Weighted integral inequalities in Orlicz martingale classes
type: Item
instance of: scholarly article
title: Weighted integral inequalities in Orlicz martingale classes
zbMATH Open document ID: 1242.60043
author: Wei Chen
author: Peide Liu
published in: Science China. Mathematics
publication date: 2012-01-16T00:00:00Z
review text: Let \(\phi\) be a positive nondecreasing right continuous function on \(\mathbb{R}^+\) with \(\phi(0+)= 0\) and \(\phi(s)\to\infty\) as \(s\to\infty\). Define the Young function \(\Phi\) by \(\Phi(t)= \int^t_0 \phi(s)\,ds\) an let \(q_\Phi= \text{inf}_{t>0} t\phi(t)/\Phi(t)\). A Young function \(\Phi\) is said to satisfy the condition \(\Delta_2)\) if, for some constant \(C\), \(\Phi(2t)\leq C\Phi(t)\), \(t> 0\). If \(\Phi_1\), \(\Phi_2\) are Young's functions. \(\Phi_1\ll \Phi_2\) means that, for some constant \(C\), \[ \sum\Phi_2\circ \Phi^{-1}_1(a_n)\leq C\Phi_2\circ\Phi^{-1}_1(\sum a_n) \] holds for every nonnegative sequence \((a_n)_n\). Let \((\Omega,{\mathcal F},\mu)\) be a complete probability space and let \({\mathcal F}_0,{\mathcal F}_1,\dots\) be a filtration such that \({\mathcal F}= \sigma({\mathcal F}_0\cup{\mathcal F}_1\cup\cdots)\). A weight \(\omega\) is a random variable such that \(\omega>0\) and \(\text{E}[\omega]<\infty\). Let \((f_n)\) \((n\geq 0)\) be a martingale of the form \(f_n= \text{E}[f|F_n]\). Let \(\omega\) be a weight and \(\Phi\) a Young function. Then writing \(f= (f_n)\in L_\Phi(\omega)\) means that \[ \int_\Omega\Phi(|f|)\omega d\mu<\infty. \] Dobb's maximal operator \(M\) is defined by \(Mf= \sup_{n\geq 0}|f_n|\). For any weight \(\omega\) and \(B\in{\mathcal F}\), let \(|B|_\omega:= \int_\Omega 1_B\omega d\mu\). The authors obtain the following result involving two Young functions. Theorem. Let \(u\), \(v\) be weights and suppose \(\Phi_1\ll \Phi_2\), \(\Phi_2\in\Delta_2\) and \(q_{\Phi_1}> 1\). Then the following three statements are equivalent: (1) there exists a positive constant \(C\) such that, for all stopping times \(\tau\), \[ \Phi^{-1}_2\Biggl(\int_{\{\tau<\infty\}} \Phi_2(|\text{E}[fv|{\mathcal F}_\tau]|)\,ud\mu\Biggr)\leq C\Phi^{-1}_1\Biggl(C\int_\Omega \Phi_1(|f|)\,vd\mu\Biggr) \] whenever \(f= (f_n)\in L_{\Phi_1}(v)\); (2) there exists a positive constant \(C\) such that, for all stopping times \(\tau\), \[ \Phi^{-…
reviewed by: Klaus Schürger
Mathematics Subject Classification ID: 60G46
Mathematics Subject Classification ID: 60G42
zbMATH DE Number: 5997898
zbMATH Keywords: martingale Orlicz setting
zbMATH Keywords: Doob's maximal operator
zbMATH Keywords: weighted integral inequality
MaRDI profile type: Publication
full work available at URL: s11425-011-4208-7
OpenAlex ID: W2003788214
cites work: Weighted weak type integral inequalities for the Hardy-Littlewood maximal operator
cites work: Q3700300
cites work: Weighted Inequalities for Maximal Operators: Linearization, Localization and Factorization
cites work: On weighted weak type maximal inequalities for martingales
cites work: Two Weight Mixed Ф-Inequalities for the Hardy Operator and the Hardy-Littlewood Maximal Operator
cites work: Two Weight Φ-Inequalities for the Hardy Operator, Hardy-Littlewood Maximal Operator, and Fractional Integrals
cites work: Q3135095
cites work: Q3803616
cites work: Weighted Norm Inequalities for the Hardy Maximal Function
cites work: Q4000065
cites work: Two-weight weak-type maximal inequalities for martingales
cites work: A characterization of a two-weight norm inequality for maximal operators
cites work: Weighted norm inequalities on spaces of homogeneous type
DOI: 10.1007/S11425-011-4208-7
description: scientific article; zbMATH DE number 5997898