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The representation of functions (or objects which are in some generalize the notion of function) as constant linear combinations of sines and cosines at integer multiples of a given frequency, as Fourier transforms or as Fourier integrals.

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an analogue of Littlewood-Paley-Rubio de Francia theory

Regarding the classic Littlewood-Paley, there is an analogue. You can look it up e.g. in "Littlewood-Paley and Multiplier Theory" by R. E. Edwards, G. I. Gaudry. Rubio de Francia's result most proba …
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