{"title":"Weak-Type (1,1) Inequality for Discrete Maximal Functions and Pointwise Ergodic Theorems Along Thin Arithmetic Sets","authors":"Leonidas Daskalakis","doi":"10.1007/s00041-024-10093-z","DOIUrl":null,"url":null,"abstract":"<p>We establish weak-type (1, 1) bounds for the maximal function associated with ergodic averaging operators modeled on a wide class of thin deterministic sets <i>B</i>. As a corollary we obtain the corresponding pointwise convergence result on <span>\\(L^1\\)</span>. This contributes yet another counterexample for the conjecture of Rosenblatt and Wierdl from 1991 asserting the failure of pointwise convergence on <span>\\(L^1\\)</span> of ergodic averages along arithmetic sets with zero Banach density. The second main result is a multiparameter pointwise ergodic theorem in the spirit of Dunford and Zygmund along <i>B</i> on <span>\\(L^p\\)</span>, <span>\\(p>1\\)</span>, which is derived by establishing uniform oscillation estimates and certain vector-valued maximal estimates.</p>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00041-024-10093-z","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
Abstract
We establish weak-type (1, 1) bounds for the maximal function associated with ergodic averaging operators modeled on a wide class of thin deterministic sets B. As a corollary we obtain the corresponding pointwise convergence result on \(L^1\). This contributes yet another counterexample for the conjecture of Rosenblatt and Wierdl from 1991 asserting the failure of pointwise convergence on \(L^1\) of ergodic averages along arithmetic sets with zero Banach density. The second main result is a multiparameter pointwise ergodic theorem in the spirit of Dunford and Zygmund along B on \(L^p\), \(p>1\), which is derived by establishing uniform oscillation estimates and certain vector-valued maximal estimates.
作为推论，我们得到了在\(L^1\)上相应的点收敛结果。这为罗森布拉特（Rosenblatt）和维尔德（Wierdl）1991 年提出的猜想提供了另一个反例，该猜想断言沿着巴纳赫密度为零的算术集合的遍历平均数在 \(L^1\) 上的点收敛失败。第二个主要结果是邓福德（Dunford）和齐格蒙德（Zygmund）在 B on \(L^p\), \(p>1\)上提出的多参数点式遍历定理，它是通过建立均匀振荡估计和某些向量值最大估计推导出来的。
期刊介绍：
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.
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