Pointwise ergodic theorems for radial averages on simple Lie groups II

1. Statement of results, the method of proof, and some remarks 1.1. Definitions and statement of results. The present paper is a continuation of IN1], and we begin by briefly recalling the setup and the notation: G Gn SO(n, 1) is the group of orientation-preserving isometries of ndimensional real hyperbolic space Hn, n > 2. K a fixed maximal compact subgroup, mc Haar probability measure. A {at]t JR}-a one-parameter group of hyperbolic translations such that G KA+K is a Cartan decomposition. at the bi-K-invariant probability measure on G given by at mr * (at* mr, where denotes convolution. Note that or0 mr. #t lit f as ds, the uniform average of ors, 0 < s < t. We define #0 mtc. M(G,K)= the commutative convolution algebra (of bi-K-invariant complex bounded Borel measures on G) generated by o’t, > 0. (X, , 2) a standard Borel space with a Borel measurable G-action which preserves the probability measure 2. r(v)f(x) f6 f(o-lx)dr(g) the Markov operator on Lt’(X) corresponding to a probability measure v on G. Muf(x)=supt>oln(#t)f(x)l and Mf(x)=supt>olrC(trt)f(x)l, maximal functions associated with the action of at and #t in L’(X), 1 < p < oz. Finally, recall also the following definition.