Skip to main content
SpringerLink
Log in

Almost holomorphic extensions of ultradifferentiable functions

  • Published:
Journal d’Analyse Mathématique Aims and scope

Abstract

For a smooth functionf on ℝn, we construct an extensionF to ℂn with\(\bar \partial F\) vanishing to a high order on ℝn and give precise estimates of how the degree of smothness is reflected in the degree of vanishing. This analysis is used to define the\(\bar \partial \) operator on (n,n−1) forms with singularities on ℝn.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. Andersson and B. Berndtsson,Non-holomorphic functional calculus, Preprint, Göteborg, 2001.

  2. B. Berndtsson, Weighted estimates for\(\bar \partial \) in domains in C, Duke Math. J.66 (1992), 239–255.

    Article  MathSciNet  MATH  Google Scholar 

  3. A. Beurling,On quasianalyticity and general distributions, Lecture notes, Stanford, 1961.

  4. A. Beurling,Analytic continuation across a linear boundary, Acta Math.128 (1972) 153–182.

    Article  MathSciNet  MATH  Google Scholar 

  5. M. Dimassi and J. Sjöstrand,Spectral Asymptotics in the Semi-classical Limit, London Mathematical Society Lecture Notes Series, 268, Cambridge University Press, Cambridge, 1999.

    MATH  Google Scholar 

  6. B. Droste,Holomorphic approximation of ultradifferentiable functions, Math. Ann.257 (1981), 293–316.

    Article  MathSciNet  MATH  Google Scholar 

  7. E. M. Dynkin,An operator calculus based on the Cauchy-Green formula (Russian), inInvestigations on Linear Operators and the Theory of Functions, III, Zap. Nauv cn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI)30 (1972), 33–39.

  8. E. M. Dynkin, Functions with a prescribed bound for\(\partial f/\partial \bar z\), and a theorem of N. Levinson (Russian), Mat. Sb. (N. S.)89 (131) (1972), 182–190.

    MathSciNet  Google Scholar 

  9. E. M. Dynkin,The pseudoanalytic extension, J. Analyse Math.60 (1993), 45–70,

    MathSciNet  Google Scholar 

  10. L. Hörmander,On the singularities of partial differential equations, in1970 Proceedings of Functional Analysis and Related Topics (Tokyo, 1969), University of Tokyo Press, Tokyo, 1970.

    Google Scholar 

  11. L. Hörmander,The Analysis of Linear Partial Differential Operators I, 2nd edn., Springer-Verlag, Berlin, 1990.

    MATH  Google Scholar 

  12. J. Mather,On Nirenberg's proof of Malgrange's preparation theorem, Lecture Notes in Mathematics192, Springer-Verlag, Berlin, 1971.

    Google Scholar 

  13. L. Nirenberg,A proof of the Malgrange preparation theorem, Lecture Notes in Mathematics192, Springer-Verlag, Berlin, 1971.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Chalmers University of Technology and the University of Göteborg, S-412 96, Göteborg, Sweden

    Mats Andersson & Bo Berndtsson

Authors
  1. Mats Andersson
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bo Berndtsson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mats Andersson.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Andersson, M., Berndtsson, B. Almost holomorphic extensions of ultradifferentiable functions. J. Anal. Math. 89, 337–365 (2003). https://doi.org/10.1007/BF02893087

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02893087

Keywords

Search

Navigation