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arXiv · 2305.10603

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

Leonidas Daskalakis
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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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Leonidas Daskalakis. 2023-05-17. Weak-type (1,1) inequality for discrete maximal functions and pointwise ergodic theorems along thin arithmetic sets. https://arxiv.org/abs/2305.10603

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