Maximal functions and Fourier transforms
type: Item
instance of: scholarly article
title: Maximal functions and Fourier transforms
zbMATH Open document ID: 0612.42008
DOI: 10.1215/S0012-7094-86-05324-X
author: José L. Rubio de Francia
published in: Duke Mathematical Journal
publication date: 1986-01-01T00:00:00Z
review text: The aim of the paper is to obtain sharp \(L^ p\) bounds for maximal operators \(T^*f(x)=\sup_{t>0}| T_ tf(x)|,\) \((T_ tf){\hat{\;}}(\xi)=\hat f(\xi)m(t\xi)\) (where \(m\in L^{\infty}({\mathbb{R}}^ n)\) is a given multiplier) in terms only of the decay at infinity of m(\(\xi)\). In general, if \(| m(\xi)| \leq C| \xi |^{-a}\) and \(a>1/2\), it is proved that \(T^*\) is bounded in \(L^ p({\mathbb{R}}^ n)\) for \(2n/(n+2a-1)<p<(2n-2)/(n-2a).\) Moreover, if m is the Fourier transform of a compactly supported measure, then the range is \(p>(2a+1)/2a\). This covers \textit{E. M. Stein's} theorem on maximal spherical means [Proc. Nat. Acad. Sci. USA 73, 2174-2175 (1976; Zbl 0332.42018)] and extensions of it due to Greenleaf and Sogge- Stein. The proofs are rather elementary and based on \(L^ 2\) methods (g- functions) and vector valued Calderón-Zygmund theory.
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 42B15
Mathematics Subject Classification ID: 42A38
zbMATH DE Number: 3990042
zbMATH Keywords: L\({}^ p\) bounds
zbMATH Keywords: maximal operators
zbMATH Keywords: maximal spherical means
zbMATH Keywords: Calderón-Zygmund theory
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3990042