Journal of Fourier Analysis and Applications

, Volume 18, Issue 2, pp 386–409 | Cite as

On the Hausdorff Dimension of Continuous Functions Belonging to Hölder and Besov Spaces on Fractal d-Sets

  • Abel Carvalho
  • António CaetanoEmail author


The Hausdorff dimension of the graphs of the functions in Hölder and Besov spaces (in this case with integrability p≥1) on fractal d-sets is studied. Denoting by s∈(0,1] the smoothness parameter, the sharp upper bound min{d+1−s,d/s} is obtained. In particular, when passing from ds to d<s there is a change of behaviour from d+1−s to d/s which implies that even highly nonsmooth functions defined on cubes in ℝ n have not so rough graphs when restricted to, say, rarefied fractals.


Hausdorff dimension Box counting dimension Fractals d-Sets Continuous functions Weierstrass function Hölder spaces Besov spaces Wavelets 

Mathematics Subject Classification (2000)

26A16 26B35 28A78 28A80 42C40 46E35 


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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Centro I&D Matemática e AplicaçõesUniversidade de AveiroAveiroPortugal
  2. 2.Centro I&D Matemática e Aplicações, Departamento de MatemáticaUniversidade de AveiroAveiroPortugal

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