{"title":"Characterizations of Weights in Martingale Spaces","authors":"Jie Ju, Wei Chen, Jingya Cui, Chao Zhang","doi":"10.1007/s12220-024-01674-x","DOIUrl":null,"url":null,"abstract":"<p>Grafakos systematically proved that <span>\\(A_\\infty \\)</span> weights have different characterizations for cubes in Euclidean spaces in his classical text book. Very recently, Duoandikoetxea, Martín-Reyes, Ombrosi and Kosz discussed several characterizations of the <span>\\(A_{\\infty }\\)</span> weights in the setting of general bases. By conditional expectations, we study <span>\\(A_\\infty \\)</span> weights in martingale spaces. Because conditional expectations are Radon–Nikodým derivatives with respect to sub<span>\\(\\hbox {-}\\sigma \\hbox {-}\\)</span>fields which have no geometric structures, we need new ingredients. Under a regularity assumption on weights, we obtain equivalent characterizations of the <span>\\(A_{\\infty }\\)</span> weights. Moreover, using weights modulo conditional expectations, we have one-way implications of different characterizations.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"The Journal of Geometric Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s12220-024-01674-x","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
Grafakos systematically proved that \(A_\infty \) weights have different characterizations for cubes in Euclidean spaces in his classical text book. Very recently, Duoandikoetxea, Martín-Reyes, Ombrosi and Kosz discussed several characterizations of the \(A_{\infty }\) weights in the setting of general bases. By conditional expectations, we study \(A_\infty \) weights in martingale spaces. Because conditional expectations are Radon–Nikodým derivatives with respect to sub\(\hbox {-}\sigma \hbox {-}\)fields which have no geometric structures, we need new ingredients. Under a regularity assumption on weights, we obtain equivalent characterizations of the \(A_{\infty }\) weights. Moreover, using weights modulo conditional expectations, we have one-way implications of different characterizations.