On rearrangements of the Haar system in the space BMO
type: Item
instance of: scholarly article
title: On rearrangements of the Haar system in the space BMO
zbMATH Open document ID: 0836.42018
DOI: 10.1007/BF01904273
author: D. Z. Otyrba
published in: Analysis Mathematica
publication date: 1996-04-23T00:00:00Z
review text: Let \(A\) be the set of the dyadic intervals of \([0,1)\) including the empty interval \(\emptyset\) and consider the Haar system \(\chi : = \{h_I : I \in A\}\). Every bijection \(\Pi : A \to A\) generates an operator \(R_\Pi\) defined by \[ R_\Pi f : = \sum_{I \in A} \widehat f_I h_{\Pi (I)}, \quad \widehat f_I : = \int^1_0 f(t)h_I (t)dt. \] Denote by \(H\) the dyadic Hardy space, and by BMO the dyadic space dual to \(H\). The author proves that if \(\Pi : A \to A\) is a bijection and \(\Pi (\emptyset) = \emptyset\), then \[ |R_\Pi |_H \sim |R_{\Pi^{-1}} |_{BMO}. \] This improves and generalizes an earlier result by \textit{F. Schipp} [Anal. Math. 16, No. 2, 135-141 (1990; Zbl 0736.42019)].
Mathematics Subject Classification ID: 42C10
Mathematics Subject Classification ID: 42B30
zbMATH DE Number: 808264
zbMATH Keywords: rearrangement
zbMATH Keywords: bounded mean oscillation
zbMATH Keywords: Haar system
zbMATH Keywords: dyadic Hardy space
zbMATH Keywords: BMO
reviewed by: Ferenc Móricz
MaRDI profile type: Publication
cites work: On equivalence of rearrangements of the Haar system in dyadic Hardy and BMO spaces
cites work: Rearrangements of the Haar system in \(L_ p-\)spaces
description: scientific article; zbMATH DE number 808264