Convergence of weighted ergodic averages

TL;DR

This paper establishes conditions under which weighted ergodic averages converge to zero almost everywhere for contractions on L^2 spaces, extending to one-sided weighted ergodic Hilbert transforms.

Contribution

It provides new sufficient conditions for the almost everywhere convergence of weighted ergodic averages involving contractions on L^2 spaces.

Findings

Weighted ergodic averages converge to zero under specified conditions.

Conditions are applicable to one-sided weighted ergodic Hilbert transforms.

Results extend classical ergodic theorems to weighted and one-sided contexts.

Abstract

Let $(X, \mathcal{A},\mu)$ be a probability space and let $T$ be a contraction on $L^2(\mu)$. We provide suitable conditions over sequences $(w_k)$, $(u_k)$ and $(A_k)$ in such a way that the weighted ergodic limit $\lim\limits_{N\rightarrow\infty}\frac{1}{A_N}\sum_{k=0}^{N-1} w_k T^{u_k}(f)=0$ $\mu$-a.e. for any function $f$ in $L^2(\mu)$. As a consequence of our main theorems, we also deal with the so-called one-sided weighted ergodic Hilbert transforms.