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Mat. Sb., 2001, Volume 192, Number 9, Pages 17–38 (Mi msb593)  

This article is cited in 13 scientific papers (total 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 $L(X)$.

DOI: https://doi.org/10.4213/sm593

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

Bibliographic databases:

UDC: 517.5
MSC: Primary 26B15, 43A85, 53C65; Secondary 53C35
Received: 17.07.2000 and 21.05.2001

Citation: V. V. Volchkov, “Theorems on ball mean values in symmetric spaces”, Mat. Sb., 192:9 (2001), 17–38; Sb. Math., 192:9 (2001), 1275–1296

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Vit. V. Volchkov, “Local two radii theorem on the sphere”, St. Petersburg Math. J., 16:3 (2005), 453–475  mathnet  crossref  mathscinet  zmath
    2. Volchkov V.V., Volchkov V.V., “New results in integral geometry”, Complex Analysis and Dynamical Systems II, Contemporary Mathematics Series, 382, 2005, 417–432  crossref  mathscinet  zmath  isi
    3. V. V. Volchkov, “Local two-radii theorem in symmetric spaces”, Sb. Math., 198:11 (2007), 1553–1577  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    4. Ochakovskaya, OA, “Liouville-type theorems for functions with zero integrals over balls of fixed radius”, Doklady Mathematics, 76:1 (2007), 530  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    5. Vit. V. Volchkov, “Functions with ball mean values equal to zero on compact two-point homogeneous spaces”, Sb. Math., 198:4 (2007), 465–490  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    6. 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  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    7. Ochakovskaya, OA, “MAJORANTS OF FUNCTIONS WITH VANISHING INTEGRALS OVER BALLS”, Ukrainian Mathematical Journal, 60:6 (2008), 1003  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    8. 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  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    9. Ochakovskaya O.A., “On the Injectivity of the Pompeiu Transform for Integral Ball Means”, Ukrainian Math J, 63:3 (2011), 416–424  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    10. V. V. Volchkov, Vit. V. Volchkov, “Behaviour at infinity of solutions of twisted convolution equations”, Izv. Math., 76:1 (2012), 79–93  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    11. O. A. Ochakovskaya, “Theorems on ball mean values for solutions of the Helmholtz equation on unbounded domains”, Izv. Math., 76:2 (2012), 365–374  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    12. Ochakovskaya O.A., “Spherical Mean Theorems for Solutions of the Helmholtz Equation”, Dokl. Math., 85:1 (2012), 60–62  crossref  mathscinet  zmath  isi  elib  scopus
    13. Volchkov V.V., Volchkov V.V., “A uniqueness theorem for the non-Euclidean Darboux equation”, Lobachevskii J. Math., 38:2, SI (2017), 379–385  crossref  mathscinet  zmath  isi  scopus
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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