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.)
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