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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. 42234247. doi:10.1090/tran/7126

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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 αH¨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 βH¨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 selfaffine functions and generic αH¨older continuous functions. Finally, let {B(t) : t ∈ [0, 1]} be a twodimensional 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 nondecreasing in each coordinate. It remains open whether 1/3 is best possible.
Item Type:  Journal Article  

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:  00029947  
Official Date:  June 2018  
Dates: 


Volume:  370  
Page Range:  pp. 42234247  
DOI:  10.1090/tran/7126  
Status:  Peer Reviewed  
Publication Status:  Published  
Access rights to Published version:  Restricted or Subscription Access  
Related URLs:  
Open Access Version: 
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