Almost everywhere convergence of convolution products

Details Almost everywhere convergence of convolution products Almost everywhere convergence of convolution products

2011-04-16

arxiv.org/abs/1104.3237v1

Mathematics

Classical Analysis and ODEs

13 pages, to appear In the Canadian Mathematical Bulletin

Scientific paper

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.

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