Some remarks on Banach spaces in which martingale difference sequences are unconditional
type: Item
instance of: scholarly article
title: Some remarks on Banach spaces in which martingale difference sequences are unconditional
zbMATH Open document ID: 0533.46008
DOI: 10.1007/BF02384306
published in: Arkiv för Matematik
publication date: 1983-01-01T00:00:00Z
review text: It is proved that the boundness of the vector-valued Hilbert-transform on the circle \({\mathbb{T}}\), \(H\otimes Id_ X\) on the space \(L^ p\!_ X({\mathbb{T}})\) implies that the Banach space X has the so called UMD- property, which means that X-valued martingale difference sequences are unconditional in \(L^ p\!_ X (1<p<\infty)\). This result, providing the converse of a theorem due to D. Burkholder ad T. McConnell, is already of interest in the scalar case, since it deduces Paley's theorem from the M. Riesz theorem on conjugate functions. In the second part of the paper, the problem whether or not superreflexive lattices need to have the UMD- property is solved (by constructing a counterexample).
Mathematics Subject Classification ID: 46B42
Mathematics Subject Classification ID: 46B10
Mathematics Subject Classification ID: 60G46
Mathematics Subject Classification ID: 46E40
Mathematics Subject Classification ID: 46B15
Mathematics Subject Classification ID: 42A50
zbMATH DE Number: 3846212
zbMATH Keywords: boundness of the vector-valued Hilbert-transform
zbMATH Keywords: UMD-property
zbMATH Keywords: martingale difference sequences
zbMATH Keywords: superreflexive lattices
author: Jean Bourgain
MaRDI profile type: Publication
cites work: Unconditional bases and martingales in LP(F)
cites work: A geometrical characterization of Banach spaces in which martingale difference sequences are unconditional
cites work: Q3924889
cites work: Super-Reflexive Banach Spaces
cites work: Q4188284
cites work: Q4076339
cites work: Martingales with values in uniformly convex spaces
cites work: Some applications of the complex interpolation method to Banach lattices
description: scientific article; zbMATH DE number 3846212