Martingale Orlicz-Hardy spaces
type: Item
instance of: scholarly article
title: Martingale Orlicz-Hardy spaces
zbMATH Open document ID: 1260.60082
author: Takashi Miyamoto
author: Eiichi Nakai
author: Gaku Sadasue
published in: Mathematische Nachrichten
publication date: 2012-05-14T00:00:00Z
Mathematics Subject Classification ID: 60G46
Mathematics Subject Classification ID: 42B30
Mathematics Subject Classification ID: 42B35
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 6034382
zbMATH Keywords: Martingale
zbMATH Keywords: Orlicz-Hardy space
zbMATH Keywords: atomic decomposition
zbMATH Keywords: Hardy space
zbMATH Keywords: Campanato space
zbMATH Keywords: bounded mean oscillation
MaRDI profile type: Publication
full work available at URL: mana.201000109
OpenAlex ID: W1970708208
cites work: Extrapolation and interpolation of quasi-linear operators on martingales
cites work: Boundedness and convergence of martingales in rearrangement-invariant function spaces
cites work: Q5539962
cites work: Some Orlicz-norm inequalities for martingales
cites work: An atomic decomposition of the predual of BMO(\(\rho\) )
cites work: Martingale Hardy spaces for \(0<p\leq 1\)
cites work: Martingale Hardy spaces and their applications in Fourier analysis
cites work: \((C,\alpha)\) summability of Walsh-Fourier series
cites work: Atomic decompositions of Lorentz martingale spaces and applications.
DOI: 10.1002/MANA.201000109
review text: Suppose \(\left( \Omega, \mathcal{F}, P \right)\) is a probability space. Let \(\mathcal{M}\) be the set of all martingales \(f=\{f_n\}_{n\geqslant 0}\) relative to a non-decreasing sequence of \(\sigma\)-subalgebras \(\{\mathcal{F}_n\}_{n\geqslant 0}\) generating the \(\sigma\)-algebra \(\mathcal{F}\). For a martingale \(f \in \mathcal{M}\), the authors define the maximal function \(f^*\), the quadratic variation \(S(f)\), and the conditional quadratic variation \(s(f)\). Let \(L_\Phi\) be the Orlicz space associated with a non-convex Orlicz function \(\Phi\) satisfying certain additional conditions. Then, the authors use \(f^*\), \(S(f)\) and \(s(f)\) to define five martingale Hardy-Orlicz subspaces of \(\mathcal{M}\). Their results include an atomic decomposition for some of these Hardy-Orlicz spaces. In particular, if the sequence of \(\sigma\)-algebras \(\{\mathcal{F}_n\}_{n\geqslant 0}\) is a regular stochastic basis, then all these five spaces coincide, and so one gets an atomic decomposition for all five spaces. As another application of the atomic decomposition, the authors show that the dual of one of these spaces is a generalized martingale Campanato space \(\mathcal{L}_{2,\phi}\) with a certain relation between functions \(\Phi\) and \(\phi\). Finally, a John-Nirenberg-type inequality for \(\mathcal{L}_{2,\phi}\) is proved under the assumption that \(\{\mathcal{F}_n\}_{n\geq 0}\) is regular.
reviewed by: Maria Gordina
description: scientific article; zbMATH DE number 6034382