A generalization of maximal functions on compact semisimple Lie groups (Q811585)

Let \(G\) be a compact semisimple Lie group with finite centre. For each positive number \(s\), let \(\mu_{sH}\) denote the \(\hbox{Ad}(G)\)-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 \({\mathcal M}_ H\) on \({\mathcal C}(G)\). Then we estimate the Fourier transform of \(\mu_{sH}\) and of some derived distributions. Our result leads to the boundedness of \({\mathcal M}_ H\) on \(L^ p(G)\), for all \(p\) greater than some index \(p_ 0\) in (1,2). This generalizes a recent result of \textit{M. Cowling} and \textit{C. Meaney} [Trans. Am. Math. Soc. 315, 811-822 (1989; Zbl 0689.42018)].