{"title":"Characterizations of $A_\\infty$ Weights in Ergodic Theory","authors":"Wei Chen, Jingyi Wang","doi":"arxiv-2409.08896","DOIUrl":null,"url":null,"abstract":"We establish a discrete weighted version of Calder\\'{o}n-Zygmund\ndecomposition from the perspective of dyadic grid in ergodic theory. Based on\nthe decomposition, we study discrete $A_\\infty$ weights. First,\ncharacterizations of the reverse H\\\"{o}lder's inequality and their extensions\nare obtained. Second, the properties of $A_\\infty$ are given, specifically\n$A_\\infty$ implies the reverse H\\\"{o}lder's inequality. Finally, under a\ndoubling condition on weights, $A_\\infty$ follows from the reverse H\\\"{o}lder's\ninequality. This means that we obtain equivalent characterizations of\n$A_{\\infty}$. Because $A_{\\infty}$ implies the doubling condition, it seems\nreasonable to assume the condition.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"118 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Probability","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.08896","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
We establish a discrete weighted version of Calder\'{o}n-Zygmund
decomposition from the perspective of dyadic grid in ergodic theory. Based on
the decomposition, we study discrete $A_\infty$ weights. First,
characterizations of the reverse H\"{o}lder's inequality and their extensions
are obtained. Second, the properties of $A_\infty$ are given, specifically
$A_\infty$ implies the reverse H\"{o}lder's inequality. Finally, under a
doubling condition on weights, $A_\infty$ follows from the reverse H\"{o}lder's
inequality. This means that we obtain equivalent characterizations of
$A_{\infty}$. Because $A_{\infty}$ implies the doubling condition, it seems
reasonable to assume the condition.