Quantitative weighted estimates for Rubio de Francia's Littlewood--Paley square function

TL;DR

This paper establishes sharp quantitative weighted estimates for Rubio de Francia's Littlewood--Paley square function, extending understanding of its behavior with respect to weights and providing endpoint weak-type bounds.

Contribution

It provides the first sharp linear dependence on the weight characteristic for this square function in the range p ≥ 3, and develops sparse domination techniques for these operators.

Findings

Linear dependence on [w]_{A_{p/2}} is sharp for p ≥ 3.

Weighted weak-type estimates are obtained at p=2.

Sparse domination is established for the square function.

Abstract

We consider the Rubio de Francia's Littlewood--Paley square function associated with an arbitrary family of intervals in $\mathbb{R}$ with finite overlapping. Quantitative weighted estimates are obtained for this operator. The linear dependence on the characteristic of the weight $[w]_{A_{p/2}}$ turns out to be sharp for $3\le p<\infty$, whereas the sharpness in the range $2<p<3$ remains as an open question. Weighted weak-type estimates in the endpoint $p=2$ are also provided. The results arise as a consequence of a sparse domination shown for these operators, obtained by suitably adapting the ideas coming from Benea (2015) and Culiuc et al. (2016).