A generalization for Fourier transforms of a theorem due to Marcinkiewicz
type: Item
instance of: scholarly article
title: A generalization for Fourier transforms of a theorem due to Marcinkiewicz
zbMATH Open document ID: 0976.42006
author: Ferenc Weisz
published in: Journal of Mathematical Analysis and Applications
publication date: 2001-12-28T00:00:00Z
review text: The author defines properly the Marcinkiewicz means of the Fourier transforms of tempered distributions and proves that its maximal operator is bounded from the Hardy-Lorentz space \(H_{p,q}({\mathbb{R}}^2)\) to the Lorentz space \(L_{p,q}({\mathbb{R}}^2)\) for all \(p_0<p\leq\infty\) and \(0<q\leq \infty,\) where \(p_0<1\) depends on a parameter in the definition of the means, and is also of weak type \((1,1).\) As a consequence, the author proves that the Marcinkiewicz means of a function \(f\in L_1({\mathbb{R}}^2)\) converge a.e. to the function in question, which is a generalization of a summability result due to Marcinkiewicz and to Zhizhiashvili. Then the author shows that the Marcinkiewicz means are uniformly bounded on the Hardy-Lorentz space \(H_{p,q}({\mathbb{R}}^2)\) for the same \(p\) and \(q,\) and so they converge in the \(H_{p,q}({\mathbb{R}}^2)\) norm. Similar results for the Riesz transforms are also given. The author uses the same techniques as in his previous papers.
reviewed by: Elijah Liflyand
Mathematics Subject Classification ID: 42B10
Mathematics Subject Classification ID: 46F12
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 42B30
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 1563086
zbMATH Keywords: Hardy-Lorentz space
zbMATH Keywords: Marcinkiewicz means
zbMATH Keywords: Riesz transform
zbMATH Keywords: Fourier transform
zbMATH Keywords: maximal operator
zbMATH Keywords: atomic decomposition
zbMATH Keywords: tempered distribution
MaRDI profile type: Publication
full work available at URL: jmaa.2000.7094
OpenAlex ID: W1972862152
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DOI: 10.1006/JMAA.2000.7094
description: scientific article; zbMATH DE number 1563086