An extension of the Córdoba-Fefferman theorem on the equivalence between the boundedness of certain classes of maximal and multiplier operators
type: Item
instance of: scholarly article
title: An extension of the Córdoba-Fefferman theorem on the equivalence between the boundedness of certain classes of maximal and multiplier operators
zbMATH Open document ID: 1178.42019
author: Alexander Stokolos
author: Paul Hagelstein
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2008-11-06T00:00:00Z
review text: Let the geometric maximal operator \(M_\theta f(x)=\sup_{x\in R}\frac{1}{|R|}\int_R|f(y)|\,dy\) and the multiplier operator \(T_\theta\), \(\widehat{T_\theta f}(\xi)=\chi_{P_\theta}(\xi)\widehat f(\xi)\), where \(\theta_1>\theta_1>\dots\) is a decreasing sequence of angles between 0 and \(\pi/2\), and the supremum in the first formula extends over the collection of rectangle in the plane of arbitrary eccenfrity oriented in one of the directions \(\theta_i\). Extending a result of \textit{A. Córdoba} and \textit{R. Fefferman} [Proc. Natl. Acad. Sci. USA 74, 423--425 (1977; Zbl 0342.42003)], there is proved the following theorem: (a) if \(M_\theta\) is bounded on \(L^p(\mathbb{R}^2)\) for some \(1<p<\infty\), then \(M_\theta\) and \(T_\theta\) are bounded on \(L^q(\mathbb{R}^2)\) for all \(1<q<\infty\), (b) if \(M_\theta\) satisfies the tauberian condition \(|\{x:M_\theta\chi_E(x)>1/2\}\leq C|E|\), then \(M_\theta\) and \(T_\theta\) are bounded on \(L^q(\mathbb{R}^2)\) for all \(1<q<\infty\).
reviewed by: Julian Musielak
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 42B15
zbMATH DE Number: 5362117
zbMATH Keywords: harmonic analysis
zbMATH Keywords: geometric maximal operator
zbMATH Keywords: multiplier operator
MaRDI profile type: Publication
full work available at URL: j.crma.2008.09.010
OpenAlex ID: W2020415452
cites work: Kakeya sets and directional maximal operators in the plane
cites work: Kakeya sets in Cantor directions
cites work: On the equivalence between the boundedness of certain classes of maximal and multiplier operators in Fourier analysis
cites work: Tauberian conditions for geometric maximal operators
cites work: Q5607484
DOI: 10.1016/J.CRMA.2008.09.010
description: scientific article; zbMATH DE number 5362117