Weak-Type (1,1) Inequality for Discrete Maximal Functions and Pointwise Ergodic Theorems Along Thin Arithmetic Sets

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2024-06-12 DOI:10.1007/s00041-024-10093-z

Leonidas Daskalakis

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.

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离散极值函数的弱型 (1,1) 不等式和沿薄算术集的点式遍历定理

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。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

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.