On a maximal function on compact Lie groups

Abstract:

Suppose that $G$ is a compact Lie group with finite centre. For each positive number $s$ we consider the $\operatorname {Ad}(G)$-invariant probability measure ${\mu _s}$ carried on the conjugacy class of $\exp (s{H_\rho })$ in $G$. This one-parameter family of measures is used to define a maximal function $\mathcal {M} f$, for each continuous function $f$ on $G$. Our theorem states that there is an index ${p_0}$ in $(1,2)$, depending on $G$, such that the maximal operator $\mathcal {M}$ is bounded on ${L^p}(G)$ when $p$ is greater than ${p_0}$. 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.