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

• Di Plinio, Francesco ^[1] ; Flórez-Amatriain, Mikel ^[2] ; Parissis, Ioannis ^[3] ; Roncal, Luz ^[2]
  1. [1] Università degli Studi di Napoli Parthenope

     Università degli Studi di Napoli Parthenope

     Nápoles, Italia

     
  2. [2] Basque Center for Applied Mathematics

     Basque Center for Applied Mathematics

     Bilbao, España

     
  3. [3] Universidad del País Vasco/Euskal Herriko Unibertsitatea

     Universidad del País Vasco/Euskal Herriko Unibertsitatea

     Leioa, España

     

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• Localización: Publicacions matematiques, ISSN 0214-1493, Vol. 69, Nº 2, 2025, págs. 445-471
• Idioma: inglés
• DOI: 10.5565/publmat6922510
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• Resumen

  • 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.

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