Abstract. We prove a multiparameter $L\log ^{k}L$ generalization of the Báez--Duarte Abelian ergodic theorem for positive linear contractions on $L_{1}$, which allows the application of the local convergence principle of Sucheston's type. Next we establish a new weighted Abelian ratio ergodic theorem for Dunford--Schwartz operators on $L_{1}$ with modulation by Besicovitch sequences. Moreover, this (one-parameter) result is generalized to the case of multiparameter operator averages, which allows the application of Fava's maximal ergodic inequality.

DOI: 10.14232/actasm-019-757-4

AMS Subject Classification (1991): 47A35, 40H05, 40G10

Keyword(s): local convergence principle, Chacon's theorem, Báez-Duarte's theorem, Fava's theorem, Abelian ratio ergodic theorem, positive linear contraction, Besicovitch sequence, modulated Abelian ratio ergodic theorem

received 21.1.2019, revised 25.11.2019, accepted 4.3.2020. (Registered under 7/2019.)