BMO estimates for lacunary series
type: Item
instance of: scholarly article
title: BMO estimates for lacunary series
zbMATH Open document ID: 0726.42007
DOI: 10.1007/BF02387383
author: Elizabeth Kochneff
author: Kecheng Zhou
author: Yoram Sagher
published in: Arkiv för Matematik
publication date: 1990-01-01T00:00:00Z
review text: Let \(0<p<\infty\) and \(f=\sum_{k}c_ k\psi_{n_ k}\) be a lacunary Walsh series or trigonometric series with \(C\equiv \{\sum_{k\neq 0}| c_ k|^ 2\}^{1/2}<\infty.\) It is known that if I is an interval sufficiently large then the local \(L^ p\) norm of f, averaged on I, is equivalent to C. Here the authors show that when \(p=\infty\), the \(L^ p\) norm must be replaced by the BMO norm (dyadic BMO in the Walsh case, classical BMO in the trigonometric case). Namely, they prove there are constants, A, B (which depend only on the lacunarity constant q) such that \(AC\leq \| f\|_{BMO}\leq BC.\) They give an example to show dyadic BMO cannot be replaced by BMO in the Walsh case. They also extend both the \(L^ p\) and the BMO inequalities to multiple Walsh series and multiple trigonometric series. The proofs are clear and well presented, decomposing the sums into managable pieces which are carefully estimated by classical methods.
Mathematics Subject Classification ID: 42A55
Mathematics Subject Classification ID: 42C10
Mathematics Subject Classification ID: 42B05
zbMATH DE Number: 4198746
zbMATH Keywords: lacunary trigonometric series
zbMATH Keywords: \(L^ p\) inequalities
zbMATH Keywords: lacunary Walsh series
zbMATH Keywords: \(L^ p\) norm
zbMATH Keywords: BMO norm
zbMATH Keywords: dyadic BMO
zbMATH Keywords: BMO inequalities
zbMATH Keywords: multiple Walsh series
zbMATH Keywords: multiple trigonometric series
reviewed by: William R. Wade
MaRDI profile type: Publication
cites work: On functions of bounded mean oscillation
cites work: Q3855655
cites work: Q4712842
full work available at URL: https://doi.org/10.1007/bf02387383
OpenAlex ID: W2009262898
description: scientific article; zbMATH DE number 4198746