fornæss , A construction of peak functions on weakly pseudoconvex domains , Ann. of Math. , 107 ( 1978 ), pp. 555 - 568 . MR 492400 | Zbl 0392.32004 · Zbl 0392.32004 · doi:10.2307/1971128
[5] C.O. Kiselman , On entire functions of exponential type and indicators of analytic functionals , Acta Math. 117 , ( 1967 ), pp. 1 - 35 . MR 210940 | Zbl 0152.07602 · Zbl 0152.07602 · doi:10.1007/BF02395038
[6] J.J. Kohn - L. Nirenberg , A pseudoconvex demain not admitting a holomorphic support function , Math. Ann. 201 ( 1973 ), pp. 265 - 268 . MR 330513 | Zbl 0248.32013 · Zbl 0248.32013 · doi:10.1007/BF01428194
[7] E.E. Levi , Sulle ipersurperfici dello spazio a 4 dimensioni che possono essere frontiera del campo di esistenza di una funzione analitica di due variabili complesse , Ann. Mat. Pura Appl. , 18 , s. III ( 1911 ), pp. 69 - 79 . JFM 42.0449.02 · JFM 42.0449.02
[8] H. Lewy , On the local character of the solutions of an atypical linear differential equation in three variables and a related theorem for regular functions of two complex variables , Ann. of Math. , 74 ( 1956 ), pp. 514 - 522 . MR 81952 | Zbl 0074.06204 · Zbl 0074.06204 · doi:10.2307/1969599
[9] C. Rea , Extension holomorphe bilatérale des fonctions de Cauchy-Riemann données sur une hypersurface différentiable de C2 , C. R. Acad. Sc. Paris , Sér. A , 294 , ( 1982 ), pp. 577 - 579 . MR 663083 | Zbl 0496.32013 · Zbl 0496.32013
[10] C. Rea , The Cauchy problem for the \partial operator , Boll. UMI ( 6 ), 1-A ( 1982 ), pp. 443 - 449 . Zbl 0525.32018 · Zbl 0525.32018
[11] C. Rea , Prolongement holomorphe des fonctions CR, conditions suffisantes , C. R. Acad. Sc. Paris , 297 ( 1983 ), pp. 163 - 166 . MR 725396 | Zbl 0568.32011 · Zbl 0568.32011
[12] L.I. Ronkin , Introduction to the theory of entire functions of several variables , Tr. of Math. Monographs , 44 , A.M.S . ( 1974 ). MR 346175 | Zbl 0286.32004 · Zbl 0286.32004
[13] B. Stensønes - Henriksen , Ph. D. Thesis , Princeton University .
[14] A. Bogges - J. Pitts , CR extension near a point of higher type , Duke Math. J. , 52 ( 1985 ), pp. 67 - 102 . Article | MR 791293 | Zbl 0573.32019 · Zbl 0573.32019 · doi:10.1215/S0012-7094-85-05206-8
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.