Pointwise localization and sharp weighted bounds for Rubio de Francia square functions
Di Plinio, Francesco (Università di Napoli. Dipartimento di Matematica e Applicazioni)
Flórez Amatriain, Mikel (Basque Center for Applied Mathematics (BCAM))
Parissis, Ioannis (Universidad del País Vasco. Departamento de Matemáticas)
Roncal, Luz (Basque Center for Applied Mathematics (BCAM))

Date:	2025
Abstract:	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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Language:	Anglès
Document:	Article ; recerca ; Versió publicada
Subject:	Rubio de Francia square function ; Sparse domination ; Sparse square functions ; Exponential square integrability ; Sharp weighted bounds ; Time-frequency analysis ; Localization principles
Published in:	Publicacions matemàtiques, Vol. 69 Núm. 2 (2025) , p. 445-471 (Articles) , ISSN 2014-4350

Adreça original: https://raco.cat/index.php/PublicacionsMatematiques/article/view/10000006179
DOI: 10.5565/PUBLMAT6922510

27 p, 523.3 KB

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