Multiparameter ergodic Cesàro-α averages

A. L. Bernardis ; R. Crescimbeni ; C. Ferrari Freire

Colloquium Mathematicae, Tome 139 (2015), p. 15-29 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

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Publié le : 2015-01-01

Zbl 1323.47005

EUDML-ID : urn:eudml:doc:284060