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Non-commutative ergodic averages of balls and spheres over Euclidean spaces

Published online by Cambridge University Press:  14 June 2018

GUIXIANG HONG
GUIXIANG HONG*
Affiliation:
School of Mathematics and Statistics, Wuhan University and Hubei Key Laboratory of Computational Science, Wuhan University, Wuhan 430072, China email guixiang.hong@whu.edu.cn
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Abstract

In this paper, we establish a non-commutative analogue of Calderón’s transference principle, which allows us to deduce the non-commutative maximal ergodic inequalities from the special case—operator-valued maximal inequalities. As applications, we deduce the non-commutative Stein–Calderón maximal ergodic inequality and the dimension-free estimates of the non-commutative Wiener maximal ergodic inequality over Euclidean spaces. We also show the corresponding individual ergodic theorems. To show Wiener’s pointwise ergodic theorem, following a somewhat standard way we construct a dense subset on which pointwise convergence holds. To show Jones’ pointwise ergodic theorem, we use again the transference principle together with the Littlewood–Paley method, which is different from Jones’ original variational method that is still unavailable in the non-commutative setting.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 2 , February 2020 , pp. 418 - 436
Copyright
© Cambridge University Press, 2018 

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