ACTA MATHEMATICA UNIVERSITATIS COMENIANAE

Vol. 69,   2   (2000)
pp.   233-240

VECTOR ERGODIC THEOREM IN $L(X)\log L(X)$
K. EL BERDAN

Abstract.  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 $f\in L^p(X)$, $1<p<\infty$, 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.

AMS subject classification.  47A35
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