Level sets of Hölder functions and Hausdorff measures.
type: Item
instance of: scholarly article
title: Level sets of Hölder functions and Hausdorff measures.
zbMATH Open document ID: 1035.26009
DOI: 10.4171/ZAA/1103
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 2002-01-01T00:00:00Z
review text: Let \(f\) be a \(C^n(I)\) \((1\leq n<\infty)\) function on a closed interval \(I\subset\mathbb{R}\), \(f^{(0)}= f\), \(f^{(i)}\) \((1\leq i\leq n)\) the \(i\)th derivative and \(Z_{(f,n)}= \{x\in I: f^{(i)}(x)= 0\) for all \(1\leq i\leq n\}\) the so-called zero-derivative set. If \(0<\alpha\leq 1\), \(C^{n,\alpha}(I)\) denotes the space of \(C^n(I)\) functions with Hölder \(n\)th derivatives. In the paper are investigated some connections between Hausdorff measures, Hölder functions and analytic sets in terms of images of zero-derivative sets and level sets. The author characterizes subsets \(M\subset\mathbb{R}\) which are (1) the image under some \(C^{n,\alpha}\) function \(f\) of the set \(Z_{(f,n)}\); (2) the set of points where the level sets of some \(C^{n,\alpha}\) function are perfect; (3) the set of points where the level sets of some \(C^{n,\alpha}\) functions are uncountable. The paper generalizes, in some sense, results of \textit{E. D'Aniello} and \textit{U. B. Darji} [``\(C^n\) functions, Hausdorff measures and analytic sets'', Adv. Math. 164, No. 1, 117--143 (2001; Zbl 0998.28003)].
Mathematics Subject Classification ID: 26A24
Mathematics Subject Classification ID: 26A30
Mathematics Subject Classification ID: 28A78
zbMATH DE Number: 1827606
zbMATH Keywords: Hausdorff measures
zbMATH Keywords: Hölder functions
zbMATH Keywords: zero-derivative sets
zbMATH Keywords: level sets
author: Emma D'Aniello
reviewed by: Pavel Kostyrko
MaRDI profile type: Publication
cites work: On the Complementary Intervals of a Linear Closed Set of Zero Lebesgue Measure
cites work: Q4331836
cites work: The Level Structure of a Residual Set of Continuous Functions
cites work: Differentiability Through Change of Variables
cites work: On the existence of \(C^1\) functions with perfect level sets
cites work: \(C^n\) functions, Hausdorff measures, and analytic sets
cites work: Level sets of a typical 𝐶ⁿ function
cites work: Q5565773
cites work: Q5541833
cites work: Q3222400
cites work: Homeomorphisms of an interval and smoothness of a function
full work available at URL: https://doi.org/10.4171/zaa/1103
OpenAlex ID: W2064680208
Recommended article: \(C^n\) functions, Hausdorff measures, and analytic sets
Recommended article: On the existence of \(C^1\) functions with perfect level sets
Recommended article: Q3418368
Recommended article: Level sets of a typical 𝐶ⁿ function
Recommended article: Points of non-differentiability of typical Lipschitz functions
Recommended article: Distribution of interpolation points
Recommended article: Q3608054
Recommended article: Derivatives at the boundary for analytic Lipschitz functions
Recommended article: The zero set of a real analytic function
Recommended article: Ulam-Zahorski Problem on Free Interpolation by Smooth Functions
description: scientific article; zbMATH DE number 1827606