Geodesic
Theorems on ball mean values in symmetric spaces
Sbornik. Mathematics, Tome 192 (2001) no. 9, pp. 1275-1296

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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 $L(X)$. 
@article{SM_2001_192_9_a1,
     author = {V. V. Volchkov},
     title = {Theorems on ball mean values in symmetric spaces},
     journal = {Sbornik. Mathematics},
     pages = {1275--1296},
     publisher = {mathdoc},
     volume = {192},
     number = {9},
     year = {2001},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2001_192_9_a1/}
}
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V. V. Volchkov. Theorems on ball mean values in symmetric spaces. Sbornik. Mathematics, Tome 192 (2001) no. 9, pp. 1275-1296. http://geodesic.mathdoc.fr/item/SM_2001_192_9_a1/