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 $1/2 \leq V(\alpha )\leq 1/(2-\alpha )$ and asked whether the upper bound is sharp. We show that in fact $V(\alpha )=\max \{1/2,\alpha \}$. Let $\dim _{\mathcal {H}}$ and $\overline {\dim }_{\mathcal {M}}$ 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 $\overline {\dim }_{\mathcal {M}} A>\max \{1-\alpha ,\alpha \}$ and $B\colon A\to \mathbb {R}$ is of bounded variation. Furthermore, almost surely, there exists no set $A\subset [0,1]$ such that $\overline {\dim }_{\mathcal {M}} A>1-\alpha$ and $B\colon A\to \mathbb {R}$ 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 $\dim _{\mathcal {H}} D\geq 1/3$ and $\mathbf {B}\colon D\to \mathbb {R}^2$ is non-decreasing in each coordinate. It remains open whether $1/3$ is best possible.