Skip to main content
SpringerLink
Log in

Hyperfunctions and (analytic) hypoellipticity

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

In this work we discuss the problem of smooth and analytic regularity for hyperfunction solutions to linear partial differential equations with analytic coefficients. In particular we show that some well known “sum of squares” operators, which satisfy Hörmander’s condition and consequently are hypoelliptic, admit hyperfunction solutions that are not smooth (in particular they are not distributions).

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. Baouendi M.S., Goulaouic C.: Nonanalytic-hypoellipticity for some degenerate elliptic equations. Bull. AMS 78, 483–486 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  2. Caetano, P.A.S., Cordaro, P.D.: Gevrey solvability and Gevrey regularity in differential complexes associated to locally integrable structures. Trans. AMS (2009, in press)

  3. Christ M.: Certain sum of squares of vector fields fail to be analytic-hypoelliptic. Commun. PDE 16, 1695–1707 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  4. Cordaro P.D., Trépreau J.-M.: On the solvability of linear partial differential equations in spaces of hyperfunctions. Ark. Mat. 36, 41–71 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  5. De Wilde M.: Closed Graph Theorems and Webbed Spaces. Pitman, London (1978)

    MATH  Google Scholar 

  6. Hanges N., Himonas A.A.: Singular solutions for sums of squares of vector fields. Commun. PDE 16, 1503–1511 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  7. Hörmander L.: Hypoelliptic second order differential equations. Acta Math. 119, 147–171 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  8. Hörmander L.: The Analysis of Linear Partial Differential Operators I. Springer, New York (1983)

    MATH  Google Scholar 

  9. Komatsu H.: Ultradistributions I. Structure theorems and a characterization. J. Fac. Sci. Univ. Tokyo Sec. IA 20, 25–105 (1973)

    MATH  MathSciNet  Google Scholar 

  10. Matsuzawa T.: Hypoellipticity in ultradistribution spaces. J. Fac. Sci. Univ. Tokyo 34, 779–790 (1987)

    MATH  MathSciNet  Google Scholar 

  11. Matsuzawa T.: Gevrey hypoellipticity for Grushin operators. Publ. Res. Inst. Math. Sci. 33(5), 775–799 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  12. Okaji T.: Gevrey-hypoellipticity operators which are not \({\mathcal{C}^\infty}\) -hypoelliptic. J. Math. Kyoto Univ. 28(2), 311–322 (1988)

    MATH  MathSciNet  Google Scholar 

  13. Oleinik O.A.: On the analyticity of solutions of partial differential equations and systems. Astérisque 2/3, 272–285 (1973)

    Google Scholar 

  14. Petrowsky I.G.: Sur l’analyticité des solutions des systèmes d’équations différentielles. Mat. Sb. 5, 3–70 (1939)

    MATH  MathSciNet  Google Scholar 

  15. Lai P.T., Robert D.: Sur un problème aux valeurs propres non-linéaire. Isr. J. Math. 36, 169–186 (1980)

    Article  MATH  Google Scholar 

  16. Rodino L.: Linear Partial Differential Operators in Gevrey Spaces. World Cientific, Singapure (1993)

    MATH  Google Scholar 

  17. Schapira P.: Théorie des Hyperfonctions. Lecture Notes in Math, vol. 126. Springer, New York (1970)

    Google Scholar 

  18. Treves F.: Analytic-hypoelliptic partial differential operators of principal type. Commun. Pure Appl. Math. 24, 537–570 (1971)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Universidade de São Paulo, São Paulo, SP, Brazil

    Paulo D. Cordaro

  2. Department of Mathematics, Lehman College, CUNY, Bronx, NY, 10468, USA

    Nicholas Hanges

Authors
  1. Paulo D. Cordaro
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nicholas Hanges
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nicholas Hanges.

Additional information

This research project was supported by the NSF Grant INT 0227100. P. D. Cordaro was also partially supported by CNPq and Fapesp.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cordaro, P.D., Hanges, N. Hyperfunctions and (analytic) hypoellipticity. Math. Ann. 344, 329–339 (2009). https://doi.org/10.1007/s00208-008-0308-2

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-008-0308-2

Mathematics Subject Classification (2000)

Search

Navigation