New correction theorems in the light of a weighted Littlewood--Paley--Rubio de Francia inequality

Abstract

We prove the following correction theorem: every function $f$ on the circumference $\mathbb{T}$ that is bounded by the ${\alpha }_1$-weight $w$ (this means that $M{w}^2\le C{w}^2$) can be modified on a set $e$ with $\int\limits_{e} w \leq \eps$ so that its quadratic function built up from arbitary sequence of nonintersecting intervals in $\mathbb{Z}$ will not exceed $C \log \frac{1}{\eps} w$.