Functional Analysis and Applications Seminar 

26/10/2012, 14:30 — 15:30 — Room P3.10, Mathematics Building
António Caetano, Universidade de Aveiro

Hausdorff dimension of functions on $d$-sets

The sharp upper bound for the Hausdorff dimension of the graphs of the functions in Hölder and Besov spaces (in this case with integrability $p\geq 1$) on fractal $d$-sets is obtained: $\min \{ d+1-s,d/s \} $, where $s\in (0,1]$ denotes the smoothness parameter. In particular, when passing from $d\geq s$ to $d \lt s$ there is a change of behaviour from $ d+1-s $ to $d/s$ which implies that even highly nonsmooth functions defined on cubes in $\mathbb{R}^n$ have not so rough graphs when restricted to, say, rarefied fractals.