Geodesic
VECTOR ERGODIC THEOREM IN $L(X)\log L(X)$
Acta mathematica Universitatis Comenianae, Tome 69 (2000) no. 2 
K. EL Berdan. VECTOR ERGODIC THEOREM IN $L(X)\log L(X)$. Acta mathematica Universitatis Comenianae, Tome 69 (2000) no. 2. http://geodesic.mathdoc.fr/item/AMUC_2000_69_2_a8/
@article{AMUC_2000_69_2_a8,
     author = {K. EL Berdan},
     title = {VECTOR {ERGODIC} {THEOREM} {IN} $L(X)\log L(X)$},
     journal = {Acta mathematica Universitatis Comenianae},
     year = {2000},
     volume = {69},
     number = {2},
     url = {http://geodesic.mathdoc.fr/item/AMUC_2000_69_2_a8/}
}
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%J Acta mathematica Universitatis Comenianae
%D 2000
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%F AMUC_2000_69_2_a8

Voir la notice de l'article provenant de la source Comenius University

Let $X$ be a reflexive Banach space and $\Omega $ be a finite measure space. We prove the almost everywhere convergence of the vector multiparameter averages \frac1n_1\dots n_d\sum_0\leq k_1,\dots k_d<n_j\alpha _k_1^1\dots \alpha _k_d^dT_1^k_1\dots T_d^k_df for all $f\in L^p(X)$, $1, and where $ ( \alpha _n^j) $ are bounded Besicovitch sequences $\left( j=1,\dots ,d\right) $ with $T_1,\dots ,T_d$ are linear operators acting on $% L^1(X)$ and satisfying certain conditions. For $d=2$, we obtain more general result. Indeed, in this case, we prove the convergence a.e. for $f\in L(X)\log L(X)$. The general case ($d>2$) requires integrability of the supremum of the norm of these averages. As applications, we give new proof of Zygmund-Fava's Theorem.