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

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 d≥s 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 ℝn have not so rough graphs when restricted to, say, rarefied fractals.