Regularity of spherical means and localization of spherical harmonic expansions

Let G/K be a compact symmetric space, and let G = KAK be a Cartan decomposition of G. For f in L1(G) we define the spherical means f(g, t) =∫k ∫k f(gktk′) dk dk′, g ∈ G, t ∈ A. We prove that if f is in LP(G), 1 ⩽ p ⩽ 2, then for almost every g e G the functions t ↦ f(g, t) belong to certain Soblev spaces on A. From these regularity results for the spherical means we deduce, if G/K is a compact rank one symmetric space, a theorem on the almost everywhere localization of spherical harmonic expansions of functions in L2(G/K)