This article is cited in 13 papers

Theorems on ball mean values in symmetric spaces

V. V. Volchkov

Donetsk State University

Abstract: 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 $L(X)$.

UDC: 517.5

MSC: Primary 26B15, 43A85, 53C65; Secondary 53C35

Received: 17.07.2000 and 21.05.2001

DOI: 10.4213/sm593

 English version: Sbornik: Mathematics, 2001, 192:9, 1275–1296

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