Level Sets of Hölder Functions and Hausdorff Measures

Abstract

In this paper we investigate some connections between Hausdorff measures, Hölder functions and analytic sets in terms of images of zero-derivative sets and level sets. We characterize in terms of Hausdorff measures and descriptive complexity subsets $M\subseteq \mathbb{R}$ which are
(1) the image under some $C^{m,\alpha }$ function $f$ of the set of points where the derivatives of first $n$ orders are zero
(2) the set of points where the level sets of some $C^{m,\alpha }$ function are perfect
(3) the set of points where the level sets of some $C^{m,\alpha }$ function are uncountable.