{"title":"On strongly orthogonal martingales in UMD Banach spaces","authors":"I. Yaroslavtsev","doi":"10.37190/0208-4147.41.1.10","DOIUrl":null,"url":null,"abstract":"In the present paper we introduce the notion of strongly orthogonal martingales. Moreover, we show that for any UMD Banach space $X$ and for any $X$-valued strongly orthogonal martingales $M$ and $N$ such that $N$ is weakly differentially subordinate to $M$ one has that for any $1<p<\\infty$ \\[ \\mathbb E \\|N_t\\|^p \\leq \\chi_{p, X}^p \\mathbb E \\|M_t\\|^p,\\;\\;\\; t\\geq 0, \\] with the sharp constant $\\chi_{p, X}$ being the norm of a decoupling-type martingale transform and being within the range \\[ \\max\\Bigl\\{\\sqrt{\\beta_{p, X}}, \\sqrt{\\hbar_{p,X}}\\Bigr\\} \\leq \\max\\{\\beta_{p, X}^{\\gamma,+}, \\beta_{p, X}^{\\gamma, -}\\} \\leq \\chi_{p, X} \\leq \\min\\{\\beta_{p, X}, \\hbar_{p,X}\\}, \\] where $\\beta_{p, X}$ is the UMD$_p$ constant of $X$, $\\hbar_{p, X}$ is the norm of the Hilbert transform on $L^p(\\mathbb R; X)$, and $\\beta_{p, X}^{\\gamma,+}$ and $ \\beta_{p, X}^{\\gamma, -}$ are the Gaussian decoupling constants.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":" ","pages":""},"PeriodicalIF":0.4000,"publicationDate":"2018-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Probability and Mathematical Statistics-Poland","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.37190/0208-4147.41.1.10","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
引用次数: 0
Abstract
In the present paper we introduce the notion of strongly orthogonal martingales. Moreover, we show that for any UMD Banach space $X$ and for any $X$-valued strongly orthogonal martingales $M$ and $N$ such that $N$ is weakly differentially subordinate to $M$ one has that for any $1
期刊介绍:
PROBABILITY AND MATHEMATICAL STATISTICS is published by the Kazimierz Urbanik Center for Probability and Mathematical Statistics, and is sponsored jointly by the Faculty of Mathematics and Computer Science of University of Wrocław and the Faculty of Pure and Applied Mathematics of Wrocław University of Science and Technology. The purpose of the journal is to publish original contributions to the theory of probability and mathematical statistics.