Almost everywhere convergence of convolution products

TL;DR

This paper investigates the almost everywhere convergence of convolution-based weighted averages in dynamical systems, establishing conditions for convergence and identifying cases where it fails.

Contribution

It provides new maximal estimates and identifies conditions under which convolution product averages converge almost everywhere in $L^1$, including cases of failure.

Findings

Convergence holds under certain maximal estimates and dense classes.

Explicit examples where convergence fails are provided.

The results extend understanding of convolution averages in dynamical systems.

Abstract

Let $(X,\mathcal{B},m,\tau)$ be a dynamical system with $\ds (X,\mathcal{B},m)$ a probability space and $\ds \tau$ an invertible, measure preserving transformation. The present paper deals with the almost everywhere convergence in $\ds{L}^1(X)$ of a sequence of operators of weighted averages. Almost everywhere convergence follows once we obtain an appropriate maximal estimate and once we provide a dense class where convergence holds almost everywhere. The weights are given by convolution products of members of a sequence of probability measures $\ds\{\nu_i\}$ defined on $\ds\mathbb{Z}$. We then exhibit cases of such averages, where convergence fails.