Baire generic results for the anisotropic multifractal formalism

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• 作者：Mourad Ben Slimane ; Hnia Ben Braiek
• 关键词：Triebel anisotropic wavelet bases ; Anisotropic Hölder regularity ; Anisotropic Besov and Sobolev spaces ; Anisotropic scaling function ; Anisotropic dyadic approximation ; Baire categories ; 28A78 ; 42C40 ; 54E52
• 刊名：Revista Matem¨¢tica Complutense
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：29
• 期：1
• 页码：127-167
• 全文大小：750 KB
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• 作者单位：Mourad Ben Slimane (1)
  Hnia Ben Braiek (2)
  
  1. King Saud University, Department of Mathematics, College of Science, P. O. Box 2455, Riyadh, 11451, Saudi Arabia
  2. Laboratoire de Recherche Équations aux Dérivées Partielles et Applications, Faculté des Sciences de Tunis, Université de Tunis El Manar, 2092, Tunis, Tunisia
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Algebra
  Applications of Mathematics
  Geometry
  Mathematics
  Topology
  
• 出版者：Springer Milan
• ISSN：1988-2807

文摘

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