Geodesic
On the vector-valued extension of Littlewood--Paley--Rubio de Francia inequality for Walsh functions
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 49, Tome 503 (2021), pp. 137-153

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In the case of trigonometric system, Rubio de Francia proved the one-sided Littlewood–Paley inequality for arbitrary intervals and for the functions in the $L^p$ spaces, $2\le p$. 
@article{ZNSL_2021_503_a8,
     author = {A. Tselishchev},
     title = {On the vector-valued extension of {Littlewood--Paley--Rubio} de {Francia} inequality for {Walsh} functions},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {137--153},
     publisher = {mathdoc},
     volume = {503},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2021_503_a8/}
}
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A. Tselishchev. On the vector-valued extension of Littlewood--Paley--Rubio de Francia inequality for Walsh functions. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 49, Tome 503 (2021), pp. 137-153. http://geodesic.mathdoc.fr/item/ZNSL_2021_503_a8/