Weak-type (1,1) inequality for discrete maximal functions and pointwise ergodic theorems along thin arithmetic sets

2023/05/17 by Leonidas Daskalakis, Daskalakis, Leonidas · 1 citation
Mathematics · #37A30 #37A46 #42B25 #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2305.10603

openalex publication_date 2023/05/17 · openalex created_date 2023/05/21 · openalex updated_date 2026/07/28

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 L¹. This contributes yet another counterexample for the conjecture of Rosenblatt and Wierdl from 1991 asserting the failure of pointwise convergence on L¹ 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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• Pointwise convergence of ergodic averages along quadratic bracket polynomials

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