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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
Article

Abstract

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).

Keywords

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 

Notes

Acknowledgments

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.

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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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