Cesàro summability of two-parameter trigonometric-Fourier series
type: Item
instance of: scholarly article
title: Cesàro summability of two-parameter trigonometric-Fourier series
zbMATH Open document ID: 0878.42007
author: Ferenc Weisz
published in: Journal of Approximation Theory
publication date: 1998-01-08T00:00:00Z
review text: Denote by \(\sigma^*f\) the maximal operator of the unrestricted Cesàro means of the double Fourier series of a distribution \(f\) on \(\mathbb{T}^2\), where \(\mathbb{T}:=[-\pi,\pi)\). The author proves that \(\sigma^*\) is bounded from the Hardy-Lorentz space \(H_{p,q}(\mathbb{T}\times\mathbb{T})\) into the Lorentz space \(L_{p,q}(\mathbb{T}^2)\), provided \(3/4<p\leq\infty\) and \(0<q\leq \infty\). Reviewer's remark: The special case that \(\sigma^*\) is bounded from the hybrid Hardy space \(H^\#_1(\mathbb{T}\times\mathbb{T})\) into weak-\(L^1(\mathbb{T}^2)\) as well as its consequence that the Cesàro means of a function \(f\in H^\#_1(\mathbb{T}\times\mathbb{T})\supset L\log^+L(\mathbb{T}^2)\) converge unrestrictedly to \(f\) almost everywhere, was proved by the reviewer [Stud. Math. 116, No. 1, 89-100 (1995; Zbl 0842.47020)].
reviewed by: Ferenc Móricz
Mathematics Subject Classification ID: 42B08
zbMATH DE Number: 1042512
zbMATH Keywords: maximal operator
zbMATH Keywords: Cesàro means
zbMATH Keywords: double Fourier series
MaRDI profile type: Publication
full work available at URL: jath.1996.3070
OpenAlex ID: W2050860354
cites work: Q4039801
cites work: Q4112938
cites work: A Maximal Function Characterization of the Class H p
cites work: Some recent developments in Fourier analysis and 𝐻^{𝑝}-theory on product domains
cites work: Extensions of Hardy spaces and their use in analysis
cites work: Q5618011
cites work: Q3733431
cites work: \(H^p\) spaces of several variables
cites work: Calderón-Zygmund theory for product domains: H p spaces
cites work: Q3864168
cites work: H p theory for the poly-disc
cites work: Interpolation between Hardy spaces on the bidisc
cites work: Cesaro Summability of Double Walsh-Fourier Series
cites work: Q4120988
cites work: Q5607484
cites work: Cesàro summability of multi-dimensional trigonometric-Fourier series
cites work: Cesàro summability of one- and two-dimensional trigonometric-Fourier series
cites work: Cesàro summability of two-parameter Walsh-Fourier series
cites work: Martingale Hardy spaces and their applications in Fourier analysis
cites work: On the atomic decomposition for Hardy spaces
DOI: 10.1006/JATH.1996.3070
description: scientific article; zbMATH DE number 1042512