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

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 $L^p(w)$, $p\ge 2$ norm inequalities for $T^{\Omega }$. In particular, the obtained weak $L^p(w)$ norm bounds are new for $p\ge 2$ 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 $T^{\Omega }$.

28 pages, final version incorporates the comments of the referees; to appear in Publ. Mat