Oscillation of Functions in the Hölder Class

Abstract

We study the size of the set of points where the α-divided difference of a function in the Holder class Λα is bounded below by a fixed positive constant. Our results are obtained from their discrete analogues which can be stated in the language of dyadic martingales. Our main technical result in this setting is a sharp estimate of the Hausdorff measure of the set of points where a dyadic martingale with bounded increments has maximal growth.