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

Let Hωf be the Fourier restriction of f ∈ L2 (R) to an interval ω ⊂ R. If Ω is an arbitrary collection of pairwise disjoint intervals, the square function of {Hωf : ω ∈ Ω} is termed the Rubio de Francia square function T Ω RF. This article proves a pointwise bound for T
Ω RF by a sparse operator involving local L2 -averages. A pointwise bound for the smooth version of T Ω RF by a sparse square function is also proved. These pointwise localization principles lead to quantified Lp(w), p > 2, and weak Lp(w), p ≥ 2, norm inequalities for T Ω RF. In particular, the obtained weak Lp(w)-norm bounds are new for p ≥ 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 RF is bounded on L2 (w) if and only if w ∈ A1. The conjecture is verified for radially decreasing even A1-weights, and in full generality for the Walsh group analogue of T Ω RF.