Pointwise ergodic theorems for radial averages on the Heisenberg group

Abstract

LetH=Hn=Cn×R denote the Heisenberg group, and letσrdenote the normalized Lebesgue measure on the sphere {(z, 0):|z|=r}. Let (X, B, m) be a standard Borel probability space on whichHacts measurably and ergodically by measure preserving transformations, and letπ(σr) denote the operator canonically associated withσronLp(X). We prove maximal and pointwise ergodic theorems inLp, for radial averagesσron the Heisenberg groupHn,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. © 1997 Academic Press.