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For any function $f$ defined on the set of integer $\mathbb{Z}$, we define its Fourier transform as the following periodic function: $$ \mathbb{F}f(\xi)=\sum_{n\in\mathbb{Z}}f(n)e^{-2\pi i n\xi} $$ For any periodic function $g$ (i.e. $g$ is defined on $\mathbb{T}$), we define its inversion Fourier transform as the following function on $\mathbb{Z}$: $$ \mathbb{F}^{-1}g(n)=\int_0^1g(\xi)e^{2\pi i n\xi}d\xi $$

Note that the above definitions are opposite to the usual definitions in textbooks. I want to know if there is an analogue of Littlewood-Paley (or Rubio de Francia)theory. More precisely, for $j\in\mathbb{N}$, let $$ S_jf(n)=\mathbb{F}^{-1}(\mathbb{F}f(\cdot)\phi_j(\cdot))(n) $$ where $\phi_j$ is supported near $2^{-j}$, or more generally the supports of $\phi_j$'s form a partition of $[0,1]$. What can we say about $l^p$ boundedness of the operator $Tf(n)=(\sum_{j=1}^{\infty}|S_jf(n)|^2)^{1/2}$ ?

I noticed Bourgain's paper On square functions on the trigonometric system. The review of this paper in MathSciNet MR0847119 (87m:42008) indicates that a Littlewood-Paley-Rubio de Francia theorem holds for the usual Fourier series. Since the definitions I give are opposite to the usual definitions, I don't know if the corresponding theorem still holds.

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  1. 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.
  2. Rubio de Francia's result most probably can be transplanted as well. Gillespie and Torrea (in Transference of a Littlewood-Paley-Rubio inequality and dimension free estimates), mentioned that they were about to make it, but I haven't followed the progress in this direction. They made it for compact connected abelian groups though, confirming the Rubio de Francia's conjecture.
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  • $\begingroup$ Wow, the reference you provided is very useful. I will take a good look at them. Thank you. Feel free to add more reference if you happen to see some. $\endgroup$ – Tony B Jun 15 '15 at 18:38

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