Two ergodic theorems for convex combinations of commuting isometries

Abstract

Let ( X , F , μ ) (X,\mathcal {F},\mu ) be a measure space. In this paper we obtain L p {L^p} estimates for the supremum of the Cesàro averages of combinations of commuting isometries of L p ( X , F , μ ) {L^p}(X,\mathcal {F},\mu ) . In particular, we show that a convex combination of two invertible commuting isometries of L p ( X , F , μ ) , p {L^p}(X,\mathcal {F},\mu ),p fixed, 1 > p > ∞ , p ≠ 2 1 > p > \infty ,p \ne 2 , admits of adominated estimate with constant P / ( p − 1 ) P/(p - 1) . We also show that a convex combination of an arbitrary number of commuting positive invertible isometries of L 2 ( X , F , μ ) {L^2}(X,\mathcal {F},\mu ) admits of a dominated estimate with constant 2.