Skip to main content
SpringerLink
Log in

A lower bound for systems with double characteristics

  • Published:
Journal d’Analyse Mathématique Aims and scope

Abstract

We give necessary and sufficient conditions for a lower bound with a gain of 3/2 derivatives (the so-called Weak-Hörmander inequality) for a class of systems with double characteristics.

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. L. Boutet de Monvel,Hypoelliptic operators with double characteristics and related pseudo-differential operators, Comm. Pure Appl. Math.27 (1974), 585–639.

    Article  MATH  MathSciNet  Google Scholar 

  2. L. Boutet de Monvel, A. Grigis and B. Helffer,Paramétrixes d’opérateurs pseudo-différentiels à caractéristiques multiples, Astérisque34–35 (1976), 93–121.

    MathSciNet  Google Scholar 

  3. R. Brummelhuis,A counterexample to the, Fefferman-Phong inequality for systems, C. R. Acad. Sci. Paris, Sér. I Math.310 (1990), 95–98.

    MATH  MathSciNet  Google Scholar 

  4. R. Brummelhuis,Sur les inégalités de Gårding pour les systèmes d’opérateurs pseudodifférentiels, C. R. Acad. Sci. Paris, Sér. I Math.315 (1992), 149–152.

    MATH  MathSciNet  Google Scholar 

  5. R. Brummelhuis,On Melin’s inequality for systems, Comm. Partial Differential Equations26 (2001), 1559–1606.

    Article  MATH  MathSciNet  Google Scholar 

  6. R. Brummelhuis and J. Nourrigat,A necessary and sufficient condition for Melin’s inequality for a class of systems, J. Analyse Math.85 (2001), 195–211.

    Article  MATH  MathSciNet  Google Scholar 

  7. L. Gårding,Dirichlet’s problem for linear elliptic partial differential equations, Math. Scand.1 (1953), 55–72.

    MATH  MathSciNet  Google Scholar 

  8. B. Helffer,Sur l’hypoellipticité des opérateurs pseudodifférentiels à caractéristiques multiples, Mém. Soc. Math. France51–52 (1977), 13–61.

    MathSciNet  Google Scholar 

  9. B. Helffer,Théorie spectrale pour des opératueur globalement elliptiques, Astérisque112 (1984).

  10. L. Hörmander,The Cauchy problem for differential operators with double characteristics, J. Analyse Math.32 (1977), 118–196.

    MATH  MathSciNet  Google Scholar 

  11. L. Hörmander,The Analysis of Linear Partial Differential Operators, Vols. I–III, Springer-Verlag, Berlin, 1983/85.

    Google Scholar 

  12. P. D. Lax and L. Nirenberg,On stability of difference schemes: a sharp form of Gårding’s inequality, Comm. Pure Appl. Math.19 (1966), 473–492.

    Article  MATH  MathSciNet  Google Scholar 

  13. A. Melin,Lower bounds for pseudodifferential operators, Ark. Mat.9 (1971), 117–140.

    Article  MATH  MathSciNet  Google Scholar 

  14. A. Mohamed,Étude spectrale d’opérateurs hypoelliptiques à caractéristiques multiples. II, Comm. Partial Differential Equations8 (1983), 247–316.

    Article  MATH  MathSciNet  Google Scholar 

  15. C. Parenti and A. Parmeggiani,Lower bounds for systems with double characteristics, J. Analyse Math.86 (2002), 49–91.

    MATH  MathSciNet  Google Scholar 

  16. A. Parmeggiani,A class of counterexamples to the Fefferman-Phong inequality for systems, Comm. Partial Differential Equations29 (2004), 1281–1303.

    Article  MATH  MathSciNet  Google Scholar 

  17. A. Parmeggiani and M. Wakayama,Oscillator representations and systems of ordinary differential equations, Proc. Nat. Acad. Sci. U.S.A.98 (2001), 26–30.

    Article  MATH  MathSciNet  Google Scholar 

  18. A. Parmeggiani and M. Wakayama,Non-commutative harmonic oscillators. I, Forum Math.14 (2002), 539–604.

    Article  MATH  MathSciNet  Google Scholar 

  19. A. Parmeggiani and M. Wakayama,Non-commutative harmonic oscillators, II, Forum Math.14 (2002), 669–690.

    Article  MATH  MathSciNet  Google Scholar 

  20. M. A. Shubin,Pseudodifferential Operators and Spectral Theory, Springer-Verlag, Berlin, 1987.

    MATH  Google Scholar 

  21. J. Sjöstrand,Parametrices for pseudodifferential operators with multiple characteristics, Ark. Mat.12 (1974), 85–130.

    Article  MATH  MathSciNet  Google Scholar 

  22. M. Taylor,Pseudodifferential Operators, Princeton University Press, 1981.

Download references

Author information

Authors and Affiliations

  1. Dipartmento di Matematica, Politecnico di Torino, C.So Duca degli Abruzzi 24, 10129, Torino, Italy

    Fabio Nicola

Authors
  1. Fabio Nicola
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Nicola, F. A lower bound for systems with double characteristics. J. Anal. Math. 96, 297–311 (2005). https://doi.org/10.1007/BF02787833

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02787833

Keywords

Search

Navigation