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- W2964226507 abstract "For 0 < α < 1 let V (α) denote the supremum of the numbers vsuch 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 setA ⊂ [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." @default.
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- W2964226507 date "2015-04-21" @default.
- W2964226507 modified "2023-09-27" @default.
- W2964226507 title "Restrictions of Hölder continuous functions" @default.
- W2964226507 hasPublicationYear "2015" @default.
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