Geodesic
Littlewood–Paley theorem for arbitrary intervals: weighted estimates
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 36, Tome 355 (2008), pp. 180-198 
S. V. Kislyakov. Littlewood–Paley theorem for arbitrary intervals: weighted estimates. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 36, Tome 355 (2008), pp. 180-198. http://geodesic.mathdoc.fr/item/ZNSL_2008_355_a7/
@article{ZNSL_2008_355_a7,
     author = {S. V. Kislyakov},
     title = {Littlewood{\textendash}Paley theorem for arbitrary intervals: weighted estimates},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {180--198},
     year = {2008},
     volume = {355},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2008_355_a7/}
}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

Suppose $1 and $b$ is a weight on $\mathbb R$ such that $b^{-\frac1{r-1}}$ satisfies the Muckenhoupt condition $A_{r'/2}$ ($r'$ is the exponent conjugate to $r$). If $f_j$ are functions whose Fourier transforms are supported on mutually disjoint intervals, then $$ \Bigl\Vert\sum_j f_j\Bigr\Vert_{L^p(\mathbb R,b)}\le C\Bigl\Vert\Bigl(\sum_j|f_j|^2\Bigr)^{1/2}\Bigr\Vert_{L^p(\mathbb R,b)} $$ for $0. Bibl. – 9 titles.

[1] J. L. Rubio de Francia, “A Littlewood–Paley inequality for arbitrary intervals”, Rev. Mat. Iberoamer., 1 (1985), 1–13 | MR

[2] R. R. Coifman, C. Fefferman, “Weighted norm inequalities for maximal functions and singular integrals”, Studia Math., 51 (1974), 241–250 | MR | Zbl

[3] J. Garcia-Cuerva, J. L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland, Amsterdam, 1985 | MR | Zbl

[4] J. Bourgain, “On square functions on the trigonometric system”, Bull. Soc. Math. Belg., 37:1 (1985), 20–26 | MR | Zbl

[5] S. V. Kislyakov, D. V. Parilov, “O teoreme Litlvuda–Peli dlya proizvolnykh intervalov”, Zap. nauchn. semin. POMI, 327, POMI, SPb., 2005, 98–114 | MR | Zbl

[6] J. Garcia-Cuerva, Weighted $H^p$-spaces, Dissertationes Math. (Rozprawy Mat.), 162, 1979, 63 pp. | MR | Zbl

[7] J.-O. Strömberg, A. Torchinsky, Weighted Hardy spaces, Lect. Notes Math., 1381, Springer-Verlag, Berlin etc., 1980 | MR

[8] S. V. Kislyakov, “Interpolation of $H^p$-spaces: some recent developments”, Function spaces, interpolation spaces, and related topics (Haifa, 1995), Isr. Math. Conf. Proc., 13, Bar-Ilan Univ., Ramat Gan, 1999, 102–140 | MR | Zbl

[9] A. Zigmund, Trigonometricheskie ryady, T. 1, Mir, M., 1965 | MR