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

TL;DR

This paper proves that a certain discrete maximal function associated with a set defined by a function h is of weak type (1,1), leading to a pointwise ergodic theorem along that set, advancing understanding of discrete harmonic analysis.

Contribution

The paper establishes weak type (1,1) bounds for discrete rough maximal functions defined via a function h, a novel result in discrete harmonic analysis.

Findings

Proves weak type (1,1) inequality for the discrete maximal function.

Derives a pointwise ergodic theorem along the set defined by h.

Extends classical results to a broader class of discrete sets.

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 $\mathbf{N}_{h}$.