Zygmund graphs are thin for doubling measures

TL;DR

This paper extends the known result that Lipschitz graphs are thin for doubling measures to the broader Zygmund class, which lies between Lipschitz and Hölder functions, by analyzing their second order divided differences.

Contribution

It proves that Zygmund graphs are also thin for doubling measures, expanding the class of functions with this property beyond Lipschitz functions.

Findings

Zygmund functions have bounded second order divided differences.

Graphs of Zygmund functions are thin for doubling measures.

Extension of thinness property from Lipschitz to Zygmund class.

Abstract

The Zygmund functions form an intermediate class between Lipschitz and H\"older functions; their second order divided differences are uniformly bounded. It is well known that for $d \geq 1$ the graph of any Lipschitz function $f:\R^d \rightarrow \R$ is thin for doubling measures, and we extend this result to the Zygmund class.