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

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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 $C\log(\frac1\varepsilon)w$. 
@article{ZNSL_2011_389_a12,
     author = {D. M. Stolyarov},
     title = {New correction theorems in the light of a~weighted {Littlewood--Paley--Rubio} de {Francia} inequality},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {232--251},
     publisher = {mathdoc},
     volume = {389},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2011_389_a12/}
}
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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/