Multiparameter ergodic Cesàro-α averages

A. L. Bernardis, R. Crescimbeni, and C. Ferrari Freire. "Multiparameter ergodic Cesàro-α averages." Colloquium Mathematicae 140.1 (2015): 15-29. <http://eudml.org/doc/284060>.

@article{A2015,
abstract = {Net (X,ℱ,ν) be a σ-finite measure space. Associated with k Lamperti operators on $L^\{p\}(ν)$, $T₁,..., T_k$, $n̅ = (n₁,..., n_k) ∈ ℕ^k$ and $α̅ = (α₁,...,α_k)$ with $0 < α_j ≤ 1$, we define the ergodic Cesàro-α̅ averages $_\{n̅,α̅\}f = 1/(∏_\{j=1\}^\{k\} A_\{n_j\}^\{α_j\}) ∑_\{i_k=0\}^\{n_k\} ⋯ ∑_\{i₁=0\}^\{n₁\} ∏_\{j=1\}^\{k\} A_\{n_j-i_j\}^\{α_j-1\} T_k^\{i_k\} ⋯ T₁^\{i₁\}f$. For these averages we prove the almost everywhere convergence on X and the convergence in the $L^\{p\}(ν)$ norm, when $n₁,..., n_k → ∞$ independently, for all $f ∈ L^\{p\}(dν)$ with p > 1/α⁎ where $α⁎ = min_\{1≤j≤ k\} α_j$. In the limit case p = 1/α⁎, we prove that the averages $_\{n̅,α̅\}f$ converge almost everywhere on X for all f in the Orlicz-Lorentz space $Λ(1/α⁎,φ_\{m-1\})$ with $φₘ(t) = t(1+log⁺t)^m$. 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-α̅ continuity of functions.},
author = {A. L. Bernardis, R. Crescimbeni, C. Ferrari Freire},
journal = {Colloquium Mathematicae},
keywords = {multiparameter ergodic theory; Cesàro averages; Orlicz-Lorentz spaces},
language = {eng},
number = {1},
pages = {15-29},
title = {Multiparameter ergodic Cesàro-α averages},
url = {http://eudml.org/doc/284060},
volume = {140},
year = {2015},
}

TY - JOUR
AU - A. L. Bernardis
AU - R. Crescimbeni
AU - C. Ferrari Freire
TI - Multiparameter ergodic Cesàro-α averages
JO - Colloquium Mathematicae
PY - 2015
VL - 140
IS - 1
SP - 15
EP - 29
AB - Net (X,ℱ,ν) be a σ-finite measure space. Associated with k Lamperti operators on $L^{p}(ν)$, $T₁,..., T_k$, $n̅ = (n₁,..., n_k) ∈ ℕ^k$ and $α̅ = (α₁,...,α_k)$ with $0 < α_j ≤ 1$, we define the ergodic Cesàro-α̅ averages $_{n̅,α̅}f = 1/(∏_{j=1}^{k} A_{n_j}^{α_j}) ∑_{i_k=0}^{n_k} ⋯ ∑_{i₁=0}^{n₁} ∏_{j=1}^{k} A_{n_j-i_j}^{α_j-1} T_k^{i_k} ⋯ T₁^{i₁}f$. For these averages we prove the almost everywhere convergence on X and the convergence in the $L^{p}(ν)$ norm, when $n₁,..., n_k → ∞$ independently, for all $f ∈ L^{p}(dν)$ with p > 1/α⁎ where $α⁎ = min_{1≤j≤ k} α_j$. In the limit case p = 1/α⁎, we prove that the averages $_{n̅,α̅}f$ converge almost everywhere on X for all f in the Orlicz-Lorentz space $Λ(1/α⁎,φ_{m-1})$ with $φₘ(t) = t(1+log⁺t)^m$. 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-α̅ continuity of functions.
LA - eng
KW - multiparameter ergodic theory; Cesàro averages; Orlicz-Lorentz spaces
UR - http://eudml.org/doc/284060
ER -