A multidimensional analog of a theorem due to Zygmund
type: Item
instance of: scholarly article
title: A multidimensional analog of a theorem due to Zygmund
zbMATH Open document ID: 0915.26003
DOI: 10.1007/BF02355081
author: V. A. Okulov
published in: Mathematical Notes
publication date: 1999-07-07T00:00:00Z
review text: In [Prace Mat.-Fiz. 33, 125-132 (1924; JFM 52.0274.03)] \textit{A. Zygmund} proved that if \[ \int^\pi_0 {\omega(t)\over t} dt= \infty, \] then there exists a continuous function \(f\) such that \(\omega(f, \delta)= O(\omega(\delta))\) and the adjoint function \(\overline f\) is even unbounded. He also proved that if \(f\in C(T)\), then \[ \omega(\overline f, \delta)\leq A\Biggl\{ \int^\delta_0 {\omega(f,t)\over t} dt+ \delta\int^\pi_\delta {\omega(f,t)\over t^2} dt\Biggr\}.\tag{1} \] \textit{N. K. Bari} and \textit{S. B. Stechkin} [Trudy Moskovsk. Mat. Obsc. 5, 483-522 (1956; Zbl 0072.05702)] verified that the estimate given in (1) is asymptotically sharp. The author proves the multidimensional analogs of the above-stated results.
Mathematics Subject Classification ID: 26B05
Mathematics Subject Classification ID: 42B99
zbMATH DE Number: 1238048
zbMATH Keywords: modulus of continuity
zbMATH Keywords: theorem of Zygmund
zbMATH Keywords: asymptotically sharp estimate
zbMATH Keywords: JFM 52.0274.03
zbMATH Keywords: continuous function
zbMATH Keywords: adjoint function
zbMATH Keywords: multidimensional analogs
reviewed by: László Leindler
MaRDI profile type: Publication
cites work: Multidimensional analogue of a theorem of Privalov
cites work: Q3816504
cites work: Conjugate functions of several variables in the class Lip alpha
description: scientific article; zbMATH DE number 1238048