Hardy spaces on the plane and double Fourier transforms.
type: Item
instance of: scholarly article
title: Hardy spaces on the plane and double Fourier transforms.
zbMATH Open document ID: 1055.42503
author: Dang Vu Giang
author: Ferenc Móricz
published in: The Journal of Fourier Analysis and Applications
publication date: 1998-05-14T00:00:00Z
full work available at URL: https://eudml.org/doc/59493
review text: In this paper, very accurate consideration of the interplay between \(H(\mathbf R^2)\), the space of integrable functions with integrable Riesz transforms, \[ H(\mathbf R^2\times \mathbf R^2)=\{f\in L(\mathbf R^2): H_1f,H_2f,H_1H_2f\in L(\mathbf R^2)\}, \] where \(H_jf\) is the Hilbert transform with respect to the \(j\)th variable, and \[ J(\mathbf R^2\times \mathbf R^2)=\{f\in L(\mathbf R^2): H_1f,H_2f\in L(\mathbf R^2)\}, \] is given. Theorem 2, the main result of this paper, gives simple sufficient conditions for a function of two variables to be the double Fourier transform of a function in \(L(\mathbf R^2)\) or \(H(\mathbf R^2\times \mathbf R^2)\). The author gives an extension to double Fourier series. For two open problems, conjectures are formulated. Note that a result analogous to Theorem 2 was proved in a paper by the reviewer [Anal. Math. 19, No. 2, 151--168 (1993; Zbl 0794.42006)], not only in the one-dimensional case but for any dimension.
Mathematics Subject Classification ID: 42B30
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 42B10
Mathematics Subject Classification ID: 44A15
zbMATH DE Number: 1153121
zbMATH Keywords: Riesz transform
zbMATH Keywords: Hilbert transform
zbMATH Keywords: double Fourier transform
reviewed by: Elijah Liflyand
MaRDI profile type: Publication
OpenAlex ID: W2070140288
DOI: 10.1007/S00041-001-4040-5
description: scientific article; zbMATH DE number 1153121