New York Journal of Mathematics
Volume 28 (2022), 1448-1462


Grigori A. Karagulyan, Michael T. Lacey, and Vahan A. Martirosyan

On the convergence of multiple ergodic means

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Published: October 22, 2022.
Keywords: Ergodic theorems, strong maximal function, multiple ergodic sums.
Subject [2010]: 37A30, 37A46, 42B25.

In this paper, we consider multiple ergodic means generated by a finite collection of measure-preserving transformations. We prove an almost sure convergence of such means for the functions of the logarithmic Orlicz space associated with the rank of the transformations. It is also shown the optimality of this function class in this convergence property. Our result extends a theorem by Dunford and Zygmund, where the convergence was claimed in the logarithmic space associated with the number of transformations. We prove our result by establishing a weak type inequality for certain maximal functions in Euclidean space.


Research of Lacey was supported in part by grant from the US National Science Foundation, DMS-1949206, and the research of Karagulyan was supported by the Science Committee of RA, in the frames of the research project 21AG-1A045.

Author information

Grigori A. Karagulyan:
Institute of Mathematics NAS RA
24/5 Marshal Baghramian ave
Yerevan, 0019, Republic of Armenia, and
Faculty of Mathematics and Mechanics
Yerevan State University
Alex Manoogian, 1, Yerevan, 0025, Republic of Armenia


Michael T. Lacey:
School of Mathematics
Georgia Institute of Technology
Atlanta, GA 30332, USA


Vahan A. Martirosyan:
Faculty of Mathematics and Mechanics
Yerevan State University
Alex Manoogian, 1, 0025, Yerevan, Armenia