Restrictions of Hölder continuous functions

Abstract

For $0<\alpha <1$ let $V(\alpha )$ denote the supremum of the numbers $v$ such that every $\alpha$-Hölder continuous function is of bounded variation on a set of Hausdorff dimension $v$. Kahane and Katznelson (2009) proved the estimate $1/2\le V(\alpha )\le 1/(2-\alpha )$ and asked whether the upper bound is sharp. We show that in fact $V(\alpha )=\max \{1/2,\alpha \}$. Let ${\dim }_H$ and ${\overline{\dim }}_M$ denote the Hausdorff and upper Minkowski dimension, respectively. The upper bound on $V(\alpha )$ is a consequence of the following theorem. Let $\{B(t):t\in [0,1]\}$ be a fractional Brownian motion of Hurst index $\alpha$. Then, almost surely, there exists no set $A\subset [0,1]$ such that ${\overline{\dim }}_{M}A>\max \{1-\alpha ,\alpha \}$ and $B:A\to \mathbb{R}$ is of bounded variation. Furthermore, almost surely, there exists no set $A\subset [0,1]$ such that ${\overline{\dim }}_{M}A>1-\alpha$ and $B:A\to \mathbb{R}$ is $\beta$-Hölder continuous for some $\beta >\alpha$. The zero set and the set of record times of $B$ witness that the above theorems give the optimal dimensions. We also prove similar restriction theorems for deterministic self-affine functions and generic $\alpha$-Hölder continuous functions. Finally, let $\{\mathbf{B}(t):t\in [0,1]\}$ be a two-dimensional Brownian motion. We prove that, almost surely, there is a compact set $D\subset [0,1]$ such that ${\dim }_{H}D\ge 1/3$ and $\mathbf{B}:D\to {\mathbb{R}}^2$ is non-decreasing in each coordinate. It remains open whether $1/3$ is best possible.

arXiv comment24 pages, 4 figures, final version