New upper bounds of the Hausdorff dimensions of graphs of continuous functions

Abstract

Let $f$ be a real function, continuous over ${\mathbb R}^d$, $d\geq1$, and let $\Gamma(f,D)\,{\hbox{\sun}}\,\{(x,f(x))|x\in D\}$ be its graph over a domain $D\subseteq {\mathbb R}^d$ and $\Gamma{f}$ its graph over ${\mathbb R}^d$. The question naturally arises about which (and how) smoothness properties of $f$ are related to the Hausdorff dimension of $\Gamma(f)$. It is a well-known result that if $f$ is locally $\alpha$-Hölder, which we denote by $f\in\Fspaceloc{C}{\alpha}$, for some $\alpha>0$, then $\mathrm{dim}_{\mathcal{H}}(\Gamma(f))\leq \max(d,d+1-\alpha)$. Of course, in this inequality, one can replace $\alpha$ by the Höolder exponent of $f$, i.e. by $\sup\{s:f\in\Fspaceloc{C}{s}\}$. The question of the sharpness of the obtained bound yields different outputs depending on $f$: one can show that the equality is satisfied over some classes of functions (see [5] and [1, chapter 8]) but it is not difficult to find examples for which the inequality is strict. One should also mention that in the simple case of the well-known Weierstrass function, neither the equality nor a smaller upper bound could be proved (see [11] for related results).