Level sets of Hölder functions and Hausdorff measures. (Q1848664)

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)].