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

@article{Nazarov2009ACZ,
  title={A Calderon Zygmund decomposition for multiple frequencies and an application to an extension of a lemma of Bourgain},
  author={Fedor Nazarov and Richard Oberlin and Christoph Thiele},
  journal={arXiv: Classical Analysis and ODEs},
  year={2009}
}
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. 

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