2-Microlocal Besov Spaces

  • Henning Kempka
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


We introduce 2-microlocal Besov spaces which generalize the 2-microlocal spaces \({C}_{{x}_{0}}^{s,s^{\prime}}(\mathbb{R}n)\) by Bony. We give a unified Fourier-analytic approach to define generalized 2-microlocal Besov spaces and we present a wavelet characterization for them. Wavelets provide a powerful tool for studying global and local regularity properties of functions. Further, we prove a characterization with wavelets for the local version of the 2-microlocal Besov spaces and we give first connections and generalizations to local regularity theory.


Besov Space Local Space Local Regularity Daubechies Wavelet Open Neighborhood Versus 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    P. Andersson: Two-microlocal spaces, local norms and weighted spaces. Göteborg, Paper 2 in Ph.D. Thesis (1997), 35–58.Google Scholar
  2. 2.
    B. Beauzamy: Espaces de Sobolev et de Besov d’ordre définis sur L p . C.R. Acad. Sci. Paris (Ser. A) 274 (1972), 1935–1938.MATHMathSciNetGoogle Scholar
  3. 3.
    O. V. Besov: Equivalent normings of spaces of functions of variable smoothness. (Russian) Tr. Mat. Inst. Steklova 243 (2003), Funkts. Prostran., Priblizh., Differ. Uravn., 87–95. [Translation in Proc. Steklov Inst. Math. 243 (2003), no. 4, 80–88.]Google Scholar
  4. 4.
    J.-M. Bony: Second microlocalization and propagation of singularities for semi-linear hyperbolic equations. Taniguchi Symp. HERT, Katata (1984), 11–49.Google Scholar
  5. 5.
    I. Daubechies: Ten lectures on wavelets. Philadelphia, PA: SIAM (1992).MATHGoogle Scholar
  6. 6.
    D. E. Edmunds, H. Triebel: Function Spaces, entropy numbers, differential operators. Cambridge University Press, Cambridge (1996).MATHCrossRefGoogle Scholar
  7. 7.
    W. Farkas, H.-G. Leopold: Characterisations of function spaces of generalised smoothness. Annali di Mathematica 185 (2006), 1–62.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    S. Jaffard: Pointwise smoothness, two-microlocalisation and wavelet coefficients. Publ. Math. 35 (1991), 155–168.MATHMathSciNetGoogle Scholar
  9. 9.
    S. Jaffard, Y. Meyer: Wavelet methods for pointwise regularity and local oscillations of functions. Mem. AMS, 123 (1996).Google Scholar
  10. 10.
    G. A. Kalyabin: Characterization of spaces of generalized Liouville differentiation. Mat. Sb. Nov. Ser. 104 (1977), 42–48.Google Scholar
  11. 11.
    H. Kempka: Local characterization of generalized 2-microlocal spaces. Jenaer Schriften zur Math. Inf. 20/06 (2006).Google Scholar
  12. 12.
    H. Kempka: Generalized 2-microlocal Besov spaces. Dissertation, Jena (2008).Google Scholar
  13. 13.
    H. Kempka: Atomic, molecular and wavelet decomposition of generalized 2-microlocal Besov spaces. J. Funct. Spaces Appl. (2010) (in press).Google Scholar
  14. 14.
    H. Kempka: 2-microlocal Besov and Triebel-Lizorkin spaces of variable integrability. Rev. Mat. Complut. 22 no. 1 (2009), 227–251.MATHMathSciNetGoogle Scholar
  15. 15.
    H.-G. Leopold: On function spaces of variable order of differentiation. Forum Math. 3 (1991), 633–644.CrossRefMathSciNetGoogle Scholar
  16. 16.
    J. Lévy Véhel, S. Seuret: The 2-microlocal formalism. Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot, Proceedings of Symposia in Pure Mathematics, PSPUM, 72, part 2 (2004), 153–215.Google Scholar
  17. 17.
    Y. Meyer: Wavelets, vibrations and scalings. CRM monograph series, AMS, vol. 9 (1997).Google Scholar
  18. 18.
    Y. Meyer, H. Xu: Wavelet analysis and chirps. Appl. Comput. Harmon. Anal. 4 (1997), 366–379.MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    S. Moritoh, T. Yamada: Two-microlocal Besov spaces and wavelets. Rev. Mat. Iberoam. 20 (2004), 277–283.MATHMathSciNetGoogle Scholar
  20. 20.
    S.D. Moura: Function Spaces of generalised smoothness. Diss. Math. 398 (2001).Google Scholar
  21. 21.
    J. Peetre: New thoughts on Besov spaces. Dept. Mathematics, Duke University (1975).Google Scholar
  22. 22.
    H.-J. Schmeißer, H. Triebel: Topics in Fourier analysis and function spaces. Leipzig: Akademische Verlagsgesellschaft Geest & Portig (1987).Google Scholar
  23. 23.
    S. Seuret, J. Lévy Véhel: A time domain characterization of 2-microlocal spaces. J. Fourier Anal. Appl. 9 no. 5 (2003), 473–495.MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    H. Triebel: Theory of function spaces. Leipzig: Geest & Portig (1983).Google Scholar
  25. 25.
    H. Triebel: Theory of function spaces III. Basel: Birkhäuser (2006).MATHGoogle Scholar
  26. 26.
    A. Underberger, J. Bokobza: Les opérateurs pseudodifferentielles d’ordre variable. C.R. Acad. Sci. Paris (Ser. A) 261 (1965), 2271–2273.Google Scholar
  27. 27.
    P. Wojtaszczyk: A mathematical introduction to wavelets. Cambridge University Press, London Math. Society Student Texts 37 (1997).Google Scholar
  28. 28.
    H. Xu: Généralisation de la théorie des chirps à divers cadres fonctionnels et application à leur analyse par ondelettes. Ph.D. thesis, Université Paris IX Dauphine (1996).Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Mathematical InstituteFriedrich Schiller UniversityJenaGermany

Personalised recommendations