Multiparameter ergodic Cesàro-$\alpha $ averages

A. L. Bernardis¹ ; R. Crescimbeni² ; C. Ferrari Freire²

¹IMAL (CONICET-UNL) Facultad de Ingeniería Química (UNL) Colectora Ruta Nac. No. 168, Paraje El Pozo 3000 Santa Fe, Argentina
²Departamento de Matemática Universidad Nacional del Comahue Buenos Aires 1400 8300 Neuquén, Argentina

Colloquium Mathematicum, Tome 140 (2015) no. 1, pp. 15-29

Résumé

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.

Détail

DOI : 10.4064/cm140-1-3

Keywords: net mathcal sigma finite measure space associated lamperti operators ldots bar ldots mathbb bar alpha alpha ldots alpha alpha leq define ergodic ces ro bar alpha averages mathcal bar bar alpha frac prod alpha sum cdots sum prod j i alpha j cdots these averages prove almost everywhere convergence convergence norm ldots infty independently alpha * where alpha * min leq leq alpha limit alpha * prove averages mathcal bar bar alpha converge almost everywhere orlicz lorentz space varlambda alpha * varphi m varphi log obtain result limit study inequalities composition operators restricted weak type i another application these inequalities study strong ces ro bar alpha continuity functions