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We introduce a Calderon Zygmund decomposition such that the bad function hasvanishing integral against a number of pure frequencies. Then we prove avariation norm variant of a maximal inequality for several frequencies due toBourgain. To obtain the full range of Lp estimates we apply the multi frequencyCalderon Zygmund decomposition.

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