On a maximal function on compact Lie groups

Suppose that G is a compact Lie group with finite centre. For each positive number s we consider the Ad(G)-invariant probability measure /,u carried on the conjugacy class of exp(sHp) in G. This one-parameter family of measures is used to define a maximal function ,(f, for each continuous function f on G. Our theorem states that there is an index po in (1, 2), depending on G, such that the maximal operator X0 is bounded on LP(G) when p is greater than p0. When the rank of G is greater than one, this provides an example of a controllable maximal operator coming from averages over a family of submanifolds, each of codimension greater than one.