Revista Matemática Complutense

, Volume 29, Issue 1, pp 127–167 | Cite as

Baire generic results for the anisotropic multifractal formalism

  • Mourad Ben SlimaneEmail author
  • Hnia Ben Braiek


Ben Slimane (Math Proc Camb Philos Soc 124:329–363, 1998) has constructed specific anisotropic selfsimilar functions as counter-examples for the isotropic multifractal formalism. An anisotropic multifractal formalism has been formulated and its validity for anisotropic selfsimilar functions has been proved. In this paper, using Triebel anisotropic wavelet decompositions, we first obtain lower bounds of the anisotropic scaling function and upper bounds of the u-spectrum of singularities valid for all functions. We then investigate the generic validity, in the sense of Baire’s categories, of the anisotropic formalism in some anisotropic functional spaces. We thus extend in the anisotropic setting some results of Jaffard (J Math Pure Appl 79:525–552, 2000, Ann Appl Probab 10:313–329, 2000) and Jaffard and Meyer (Memoirs of the American Mathematical Society, vol. 123, 1996).


Triebel anisotropic wavelet bases Anisotropic Hölder regularity Anisotropic Besov and Sobolev spaces Anisotropic scaling function Anisotropic dyadic approximation  Baire categories 

Mathematical Subject Classification

28A78 42C40 54E52 



The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for its funding Research Group no. RG-1435-063. Mourad Ben Slimane is thankful to Stéphane Jaffard for stimulating discussions.


  1. 1.
    Arneodo, A., Bacry, E., Muzy, J.F.: Singularity spectrum of fractal signals from wavelet analysis: exact results. J. Stat. Phys. 70, 635–674 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Arneodo, A., Bacry, E., Jaffard, S., Muzy, J.F.: Singularity spectrum of multifractal functions involving oscillating singularities. J. Fourier Anal. Appl. 4, 159–174 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bayart, F., Heurteaux, Y.: Multifractal analysis of the divergence of Fourier series: the extreme cases. Annales scientifiques de l’ENS 45(6), 927–946 (2012)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Ben Braiek, H., Ben Slimane, M.: Critère de régularité directionnelle. Directional regularity criteria. C. R. Acad. Sci. Paris Sér. I Math 349, 385–389 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ben Slimane, M.: Multifractal formalism for self-similar functions under the action of nonlinear dynamical systems. Constr. Approx. 15, 209–240 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ben Slimane, M.: Multifractal formalism and anisotropic self-similar functions: Math. Proc. Camb. Philos. Soc. 124, 329–363 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    BenSlimane, M.: Some functional equations revisited: the multifractal properties. Int. Trans. Spec. Funct. 14, 333–348 (2003)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Ben Slimane, M.: Baire generic histograms of wavelet coefficients and large deviation formalism in Besov and Sobolev spaces. J. Math. Anal. Appl. 349, 403–412 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ben Slimane, M., Ben Braiek, H.: Directional and anisotropic regularity and irregularity criteria in Triebel wavelet bases. J. Fourier Anal. Appl. 18(5), 893–914 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ben Slimane, M., Ben Braiek, H.: On the gentle properties of anisotropic Besov spaces. J. Math. Anal. Appl. 396, 21–48 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Beresnevich, V., Dickinson, D., Velani, S.: A mass transference principle and the Duffin–Schaeffer conjecture for Hausdorff measures. Ann. Math. 164, 971–992 (2006)CrossRefzbMATHGoogle Scholar
  12. 12.
    Beresnevich, V., Velani, S.: Measure theoretic laws for lim-sup sets. arXiv:math/0401118
  13. 13.
    Biermé, H., Meerschaert, M.M., Scheffler, H.P.: Operator scaling stable random fields. Stoch. Proc. Appl. 117(3), 312–332 (2007)CrossRefzbMATHGoogle Scholar
  14. 14.
    Bony, J.M.: Second microlocalization and propagation of singularities for semilinear hyperbolic equations. In: Hyperbolic Equations and Related Topics (Kata/Kyoto, 1984), pp. 11–49, Academic Press, Boston (1986)Google Scholar
  15. 15.
    Buczolich, Z., Nagy, J.: Hölder spectrum of typical monotone continuous functions. Real Anal. Exchange 26, 133–156 (2000/2001)Google Scholar
  16. 16.
    Buczolich, Z., Seuret, S.: Typical Borel measures on \([0,1]^d\) satisfy a multifractal formalism. Nonlinearity 23(11), 2905–2918 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Calderón, A.P., Torchinsky, A.: Parabolic maximal functions associted with a distribution. Adv. Math. 24, 101–171 (1977)CrossRefzbMATHGoogle Scholar
  18. 18.
    Clausel, M., Nicolay, S.: Wavelets techniques for pointwise anti-Hölderian irregularity. Constr. Approx. 33, 41–75 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Clausel, M., Vedel, B.: Explicit constructions of operator scaling Gaussian fields. Fractals 19(1), 101–111 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Cohen, A.: Wavelet methods in numerical analysis. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook for Numerical Analysis. North-Holland, Amsterdam (2002)Google Scholar
  21. 21.
    Dahmen, W.: Wavelet and multiscale methods for operator equations. Acta Numer. 6, 55–228 (1997)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Daubechies, I.: Orthonormal bases of compactly supported wavelets. Commun. Pure Appl. Math. 41, 909–996 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Daubechies, I., Lagarias, J.C.: On the thermodynamic formalism for functions. Rev. Math. Phys. 6, 1033–1070 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Devore, R.A.: Nonlinear approximation. Acta Numer. 7, 51–151 (1998)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Durand, A., Jaffard, S.: Multifractal analysis of Lévy fields. Probab. Theory Relat Fields (2011). doi: 10.1007/s00440-011-0340-0 Google Scholar
  26. 26.
    Falconer, K.J.: Fractal Geometry: Mathematical Foundations and Applications. Wiley, Toronto (1990)zbMATHGoogle Scholar
  27. 27.
    Farkas, W.: Atomic and subatomic decompositions in anisotropic function spaces. Math. Nachr. 209, 83–113 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Folland, G.B., Stein, E.M.: Hardy spaces on homogeneous groups. Mathematical Notes, 28, Princeton University Press and University of Tokyo Press, Princeton (1982)Google Scholar
  29. 29.
    Fraysse, A.: Generic validity of the multifractal formalism. SIAM J. Math. Anal. 39, 593–607 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Fraysse, A., Jaffard, S.: How smooth is almost every function in a Sobolev space? Rev. Math. Iberoam. 22(2), 663–682 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Frisch, U., Parisi, G.: Fully developped turbulence and intermittency. In: Proceedings of the International School of Physics Enrico Fermi, North Holland, pp. 84–88 (1985)Google Scholar
  32. 32.
    Garrigós, G., Tabacco, A.: Wavelet decompositions of anisotropic Besov spaces. Math. Nachr. 239, 80–102 (2002)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Garrigós, G., Hochmuth, R., Tabacco, A.: Wavelet characterizations for anisotropic Besov spaces with \(0<p<1\). Proc. Edinb. Math. Soc. 47, 573–595 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Hochmuth, R.: Wavelet characterizations for anisotropic Besov spaces. Appl. Comput. Harmon. Anal. 12, 179–208 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Hutchinson, J.: Fractals and self-similarity. Indiana Univ. Math. J. 30, 713–747 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Jaffard, S.: Pointwise smoothness, two-microlocalization and wavelet coefficients. Publ. Math. 35, 155–168 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Jaffard, S.: The spectrum of singularities of Riemanns function. Rev. Mat. Iberoam. 12, 441–460 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Jaffard, S.: Multifractal formalism for functions. Part 1: results valid for all functions and Part 2: selfsimilar functions. SIAM J. Math. Anal. 28, 944–998 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Jaffard, S.: The multifractal nature of the Lévy processes. Probab. Theory Related Fields 114, 207–227 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Jaffard, S.: On the Frisch–Parisi conjecture. J. Math. Pure Appl. 79, 525–552 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Jaffard, S.: On lacunary wavelet series. Ann. Appl. Probab. 10, 313–329 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Jaffard, S., Meyer, Y.: Wavelet methods for pointwise regularity and local oscillations of functions. In: Memoirs of the American Mathematical Society, vol. 123. American Mathematical Society, Providence (1996)Google Scholar
  43. 43.
    Lemarié, P.-G., Meyer, Y.: Ondelettes et bases hilbertiennes. Rev. Math. Iberoam. 1, 1–8 (1986)CrossRefGoogle Scholar
  44. 44.
    Lévy Véhel, J.: Fractal approaches in signal processing. Fractals 3(4), 755–775 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Mallat, S., Hwang, W.L.: Singularity detection and processing with wavelets. IEEE Trans. Inf. Theory 38(2), 617–643 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Melot, C.: Oscillating singularities in Besov spaces. J. Math. Pure. Appl. 83, 367–416 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Meyer, Y.: Ondelettes et Opérateurs I: Ondelettes. Hermann, Paris (1990)Google Scholar
  48. 48.
    Olsen, L.: Measurability of multifractal measure functions and multifractal dimension functions. Hiroshima Math. J. 29, 435–458 (1999)MathSciNetzbMATHGoogle Scholar
  49. 49.
    Rogers, C.A.: Hausdorff Measures. Cambridge University Press, London (1970)zbMATHGoogle Scholar
  50. 50.
    Schmeling, J., Seuret, S.: On measures resisting multifractal analysis. To appear in the book “Nonlinear Dynamics: New Directions, Theoretical Aspects” (in honor of V. Afraimovich), published in the series “Mathematical Method and Modeling”, SpringerGoogle Scholar
  51. 51.
    Triebel, H.: Theory of Function Spaces III. Monographs in Mathematics, 78th edn. Birkhäuser, Basel (2006)Google Scholar
  52. 52.
    Triebel, H.: Wavelet Bases in Anisotropic Function Spaces. In: Proceedings of Conference on “Function spaces, differential operators and nonlinear analysis”, Milovy, pp. 370–387, Math. Inst. Acad. Sci. Czech Republic, Prague (2004)Google Scholar
  53. 53.
    Vedel, B.: Besov characteristic of a distribution. Rev. Math. Complut. 20(2), 407–421 (2007)MathSciNetzbMATHGoogle Scholar
  54. 54.
    Xiao, Y.: Sample path properties of anisotropic Gaussian random fields. In: Khoshnevisan, D., Rassoul-Agha, F. (eds.) A Minicourse on Stochastic Partial Differential Equations . Lecture notes in Math, pp. 145–212. Springer, New York (1962)Google Scholar

Copyright information

© Universidad Complutense de Madrid 2015

Authors and Affiliations

  1. 1.King Saud University, Department of MathematicsCollege of ScienceRiyadhSaudi Arabia
  2. 2.Laboratoire de Recherche Équations aux Dérivées Partielles et ApplicationsFaculté des Sciences de Tunis, Université de Tunis El ManarTunisTunisia

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