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Monatshefte für Mathematik

, Volume 181, Issue 1, pp 205–215 | Cite as

Microlocal analysis in generalized function algebras based on generalized points and generalized directions

  • Hans Vernaeve
Article

Abstract

We develop a refined theory of microlocal analysis in the algebra \({\mathcal G}(\Omega )\) of Colombeau generalized functions. In our approach, the wave front is a set of generalized points in the cotangent bundle of \(\Omega \), whereas in the theory developed so far, it is a set of nongeneralized points. We also show consistency between both approaches.

Keywords

Algebras of generalized functions Microlocal analysis 

Mathematics Subject Classification

46F30 35A27 35D10 

References

  1. 1.
    Colombeau, J.F.: New Generalized Functions and Multiplication of Distributions. North-Holland, Amsterdam (1984)MATHGoogle Scholar
  2. 2.
    Colombeau, J.F.: Elementary Introduction to New Generalized Functions. North-Holland, Amsterdam (1985)MATHGoogle Scholar
  3. 3.
    Djapić, N., Pilipović, S., Scarpalezos, D.: Microlocal analysis of Colombeau’s generalized functions–propagation of singularities. J. Anal. Math. 75, 51-66 (1998)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Garetto, C.: Pseudodifferential Operators with Generalized Symbols and Regularity Theory. Ph.D. thesis. Univ. Torino (2004)Google Scholar
  5. 5.
    Garetto, C., Gramchev, T., Oberguggenberger, M.: Pseudodifferential operators with generalized symbols and regularity theory. Electron. J. Differ. Eqn. 116, 1-43 (2005)MathSciNetMATHGoogle Scholar
  6. 6.
    Grosser, M., Kunzinger, M., Oberguggenberger, M., Steinbauer, R.: Geometric Theory of Generalized Functions. Kluwer Academic Publishers, Dordrecht (2001)MATHGoogle Scholar
  7. 7.
    Hörmander, L.: The Analysis of Linear Partial Differential Operators, 2nd edn, vol. 1. Springer, New York (1990)Google Scholar
  8. 8.
    Hörmann, G.: Integration and microlocal analysis in Colombeau algebras of generalized functions. J. Math. Anal. Appl. 239, 332-348 (1999)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Hörmann, G., de Hoop, M.V.: Microlocal analysis and global solutions of some hyperbolic equations with discontinuous coefficients. Acta Appl. Math. 67, 173-224 (2001)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Hörmann, G., Oberguggenberger, M., Pilipović, S.: Microlocal hypoellipticity of linear partial differential operators with generalized functions as coefficients. Trans. Am. Math. Soc. 358, 3363-3383 (2006)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Oberguggenberger, M.: Multiplication of Distributions and Applications to Partial Differential Equations. Pitman Res. Not. Math. 259. Longman Scientific & Technical (1992)Google Scholar
  12. 12.
    Oberguggenberger, M., Vernaeve, H.: Internal sets and internal functions in Colombeau theory. Monatsh. Math. 341, 649-659 (2008)MathSciNetMATHGoogle Scholar
  13. 13.
    Vernaeve, H.: Pointwise characterizations in generalized function algebras. Monatsh. Math. 158, 195-213 (2009)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Wien 2015

Authors and Affiliations

  1. 1.Ghent UniversityGentBelgium

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