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 f(x, a), where x ϵ 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 po depends on the geometry of X, such that if f is in Lp(X) then there is a set of full measure in X and if x lies in this set, the function a ↦ f(x, a) has some Hölder continuity on compact subsets of the regular elements of A. 1980 Mathematics subject classification (Amer. Math. Soc.): 42 C 10, 43 A 15. Keywords and phrases: symmetric space, spherical function, Sobolev space, mean-value operator. operator. maximal torus.