Theorems on ball mean values in symmetric spaces

V. V. Volchkov

Sbornik. Mathematics, Tome 192 (2001) no. 9, pp. 1275-1296

Résumé

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 $L(X)$.

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