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Restrictions of holder continuous functions
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Angel, Omer, Balka, Richárd, Máthé, András and Peres, Y. (2018) Restrictions of holder continuous functions. Transactions of the American Mathematical Society, 370 . pp. 4223-4247. doi:10.1090/tran/7126 ISSN 0002-9947.
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Official URL: https://doi.org/10.1090/tran/7126
Abstract
For 0 < α < 1 let V (α) denote the supremum of the numbers v
such that every α-H¨older continuous function is of bounded variation on a set of Hausdorff dimension v. Kahane and Katznelson (2009) proved the estimate 1/2 ≤ V (α) ≤ 1/(2−α) and asked whether the upper bound is sharp. We show that in fact V (α) = max{1/2, α}. Let dimH and dimM denote the Hausdorff and upper Minkowski dimension, respectively. The upper bound on V (α) is a consequence of the following theorem. Let {B(t) : t ∈ [0, 1]} be a fractional Brownian motion of Hurst index α. Then, almost surely, there exists no set
A ⊂ [0, 1] such that dimMA > max{1 − α, α} and B : A → R is of bounded variation. Furthermore, almost surely, there exists no set A ⊂ [0, 1] such that dimMA > 1 − α and B : A → R is β-H¨older continuous for some β > α. 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 α-H¨older continuous functions. Finally, let {B(t) : t ∈ [0, 1]} be a two-dimensional Brownian motion. We prove that, almost surely, there is a compact set D ⊂ [0, 1] such that dimH D ≥ 1/3 and B: D → R2 is non-decreasing in each coordinate. It remains open whether 1/3 is best possible.
Item Type: | Journal Article | ||||||||
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Subjects: | Q Science > QA Mathematics | ||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||
Journal or Publication Title: | Transactions of the American Mathematical Society | ||||||||
Publisher: | American Mathematical Society | ||||||||
ISSN: | 0002-9947 | ||||||||
Official Date: | June 2018 | ||||||||
Dates: |
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Volume: | 370 | ||||||||
Page Range: | pp. 4223-4247 | ||||||||
DOI: | 10.1090/tran/7126 | ||||||||
Status: | Peer Reviewed | ||||||||
Publication Status: | Published | ||||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||||
Date of first compliant deposit: | 30 November 2016 | ||||||||
Date of first compliant Open Access: | 9 March 2018 | ||||||||
Related URLs: | |||||||||
Open Access Version: |
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