Oscillation of generalized differences of H\"older and Zygmund functions

TL;DR

This paper investigates the oscillation behavior of functions with derivatives in H"older or Zygmund classes, revealing growth patterns linked to Kolmogorov's Law and improved results for Lipschitz functions through Calderón-Zygmund operators.

Contribution

It introduces a novel analysis of oscillation for H"older and Zygmund functions using generalized differences and connects Lipschitz functions' behavior to Calderón-Zygmund operators.

Findings

Oscillation growth follows a version of Kolmogorov's Law of the Iterated Logarithm.

Lipschitz functions exhibit better oscillation behavior.

Connection established between Lipschitz functions and Calderón-Zygmund operators.

Abstract

In this paper we analyze the oscillation of functions having derivatives in the H\"older or Zygmund class in terms of generalized differences and prove that its growth is governed by a version of the classical Kolmogorov's Law of the Iterated Logarithm. A better behavior is obtained for functions in the Lipschitz class via an interesting connection with Calder\'on-Zygmund operators.