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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Restrictions of Hölder continuous functions


Authors: Omer Angel, Richárd Balka, András Máthé and Yuval Peres
Journal: Trans. Amer. Math. Soc. 370 (2018), 4223-4247
MSC (2010): Primary 26A16, 26A45, 28A78, 54E52, 60G17, 60G22, 60J65
DOI: https://doi.org/10.1090/tran/7126
Published electronically: February 8, 2018
MathSciNet review: 3811526
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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.


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Additional Information

Omer Angel
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia V6T 1Z2, Canada
MR Author ID: 667585
Email: angel@math.ubc.ca

Richárd Balka
Affiliation: Department of Mathematics, University of British Columbia, and Pacific Institute for the Mathematical Sciences, Vancouver, British Columbia V6T 1Z2, Canada
MR Author ID: 838282
Email: balka@math.ubc.ca

András Máthé
Affiliation: Mathematics Institute, University of Warwick, Coventry, CV4 7AL, United Kingdom
Email: a.mathe@warwick.ac.uk

Yuval Peres
Affiliation: Microsoft Research, 1 Microsoft Way, Redmond, Washington 98052
MR Author ID: 137920
Email: peres@microsoft.com

Keywords: Fractional Brownian motion, Hölder continuous, restriction, bounded variation, Hausdorff dimension, box dimension, Minkowski dimension, self-affine function, generic, typical, Baire category
Received by editor(s): April 19, 2015
Received by editor(s) in revised form: November 11, 2016
Published electronically: February 8, 2018
Additional Notes: The first author was supported in part by NSERC
The second and third authors were supported by the National Research, Development and Innovation Office-NKFIH, 104178
The third author was also supported by the Leverhulme Trust.
Article copyright: © Copyright 2018 American Mathematical Society