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

IF 1.2 3区数学 Q2 MATHEMATICS, APPLIED Journal of Fourier Analysis and Applications 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。

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来源期刊

Journal of Fourier Analysis and Applications 数学-应用数学

CiteScore

2.10

自引率

16.70%

发文量

审稿时长

6-12 weeks

期刊介绍： The Journal of Fourier Analysis and Applications will publish results in Fourier analysis, as well as applicable mathematics having a significant Fourier analytic component. Appropriate manuscripts at the highest research level will be accepted for publication. Because of the extensive, intricate, and fundamental relationship between Fourier analysis and so many other subjects, selected and readable surveys will also be published. These surveys will include historical articles, research tutorials, and expositions of specific topics. TheJournal of Fourier Analysis and Applications will provide a perspective and means for centralizing and disseminating new information from the vantage point of Fourier analysis. The breadth of Fourier analysis and diversity of its applicability require that each paper should contain a clear and motivated introduction, which is accessible to all of our readers. Areas of applications include the following: antenna theory * crystallography * fast algorithms * Gabor theory and applications * image processing * number theory * optics * partial differential equations * prediction theory * radar applications * sampling theory * spectral estimation * speech processing * stochastic processes * time-frequency analysis * time series * tomography * turbulence * uncertainty principles * wavelet theory and applications

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