Multiparameter ergodic Cesàro-α averages

• Autorzy: A. L. Bernardis , R. Crescimbeni , C. Ferrari Freire
• Języki publikacji: EN

Abstrakty

EN

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
$𝓡_{n̅,α̅}f = 1/(∏_{j=1}^{k} A_{n_j}^{α_j}) ∑_{i_k=0}^{n_k} ⋯ ∑_{i₁=0}^{n₁} ∏_{j=1}^{k} A_{n_j-i_j}^{α_j-1} T_k^{i_k} ⋯ T₁^{i₁}f$.
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 $𝓡_{n̅,α̅}f$ 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.

Kategorie tematyczne

• 37A30: Ergodic theorems, spectral theory, Markov operators
• 47A35: Ergodic theory
• 42B25: Maximal functions, Littlewood-Paley theory

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2015
• Tom: 140
• Numer: 1
• Strony: 15-29

Daty

wydano

2015

Twórcy

autor

A. L. Bernardis

• IMAL (CONICET-UNL), Facultad de Ingeniería Química (UNL), Colectora Ruta Nac. No. 168, Paraje El Pozo, 3000 Santa Fe, Argentina

autor

R. Crescimbeni

• Departamento de Matemática, Universidad Nacional del Comahue, Buenos Aires 1400, 8300 Neuquén, Argentina

autor

C. Ferrari Freire

• Departamento de Matemática, Universidad Nacional del Comahue, Buenos Aires 1400, 8300 Neuquén, Argentina

Identyfikatory

DOI

10.4064/cm140-1-3