Pointwise ergodic theorems for radial averages on the Heisenberg group

Abstract

Let H=Hⁿ=Cⁿ×R denote the Heisenberg group, and let σ_r denote the normalized Lebesgue measure on the sphere {(z, 0): |z|=r}. Let (X, B , m) be a standard Borel probability space on which Hacts measurably and ergodically by measure preserving transformations, and let Π(σ_r) denote the operator canonically associated with σ_r on L^p(X). We prove maximal and pointwise ergodic theorems in L^p, for radial averagessron the Heisenberg group Hⁿ,n>1. The results are best possible for actions of the reduced Heisenberg group. The method of proof is to use the spectral theory of the Banach algebra of radial measures on the group and decay estimates for its characters to establish maximal inequalities using spectral methods, in particular Littlewood-Paley-Stein square-functions and analytic interpolation.