Let G be a compact semisimple
Lie group with finite centre. For each positive number s, let μsH denote the
Ad(G)-invariant probability measure carried on the conjugacy class of exp(sH) in G.
With this one-parameter family of measures, we define the maximal operator ℳH on
𝒞(G). We then estimate the Fourier transform of μsH and of some derived
distributions. Our result leads to the boundedness of ℳH on Lp(G), for all p greater
than some index p0 in (1,2). This generalizes a recent result of M. Cowling and C.
Meaney [2].