Geodesic
On the Littlewood--Paley theorem for arbitrary intervals
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 33, Tome 327 (2005), pp. 98-114

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We extend the results of Rubio de Francia [1] and Bourgain [2] by showing that for arbitrary mutually nonintersecting intervals $\Delta_k\subset\mathbb Z_+$, arbitrary $p\in(0,2]$, and arbitrary trigonometric polynomials $f_k$ with $\mathrm{supp}\,\widehat f_k\subset\Delta_k$, we have $$ \biggl\|\sum_k f_k\biggr\|_{H^p(\mathbb T)}\le a_p\biggl\|\biggl(\sum_k|f_k|^2\biggr)^{1/2}\biggr\|_{L^p(\mathbb T)}. $$ The method is a development of that by Rubio de Francia. 
@article{ZNSL_2005_327_a6,
     author = {S. V. Kislyakov and D. V. Parilov},
     title = {On the {Littlewood--Paley} theorem for arbitrary intervals},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {98--114},
     publisher = {mathdoc},
     volume = {327},
     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2005_327_a6/}
}
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S. V. Kislyakov; D. V. Parilov. On the Littlewood--Paley theorem for arbitrary intervals. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 33, Tome 327 (2005), pp. 98-114. http://geodesic.mathdoc.fr/item/ZNSL_2005_327_a6/