Weak type (1, 1) inequalities for discrete rough maximal functions

Abstract

The aim of this paper is to show that the discrete maximal function $$\mathcal{M}_{h}f(x)=\sup_{N\in\mathbb{N}}\frac{1}{|\mathbf{N}_{h}\cap[1, N]|}\Big|\sum_{n\in \mathbf{N}_{h}\cap[1, N]}f(x-n)\Big|,\ \ \mbox{for $x\in\mathbb{Z}$},$$ is of weak type $(1,1)$, where ${\mathbf{N}}_h=\{n\in \mathbb{N}:{\exists }_{m\in \mathbb{N}}n=\lfloor h(m)\rfloor \}$ for an appropriate function $h$. As a consequence we also obtain pointwise ergodic theorem along the set ${\mathbf{N}}_h$.

arXiv commentAccepted for publication in Journal d'Analyse Mathematique