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Weak type $(1, 1)$ estimates for maximal functions along $1$-regular sequences of integers

Volume 261 / 2021

Bartosz Trojan Studia Mathematica 261 (2021), 103-108 MSC: Primary 37A44, 37A46. DOI: 10.4064/sm200702-21-12 Published online: 24 May 2021


We show the pointwise convergence of the averages \[ \mathcal A _N f(x) = \frac {1}{\# {\bf B} _N} \sum _{n \in {\bf B} _N} f(x + n) \] for $f \in \ell ^1(\mathbb Z )$ where ${\bf B} _N = {\bf B} \cap [1, N]$, and ${\bf B} $ is a $1$-regular sequence of integers, for example ${\bf B} = \{\lfloor n \log n \rfloor : n \in \mathbb N \}$.


  • Bartosz TrojanInstitute of Mathematics
    Polish Academy of Sciences
    Śniadeckich 8
    00-656 Warszawa, Poland

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