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 $L(X)$.
\Bibitem{Vol01}
\by V.~V.~Volchkov
\paper Theorems on ball mean values in symmetric spaces
\jour Sb. Math.
\yr 2001
\vol 192
\issue 9
\pages 1275--1296
\mathnet{http://mi.mathnet.ru/eng/sm593}
\crossref{https://doi.org/10.1070/SM2001v192n09ABEH000593}
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https://www.mathnet.ru/eng/sm593
https://doi.org/10.1070/SM2001v192n09ABEH000593
https://www.mathnet.ru/eng/sm/v192/i9/p17
This publication is cited in the following 13 articles:
Volchkov V.V., Volchkov V.V., “A uniqueness theorem for the non-Euclidean Darboux equation”, Lobachevskii J. Math., 38:2, SI (2017), 379–385
V. V. Volchkov, Vit. V. Volchkov, “Behaviour at infinity of solutions of twisted convolution equations”, Izv. Math., 76:1 (2012), 79–93
O. A. Ochakovskaya, “Theorems on ball mean values for solutions of the Helmholtz equation on unbounded domains”, Izv. Math., 76:2 (2012), 365–374
Ochakovskaya O.A., “Spherical Mean Theorems for Solutions of the Helmholtz Equation”, Dokl. Math., 85:1 (2012), 60–62
Ochakovskaya O.A., “On the Injectivity of the Pompeiu Transform for Integral Ball Means”, Ukrainian Math J, 63:3 (2011), 416–424
O. A. Ochakovskaya, “Precise characterizations of admissible rate of decrease of a non-trivial function with zero ball means”, Sb. Math., 199:1 (2008), 45–65
Ochakovskaya, OA, “MAJORANTS OF FUNCTIONS WITH VANISHING INTEGRALS OVER BALLS”, Ukrainian Mathematical Journal, 60:6 (2008), 1003
Volchkov, VV, “Convolution equations and the local Pompeiu property on symmetric spaces and on phase space associated to the Heisenberg group”, Journal D Analyse Mathematique, 105 (2008), 43
V. V. Volchkov, “Local two-radii theorem in symmetric spaces”, Sb. Math., 198:11 (2007), 1553–1577
Ochakovskaya, OA, “Liouville-type theorems for functions with zero integrals over balls of fixed radius”, Doklady Mathematics, 76:1 (2007), 530
Vit. V. Volchkov, “Functions with ball mean values equal to zero on compact two-point homogeneous spaces”, Sb. Math., 198:4 (2007), 465–490
Volchkov V.V., Volchkov V.V., “New results in integral geometry”, Complex Analysis and Dynamical Systems II, Contemporary Mathematics Series, 382, 2005, 417–432
Vit. V. Volchkov, “Local two radii theorem on the sphere”, St. Petersburg Math. J., 16:3 (2005), 453–475
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