Pointwise localization and sharp weighted bounds for Rubio de Francia square functions

TL;DR

This paper establishes pointwise bounds and sharp weighted inequalities for Rubio de Francia square functions, advancing understanding of their boundedness properties and providing new results for weights in harmonic analysis.

Contribution

It introduces novel pointwise localization bounds for Rubio de Francia square functions using sparse operators, leading to sharp weighted norm inequalities and partial verification of a key conjecture.

Findings

Pointwise bounds for $T^$ by sparse operators

Sharp weak $L^p(w)$ bounds for $p>2$

Verification of the conjecture for specific weights and Walsh group case

Abstract

Let $H_\omega f$ be the Fourier restriction of $f\in L^2(\mathbb{R})$ to an interval $\omega\subset \mathbb{R}$. If $\Omega$ is an arbitrary collection of pairwise disjoint intervals, the square function of $\{H_\omega f: \omega \in \Omega\}$ is termed the Rubio de Francia square function $T^\Omega$. This article proves a pointwise bound for $T^\Omega f$ by a sparse operator involving local $L^2$-averages. A pointwise bound for the smooth version of $T^\Omega$ by a sparse square function is also proved. These pointwise localization principles lead to quantified $L^p(w)$, $p>2$ and weak $L^p(w)$, $p\geq 2$ norm inequalities for $T^\Omega$. In particular, the obtained weak $L^p(w)$ norm bounds are new for $p\geq 2$ and sharp for $p>2$. The proofs rely on sparse bounds for abstract balayages of Carleson sequences, local orthogonality and very elementary time-frequency analysis techniques. The paper also contains two results related to the outstanding conjecture that $T^\Omega$ is bounded on $L^2(w)$ if and only if $w\in A_1$. The conjecture is verified for radially decreasing even $A_1$ weights, and in full generality for the Walsh group analogue of $T^\Omega$.