Dominated and pointwise ergodic theorems with weighted averages for bounded Lamperti representations of amenable groups

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:474

Article

Tempelman, Arkady^1,2  Shulman, Alexander³ 

[1] Penn State Univ, Dept Math, University Pk, PA 16802 USA

[2] Penn State Univ, Dept Stat, University Pk, PA 16802 USA

[3] Wells Fargo & Co, San Francisco, CA USA

关键词: Ergodic theorems;    Group actions;    Group representations;   

DOI  :  10.1016/j.jmaa.2019.01.013

来源: Elsevier

【 摘 要 】

Group representations by bounded Lamperti operators in the spaces L-alpha(1 <= alpha < infinity) form a wide class of representations, including representations by bounded positive operators and (when alpha not equal 2) representations by isometric operators. The Dominated and the Pointwise Ergodic Theorems (DET and PET) for Cesaro averages for the bounded Lamperti representations of amenable a-compact locally compact groups in L-alpha(1 < alpha < infinity) were proved by A. Tempelman in Proc. Amer. Math. Soc. 143 (2015) 4989-5004. By using a completely different, functional-analytical method, developed by A. Shulman in his PhD thesis in 1988, we generalize this result to weighted averages of such representations and discuss various conditions on the weights under which the DET and the PET hold. We conclude with applications of the general results to the bounded Lamperti representations of groups of polynomial growth and of the groups R-m and Z(m). (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

Free