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Multiparameter ergodic Cesàro-α averages

100%

Bernardis A., Crescimbeni R., Ferrari Freire C.

Colloquium Mathematicum

2015

tom 140

nr 1

15-29

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.

Ergodic transforms associated to general averages

100%

Aimar H., Bernardis A., Martín-Reyes F.

Studia Mathematica

2010

tom 199

nr 2

107-143

Jones and Rosenblatt started the study of an ergodic transform which is analogous to the martingale transform. In this paper we present a unified treatment of the ergodic transforms associated to positive groups induced by nonsingular flows and to general means which include the usual averages, Cesàro-α averages and Abel means. We prove the boundedness in $L^{p}$, 1 < p < ∞, of the maximal ergodic transforms assuming that the semigroup is Cesàro bounded in $L^{p}$. For p = 1 we find that the maximal ergodic transforms are of weak type (1,1). Convergence results are also proved. We give some general examples of Cesàro bounded semigroups.

Ograniczanie wyników

1 Colloquium Mathematicum

1 Studia Mathematica

2 Bernardis A.

1 Aimar H.

1 Crescimbeni R.

1 Ferrari Freire C.

1 Martín-Reyes F.

1 2015

1 2010

Wyniki wyszukiwania

Multiparameter ergodic Cesàro-α averages

Ergodic transforms associated to general averages