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

Abstract

Let $H_ωf$ be the Fourier restriction of $f\in L^2(\mathbb{R})$ to an interval $ω\subset \mathbb{R}$. If $Ω$ is an arbitrary collection of pairwise disjoint intervals, the square function of $\{H_ωf: ω\in Ω\}$ is termed the Rubio de Francia square function $T^Ω$. This article proves a pointwise bound for $T^Ωf$ by a sparse operator involving local $L^2$-averages. A pointwise bound for the smooth version of $T^Ω$ 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 $T^Ω$. 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 $T^Ω$ 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 $T^Ω$.