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

link: http://arxiv.org/abs/0912.3010
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.