A note on Littlewood-Paley decompositions with arbitrary intervals
type: Item
instance of: scholarly article
title: A note on Littlewood-Paley decompositions with arbitrary intervals
zbMATH Open document ID: 0606.42013
DOI: 10.1016/0021-9045(86)90055-9
published in: Journal of Approximation Theory
publication date: 1986-01-01T00:00:00Z
review text: For every interval I in \({\mathbb{R}}\) define an operator \(S_ I\) by setting \((S_ If)^{\wedge}=\chi_ I\hat f\), where \(\hat f\) denotes the Fourier transform of f. Let \((I_ k)_ 1^{\infty}\) be a sequence of disjoint intervals and set \(\Delta f(x)=(\sum^{\infty}_{1}| S_{I_ k}f(x)|^ 2)^{1/2}.\) \textit{J. L. Rubio de Francia} has proved that \(\Delta\) is a bounded operator on \(L^ p({\mathbb{R}})\) for \(2\leq p<\infty\) [Report No.18, Institut Mittag-Leffler (1983)]. This paper contains an alternative proof of the basic inequality in the proof of the above theorem.
Mathematics Subject Classification ID: 42A45
Mathematics Subject Classification ID: 42A50
zbMATH DE Number: 3979722
zbMATH Keywords: Littlewood-Paley decompositions
zbMATH Keywords: maximal functions
zbMATH Keywords: Fourier transform
author: Per Sjölin
MaRDI profile type: Publication
full work available at URL: 0021-9045(86)90055-9
OpenAlex ID: W2075257560
cites work: \(H^p\) spaces of several variables
cites work: A Littlewood-Paley inequality for arbitrary intervals
description: scientific article; zbMATH DE number 3979722