A GENERALIZATION OF MAXIMAL FUNCTIONS ON COMPACT SEMISIMPLE LIE GROUPS

Let G be a compact semisimple Lie group with finite centre. For each positive number 5, let μ s π denote the Ad (G)-invariant prob-ability measure carried on the conjugacy class of exp(sH) in G. With this one-parameter family of measures, we define the maximal operator J^H on W(G). We then estimate the Fourier transform of μSH and of some derived distributions. Our result leads to the boundedness of J?H on LP(G), for all p greater than some index Po in ( 1, 2). This generalizes a recent result of M. Cowling and C. Meaney [2]. Introduction. Let G be a compact semisimple Lie group of rank / with finite centre, and with its Haar measure normalized to have total mass 1. Let g denote its Lie algebra, and let ίj be a maximal toral subalgebra of g. We denote by Φ the root system of (gc, \f), and fi