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

Abstract

Group representations by bounded Lamperti operators in the spaces Lα(1≤α<∞) form a wide class of representations, including representations by bounded positive operators and (when α≠2) representations by isometric operators. The Dominated and the Pointwise Ergodic Theorems (DET and PET) for Cesàro averages for the bounded Lamperti representations of amenable σ-compact locally compact groups in Lα(1<α<∞) 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 Rm and Zm.