On the Hausdorff dimension of continuous functions belonging to H\"older and Besov spaces on fractal d-sets

Abstract

The Hausdorff dimension of the graphs of the functions in H\"older and Besov spaces (in this case with integrability p \geq 1) on fractal d-sets is studied. Denoting by s \in (0,1] the smoothness parameter, the sharp upper bound min{d+1-s,d/s} is obtained. In particular, when passing from d \geq 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 R^n have not so rough graphs when restricted to, say, \emp{rarefied} fractals.