Geodesic
New correction theorems in the light of a weighted Littlewood–Paley–Rubio de Francia inequality
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 39, Tome 389 (2011), pp. 232-251 
D. M. Stolyarov. New correction theorems in the light of a weighted Littlewood–Paley–Rubio de Francia inequality. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 39, Tome 389 (2011), pp. 232-251. http://geodesic.mathdoc.fr/item/ZNSL_2011_389_a12/
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     author = {D. M. Stolyarov},
     title = {New correction theorems in the light of a~weighted {Littlewood{\textendash}Paley{\textendash}Rubio} de {Francia} inequality},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {232--251},
     year = {2011},
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     url = {http://geodesic.mathdoc.fr/item/ZNSL_2011_389_a12/}
}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

We prove the following correction theorem: every function $f$ on the circumference $\mathbb T$ that is bounded by an $\alpha_1$-weight $w$ (this means that $Mw^2\le Cw^2$) can be modified on a set $e$ with $\int_ew<\varepsilon$ so that the quadratic function built up from $f$ with the help of an arbitary sequence of nonintersecting intervals in $\mathbb Z$ will not exceed $C\log(\frac1\varepsilon)w$.

[1] E. M. Stein, Harmonic analysis: real-variable methods, orthogonality and oscillatory integrals, Princeton University Press, 1993 | MR | Zbl

[2] S. V. Kislyakov,, “Teorema Litlvuda–Peli dlya proizvolnykh intervalov: vesovye otsenki”, Zap. nauchn. semin. POMI, 355, 2008, 180–198 | MR | Zbl

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

[4] D. S. Anisimov, S. V. Kislyakov, “Dvoinye singulyarnye integraly: interpolyatsiya i ispravlenie”, Algebra i Analiz, 16:5 (2004), 1–33 | MR | Zbl

[5] S. V. Kisliakov, “A sharp correction theorem”, Studia Mathematica, 113:2 (1995), 177–196 | MR | Zbl

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

[7] S. V. Kisliakov, “Interpolation of $H^p$-spaces: some recent developments”, Function spaces, interpolation spaces, and related topics, Israel Math. Conference Proceedings, 13, 1999, 102–140 | MR | Zbl

[8] I. Berg, I. Lefstrem, Interpolyatsionnye prostranstva. Vvedenie, Mir, 1980 | MR

[9] K. Iosida, Funktsionalnyi analiz, LKI, 2010

[10] S. V. Kislyakov, “Kolichestvennyi aspekt teorem ob ispravlenii”, Zap. nauchn. semin. LOMI, 92, 1979, 182–191 | MR | Zbl

[11] D. E. Menshov, “Ob ravnomernoi skhodimosti ryadov Fure”, Matem. sb., 11(53):1–2 (1942), 67–96 | MR | Zbl