Exponential volume growth, maximal functions on symmetric spaces, and ergodic theorems for semi-simple Lie groups

Abstract

We prove pointwise ergodic theorems for general families of radial averages on semi-simple Lie groups, as well as strong L^p-maximal inequalities for the corresponding maximal functions, for p > 1. The methods of proof make essential use of exponential volume growth on the group, and of spectral considerations utilizing the existence of spectral gaps in measure-preserving actions of simple groups of real rank at least two. We also consider the case of the action by convolutions on the symmetric space, where we establish maximal inequalities of weak-type, going beyond L^p, p > 1. The proof here is based on some new convolution inequalities which are established by direct geometric comparison arguments. These inequalities can be viewed as a quantitative version of the wave front lemma. The maximal inequalities and pointwise ergodic theorems we establish are in sharp contrast to what one would be lead to expect by comparison with the analogous averages on the Abelian groups $\mathbb{R}^n$. We explain how the contrast arises from the difference between the polynomial volume growth on $\mathbb{R}^n$ and the exponential volume growth on semi-simple groups.