Almost everywhere convergence of convolution products

Abstract

Let $(X,\mathcal{B},m,\tau )$ be a dynamical system with $\ds (X,\mathcal{B},m)$ a probability space and $\ds τ$ 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 $\ds\{ν_i\}$ defined on $\ds\mathbb{Z}$. We then exhibit cases of such averages, where convergence fails.

arXiv comment13 pages, to appear In the Canadian Mathematical Bulletin