Arkiv för Matematik

On the holomorphic extension of CR functions from non-generic CR submanifolds of ℂn

Nicolas Eisen

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Abstract

We give a holomorphic extension result for continuous CR functions on a non-generic CR submanifold N of ℂn to complex transversal wedges with edges containing N. We show that given any v∈ℂn∖(TpN+iTpN), there exists a wedge of direction v whose edge contains a neighborhood of p in N, such that any continuous CR function defined locally near p extends holomorphically to that wedge.

Article information

Source
Ark. Mat., Volume 50, Number 1 (2012), 69-87.

Dates
Received: 9 September 2009
Revised: 15 October 2010
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485907159

Digital Object Identifier
doi:10.1007/s11512-010-0140-2

Mathematical Reviews number (MathSciNet)
MR2890345

Zentralblatt MATH identifier
1272.32038

Rights
2011 © Institut Mittag-Leffler

Citation

Eisen, Nicolas. On the holomorphic extension of CR functions from non-generic CR submanifolds of ℂ n. Ark. Mat. 50 (2012), no. 1, 69--87. doi:10.1007/s11512-010-0140-2. https://projecteuclid.org/euclid.afm/1485907159


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References

  • Aĭrapetyan, R. A., Extension of CR functions from piecewise-smooth CR manifolds, Mat. Sb. 134(176) (1987), 108–118, 143 (Russian). English transl.: Math. USSR-Sb. 62 (1989), 111–120.
  • Baouendi, S., Ebenfelt, P. and Rothschild, L., Real Submanifolds in Complex Space and Their Mappings, Princeton University Press, Princeton, NJ, 1998.
  • Baouendi, S. and Rothschild, L., Cauchy–Riemann functions on manifolds of higher codimension in complex space, Invent. Math. 101 (1990), 45–56.
  • Baouendi, S. and Trèves, F., A property of the functions and distributions annihilated by a locally integrable system of complex vector fields, Ann. of Math. 113 (1981), 387–421.
  • Boggess, A., CR Manifolds and the Tangential Cauchy–Riemann Complex, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1991.
  • Boggess, A. and Polking, J. C., Holomorphic extension of CR functions, Duke Math. J. 49 (1982), 757–784.
  • Chazarain, J. and Piriou, A., Introduction à la théorie des équations aux dérivées partielles linéaires, Gauthier-Villars, Paris, 1981.
  • Eisen, N., Holomorphic extension of decomposable distributions from a CR submanifold of ℂL, Michigan Math. J. 54 (2006), 499–577.
  • Eisen, N., On the holomorphic extension of CR functions from non-generic CR submanifolds of ℂn, the positive defect case, to appear in Michigan. Math. J.
  • Hörmander, L., Linear Partial Differential Operators, 3rd printing, Springer, Berlin, 1969.
  • Jacobowitz, H., An Introduction to CR Structures, Mathematical Surveys and Monographs 32, Amer. Math. Soc., Providence, RI, 1990.
  • Lewy, H., 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. 64 (1956), 514–522.
  • Nagel, A., Smooth zero sets and interpolation sets for some algebras of holomorphic functions on strictly pseudoconvex domains, Duke Math. J. 43 (1976), 323–348.
  • Range, R. M., Holomorphic Functions and Integral Representations in Several Complex Variables, Graduate Texts in Mathematics 108, Springer, Berlin, 1986.
  • Rudin, W., Peak-interpolation sets of class $\mathcal{C}^{1}$, Pacific J. Math. 75 (1978), 267–279.
  • Sussmann, J., Orbits of families of vector fields and integrability of distributions, Trans. Amer. Math. Soc. 180 (1973), 171–188.
  • Trépreau, J. M., Sur le prolongement holomorphe des fonctions C-R défines sur une hypersurface réelle de classe ${\mathcal{C}}^{2}$ dans ℂn, Invent. Math. 83 (1986), 583–592.
  • Trépreau, J. M., Sur la propagation des singularités dans les variétés CR, Bull. Soc. Math. France 118 (1990), 403–450.
  • Trépreau, J. M., Holomorphic extension of CR functions: a survey, in Partial Differential Equations and Mathematical Physics (Copenhagen, 1995; Lund, 1995), Progr. Nonlinear Differential Equations Appl. 21, pp. 333–355, Birkhäuser, Boston, MA, 1996.
  • Trèves, F., Hypo-Analytic Structures: Local Theory, Princeton University Press, Princeton, NJ, 1992.
  • Tumanov, A. E., Extension of CR functions into a wedge from a manifold of finite type, Math. Sb. 136(178) (1988), 128–139 (Russian). English transl.: Math. USSR-Sb. 64 (1989), 129–140.
  • Tumanov, A. E., Extending CR functions from manifolds with boundaries, Math. Res. Lett. 2 (1995), 629–642.