On the vector-valued Littlewood-Paley-Rubio de Francia inequality
type: Item
instance of: scholarly article
title: On the vector-valued Littlewood-Paley-Rubio de Francia inequality
zbMATH Open document ID: 1253.46011
DOI: 10.4171/RMI/693
author: Denis Potapov
author: Quanhua Xu
author: Fedor Sukochev
published in: Revista Matemática Iberoamericana
publication date: 2012-09-06T00:00:00Z
full work available at URL: https://arxiv.org/abs/1104.2671
reviewed by: Oscar Blasco
Mathematics Subject Classification ID: 46B07
Mathematics Subject Classification ID: 46B42
Mathematics Subject Classification ID: 42B25
zbMATH DE Number: 6078125
zbMATH Keywords: Littlewood-Paley-Rubio de Francia inequality
zbMATH Keywords: UMD space
zbMATH Keywords: Banach lattices
MaRDI profile type: Publication
arXiv ID: 1104.2671
cites work: Interpolation between vector-valued Hardy spaces
cites work: Q3737987
cites work: Q4487402
cites work: The Littlewood—Paley—Rubio de Francia property of a Banach space for the case of equal intervals
review text: The authors consider Banach spaces \(X\) having the property that NEWLINE\[NEWLINE\left(\int_0^1 \int_{\mathbb R} \|\sum_j r_j(t) S_{I_j} f(x)\|^p dt dx\right)^{1/p}\leq C\left(\int_{\mathbb R} \|f(x)\|^pdx\right)^{1/p}NEWLINE\]NEWLINE for some \(2\leq p<\infty\), where \(r_j\) are the Rademacher functions and \(S_I f(x)= \int_I \hat f(\xi)e^{i\xi x}d\xi\) for an interval \(I\) and where \(I_j\) is a collection of pairwise disjoint intervals in \(\mathbb R\).NEWLINENEWLINESuch a property is called the Littlewood-Paley-Rubio de Francia property, in short \(LPR_p\) (due to the extension of the Littlewood-Paley result to general intervals due to \textit{J. L. Rubio de Francia} [Rev. Mat. Iberoam. 1, No. 2, 1--14 (1985; Zbl 0611.42005)]. It is known (see the paper by \textit{T. P. Hytönen}, \textit{J. L. Torrea} and \textit{D. V. Yakubovich} [Proc. R. Soc. Edinb., Sect. A, Math. 139, No. 4, 819--832 (2009; Zbl 1173.42010)]) that this property implies the UMD property and type 2. In the present paper the authors show that for Banach lattices if the \(2\)-concavification of \(X\), denoted \(X_{(2)}\), is a UMD Banach lattice then \(X\) has the \(LPR_p\) property. Although it is still not known whether the property is independent of \(p\) it is proved by the authors that \(LPR_p\) implies \(LPR_q\) for any \(q\geq p\).
description: scientific article; zbMATH DE number 6078125