Theorems on ball mean values in symmetric spaces

Various classes of functions on a non-compact Riemannian symmetric space X of rank 1 with vanishing integrals over all balls of fixed radius are studied. The central result of the paper includes precise conditions on the growth of a linear combination of functions from such classes; in particular, failing these conditions means that each of these functions is equal to zero. This is a considerable refinement over the well-known two-radii theorem of Berenstein-Zalcman. As one application, a description of the Pompeiu subsets of X is given in terms of approximation of their indicator functions in L(X).