A Calderon Zygmund decomposition for multiple frequencies and an application to an extension of a lemma of Bourgain

TL;DR

This paper develops a multi-frequency Calderon Zygmund decomposition and applies it to extend Bourgain's lemma, resulting in a variation norm maximal inequality for multiple frequencies with broader Lp estimates.

Contribution

It introduces a novel Calderon Zygmund decomposition tailored for multiple frequencies and extends Bourgain's lemma using this framework.

Findings

Established a multi-frequency Calderon Zygmund decomposition.

Proved a variation norm maximal inequality for several frequencies.

Extended the range of Lp estimates for these inequalities.

Abstract

We introduce a Calderon Zygmund decomposition such that the bad function has vanishing integral against a number of pure frequencies. Then we prove a variation norm variant of a maximal inequality for several frequencies due to Bourgain. To obtain the full range of Lp estimates we apply the multi frequency Calderon Zygmund decomposition.