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

TL;DR

This paper investigates the Hausdorff dimension of graphs of functions in H"older and Besov spaces on fractal d-sets, establishing sharp bounds and revealing how fractal geometry influences graph complexity.

Contribution

It provides the first sharp upper bounds for the Hausdorff dimension of such functions' graphs on fractal sets, highlighting a phase transition based on fractal dimension and smoothness.

Findings

Sharp upper bound min{d+1-s, d/s} for Hausdorff dimension

Change of behavior from d+1-s to d/s at d=s

Highly nonsmooth functions on cubes have relatively simple graphs on fractals

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.