A Calderón Zygmund decomposition for multiple frequencies and an application to an extension of a Lemma of Bourgain

Abstract

We introduce a Calder\'on 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 $L^p$ estimates we apply the multi frequency Calder\'on Zygmund decomposition.