Let X be either the d-dimensional sphere or a compact, simply connected, simple, connected Lie group. We define a mean-value operator analogous to the spherical mean-value operator acting on integrable functions on Euclidean space. The value of this operator will be written as ℳ f (x, a), where x ∈ X and a varies over a torus A in the group of isometries of X. For each of these cases there is an interval pO < p ≦ 2, where the p0 depends on the geometry of X, such that if f is in Lp (X) then there is a set full measure in X and if x lies in this set, the function a ↦ℳ f(x, a) has some Hölder continuity on compact subsets of the regular elements of A.
