Some generalizations of the Hardy-Littlewood-Paley theorem
type: Item
instance of: scholarly article
title: Some generalizations of the Hardy-Littlewood-Paley theorem
zbMATH Open document ID: 0791.42019
DOI: 10.1007/BF01206385
published in: Mathematical Notes
publication date: 1994-02-08T00:00:00Z
review text: The Hardy-Littlewood-Paley theorem on an estimate for the norm of a function in terms of its Fourier coefficients in an orthonormal system is generalized in three directions. Firstly, an estimate of the \(L^ q\)- norm of a function or the majorant of partial sums of its transposed series in terms of its Fourier coefficients and the \(L^ r\)-norms \((2< q<r\leq\infty)\) of operators of partial sums in an orthonormal system is given. Secondly, several analogous bounds for Orlicz classes \(\varphi(L)\) in the case of orthonormal systems of bounded functions are proved. Thirdly, necessary and sufficient conditions for imbedding into \(\varphi(L)\) of classes of functions \(L^ p(\varepsilon)\) with a given majorant of best approximations in \(L^ p[0;2\pi]\) in the trigonometric system.
Mathematics Subject Classification ID: 42C15
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 484258
zbMATH Keywords: Hardy-Littlewood-Paley theorem
zbMATH Keywords: Fourier coefficients
zbMATH Keywords: orthonormal system
zbMATH Keywords: Orlicz classes
zbMATH Keywords: imbedding
zbMATH Keywords: best approximations
zbMATH Keywords: trigonometric system
author: Viktor I. Kolyada
reviewed by: Alexei Lukashov
MaRDI profile type: Publication
cites work: Q5731913
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cites work: Concerning the Generalization of the Young-Hausdorff Theorem and the Hardy-Littlewood Theorems on Fourier Constants
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full work available at URL: https://doi.org/10.1007/bf01206385
OpenAlex ID: W1975739506
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description: scientific article; zbMATH DE number 484258