Multiparameter ergodic Cesàro-$\alpha $ averages
Volume 140 / 2015
Abstract
Net $(X,{\mathcal F},\nu)$ be a $\sigma$-finite measure space. Associated with $k$ Lamperti operators on $L^p(\nu)$, $T_1,\ldots, T_k$, $\bar{n}=(n_1, \ldots, n_k) \in \mathbb{N}^k$ and $\bar{\alpha}=(\alpha_1,\ldots,\alpha_k)$ with $0<\alpha_j\leq 1$, we define the ergodic Cesàro-$\bar{\alpha}$ averages $$ \mathcal{ R}_{\bar{n},\bar{\alpha}}f= \frac{1}{\prod_{j=1}^{k}A_{n_j}^{\alpha_j}} \sum_{i_k=0}^{n_k}\cdots\sum_{i_1=0}^{n_1} \prod_{j=1}^{k}A_{n_j-i_j}^{\alpha_j-1}T_k^{i_k}\cdots T_1^{i_1}f.$$ For these averages we prove the almost everywhere convergence on $X$ and the convergence in the $L^p(\nu)$ norm, when $n_1, \ldots ,n_k \to \infty$ independently, for all $f\in L^p(d\nu)$ with $p>1/\alpha_*$ where $\alpha_*=\min_{1\leq j\leq k}\alpha_j$. In the limit case $p=1/\alpha_*$, we prove that the averages $ \mathcal{ R}_{\bar{n},\bar{\alpha}}f$ converge almost everywhere on $X$ for all $f$ in the Orlicz–Lorentz space $\varLambda({1}/{\alpha_*},\varphi_{m-1})$ with $\varphi_m(t)=t(1+\log^+t)^m$. To obtain the result in the limit case we need to study inequalities for the composition of operators $T_i$ that are of restricted weak type $(p_i,p_i)$. As another application of these inequalities we also study the strong Cesàro-$\bar{\alpha}$ continuity of functions.
