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

TL;DR

This paper establishes new correction theorems for functions on the circle, leveraging weighted Littlewood--Paley--Rubio de Francia inequalities to control quadratic functions with respect to weights.

Contribution

It introduces correction theorems that relate weighted bounds to modifications of functions on the circle, extending classical inequalities in harmonic analysis.

Findings

Functions bounded by specific weights can be corrected on small sets to control their quadratic functions.

The correction depends logarithmically on the measure of the correction set.

The results extend Littlewood--Paley theory to weighted settings.

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 $Mw^2 \leq C w^2$) 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 $C \log \frac{1}{\eps} w$.