Advertisement

Spaces of Ultradifferentiable Functions of Multi-anisotropic Type

  • Chikh Bouzar
Chapter
  • 57 Downloads
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

The paper deals first with the relationship between multi-anisotropic Gevrey spaces and Denjoy-Carleman spaces and then it introduces a class of ultradifferentiable functions unifying these both spaces.

Keywords

Denjoy-Carleman spaces Gevrey spaces Ultradifferentiable functions Linear differential operators Elliptic iterates 

Notes

Acknowledgements

The author thanks the anonymous referee whose remarks helped to improve the quality of the text.

References

  1. 1.
    A. Beurling. Quasi-analyticity and general distributions. Lecture 4 and 5, AMS Summer Institute, Standford, (1961).Google Scholar
  2. 2.
    G. Björck. Linear partial differential operators and generalized distributions. Ark. Mat., Vol. 6, p. 351–407, (1966).MathSciNetCrossRefGoogle Scholar
  3. 3.
    R. P. Boas. A theorem on analytic functions of a real variable. Bull. A. M. S., Vol. 41, p. 233–236, (1935).MathSciNetCrossRefGoogle Scholar
  4. 4.
    J. Bonet, R. Meise, S. N. Melikhov. A comparison of two different ways to define classes of ultradifferentiable functions. Bull. Belg. Math. Soc. Simon Stevin, Vol. 14–3, p. 425–444, (2007).MathSciNetCrossRefGoogle Scholar
  5. 5.
    C. Bouzar, R. Chaïli. Vecteurs Gevrey de systèmes quasielliptiques. Annales de Mathématiques de l’Université de Sidi Bel Abbés, T. 5, p. 33–43, (1998).Google Scholar
  6. 6.
    C. Bouzar, R. Chaïli. Vecteurs Gevrey d’opérateurs différentiels quasi-homogènes. Bull. of the Belgian Math. Society, Vol. 9, NÂ 2, p. 299–310, (2002).Google Scholar
  7. 7.
    C. Bouzar, R. Chaïli. Régularité des vecteurs de Beurling de systèmes elliptiques. Maghreb Math. Rev., T. 9, NÂ 1–2, p. 43–53, (2000).Google Scholar
  8. 8.
    C. Bouzar, R. Chaïli. Une généralisation de la propriété des itérés. Arch. Math., Vol. 76, NÂ 1, p. 57–66, (2001).Google Scholar
  9. 9.
    C. Bouzar, R. Chaïli. Gevrey vectors of multi-quasi-elliptic systems. Proc. Amer. Math. Soc., Vol. 131–5, p. 1565–1572, (2003).MathSciNetCrossRefGoogle Scholar
  10. 10.
    C. Bouzar, R. Chaïli. Iterates of differential operators. Progress in Analysis, Vol. I, (Berlin 2001), p. 135–141, World Sci. Publ., (2003).Google Scholar
  11. 11.
    C. Bouzar, L. R. Volevich. Hypoelliptic systems connected with Newton’s polyhedron. Math. Nachr., Vol. 273, p. 14–27, (2004).MathSciNetCrossRefGoogle Scholar
  12. 12.
    C. Bouzar, A. Dali. Multi-anisotropic Gevrey regularity of hypoelliptic operators. Operator Theory : Advances and Applications, Vol. 189, p. 265–273, (2008).MathSciNetzbMATHGoogle Scholar
  13. 13.
    C. Bouzar, A. Dali. The Gevrey regularity of multi-quasielliptic operators. Annali dell Universita di Ferrara, Vol. 57, p. 201–209, (2011).MathSciNetCrossRefGoogle Scholar
  14. 14.
    R. W. Braun, R. Meise, B. A. Taylor. Ultradifferentiable functions and Fourier analysis. Results Math., T. 17, N 3–4, p. 206–237, (1990).Google Scholar
  15. 15.
    D. Calvo, M. C. Gomez-Collado. On some generalizations of Gevrey classes. Math. Nach., Vol. 284, NÂ 7, p. 856–874, (2011).Google Scholar
  16. 16.
    D. Calvo, A. Morando, L. Rodino. Inhomogeneous Gevrey classes and ultradistributions. J. Math. Anal. Appl., 297, p. 720–739, (2004).MathSciNetCrossRefGoogle Scholar
  17. 17.
    T. Carleman. Les fonctions quasi-analytiques. Gauthier-Villars, (1926).zbMATHGoogle Scholar
  18. 18.
    A. Denjoy. Sur les fonctions quasi-analytiques de variable r éelle. C. R. Acad. Sci. Paris, T. 173, p. 1320–1322, (1921).Google Scholar
  19. 19.
    J. Friberg. Multi-quasielliptic polynomials. Ann. Sc. Norm. Sup. Pisa, Cl. di Sc., Vol. 21, p. 239–260, (1967).MathSciNetzbMATHGoogle Scholar
  20. 20.
    M. Gevrey. Sur la nature analytique des solutions des é quations aux dérivées partielles. Ann. Ec. Norm. Sup. Paris, T. 35, p. 129–190, (1918).Google Scholar
  21. 21.
    S. G. Gindikin, L. R. Volevich. The method of Newton polyhedron in the theory of partial differential equations. Kluwer, (1992).zbMATHGoogle Scholar
  22. 22.
    L. Hörmander, Linear partial differential operators. Springer, (1963).CrossRefGoogle Scholar
  23. 23.
    L. Hörmander. Distribution theory and Fourier analysis. Springer, (2000).Google Scholar
  24. 24.
    H. Komatsu. A characterization of real analytic functions. Proc. Japan Acad., Vol. 36, p. 90–93, (1960).MathSciNetCrossRefGoogle Scholar
  25. 25.
    O. Liess, L. Rodino. Inhomogeneous Gevrey classes and related pseudodifferential operators. Suppl. Boll. Un. Mat. It., Vol. 3, 1-C, p. 233–323, (1984).Google Scholar
  26. 26.
    J. L. Lions, E. Magenes. Non homogenous boundary value problems and applications, Vol. 3. Springer, (1973).Google Scholar
  27. 27.
    S. Mandelbrojt. Séries adhérentes, ré gularisations des suites, Applications. Gauthier-Villars, (1952).zbMATHGoogle Scholar
  28. 28.
    G. Métivier. Propriété des itérés et ellipticité. Comm. P.D.E., Vol. 3, N 9, p. 827–876, (1978).Google Scholar
  29. 29.
    V. P. Mikhailov. On the behavior at infinity of a class of polynomials. Trudy Math. Inst. Steklov, T. 91, p. 59–81, (1967).Google Scholar
  30. 30.
    S. N. Nikolsky. The first boundary-value problem for a general linear equation. Dokl. Akad. Nauk SSSR, T. 146, p. 767–769, (1962).Google Scholar
  31. 31.
    L. Rodino, Linear partial differential operators in Gevrey spaces. World Scientific, (1993).CrossRefGoogle Scholar
  32. 32.
    L. Rodino, F. Nicola. Global pseudo-differential calculus on euclidian spaces. Springer, (2010).zbMATHGoogle Scholar
  33. 33.
    L. R. Volevich. Local properties of solutions of quasi-elliptic systems. Math. Sbornik (N. S.), T. 59 (101), p. 3–52, (1962).Google Scholar
  34. 34.
    L. R. Volevich, S. Gindikin, On a class of hypoelliptic operators. Mat. Sb., 75 (117) (3), p. 400–416, (1968).Google Scholar
  35. 35.
    L. Zanghirati. Iterati di una classe di operatori ipoellittici e classi generalizzate di Gevrey. Boll. U.M.I., Vol. 1, suppl., p. 177–195, (1980).Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Chikh Bouzar
    • 1
  1. 1.Laboratory of Mathematical Analysis and ApplicationsUniversity of OranOranAlgeria

Personalised recommendations