Resumen de On a pointwise ergodic theorem for multiparameter semigroups

Ryotaro Sato

• Let $T_i$ $(i=1,2,\dots,d)$ be commuting null preserving transformations on a finite measure space $(X,\Cal F, \mu)$ and let $1\le p<\infty$. In this paper we prove that for every $f\in L_p(\mu)$ the averages $$ A_nf(x)=(n+1)^{-d}\sum_{0\le n_i\le n}f(T_1^{n_1}T_2^{n_2}\dots T_d^{n_d}x) $$ converge a.e. on $X$ if and only if there exists a finite invariant measure $\nu$ (under the transformations $T_i$) absolutely continuous with respect to $\mu$ and a sequence $\{X_N\}$ of invariant sets with $X_N\uparrow X$ such that $\nu B>0$ for all nonnull invariant sets $B$ and such that the Radon-Nikodym derivative $v=d\nu/d\mu$ satisfies $v\in L_q (x_N,\mu)$, $1/p+1/q=1$, for each $N\ge 1$.