Advertisement

Siberian Mathematical Journal

, Volume 45, Issue 4, pp 597–605 | Cite as

Injectivity of the Spherical Mean Operator on the Conical Manifolds of Spheres

  • M. L. Agranovsky
  • E. K. Narayanan
Article

Abstract

Let f be a continuous function on ℝ n . If f has zero integral over every sphere intersecting a given subset A of ℝ n and A lies in no affine plane of dimension n -2, then f vanishes identically. The condition on the dimension of A is sharp.

spherical mean wave equation dependence domain 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Cormack A. M. and Quinto E. T., “A Radon transform on spheres through the origin in ℝn and applications to the Darboux equation,” Trans. Amer. Math. Soc., 260, 575–581 (1980).Google Scholar
  2. 2.
    Quinto E. T., “Null spaces and ranges for the classical and spherical Radon transforms,” J. Math. Anal. Appl., 90, No. 2, 408–420 (1982).CrossRefGoogle Scholar
  3. 3.
    Epstein C. L. and Kleiner B., “Spherical means in annular regions,” Comm. Pure Appl. Math., 46, No. 3, 441–451 (1993).Google Scholar
  4. 4.
    Globevnik J., “Zero integrals on circles and characterizations of harmonic and analytic functions,” Trans. Amer. Math. Soc., 317, No. 1, 313–330 (1990).Google Scholar
  5. 5.
    Helgason S., Groups and Geometric Analysis, Academic Press, Orlando etc. (1984).Google Scholar
  6. 6.
    Globevnik J., “Holomorphic extensions from open families of circles,” Trans. Amer. Math. Soc., 355, No. 5, 1921–1931 (2003).CrossRefGoogle Scholar
  7. 7.
    Globevnik J., A Decomposition of Functions with Zero Means on Circles [Preprint].Google Scholar
  8. 8.
    Courant R. and Hilbert D., Methods of Mathematical Physics. Vol. 2, Wiley, Interscience, New York (1962).Google Scholar
  9. 9.
    Folland G. B., Introduction to Partial Differential Equations, Princeton Univ. Press, Princeton, NY (1995).Google Scholar
  10. 10.
    Hörmander L., The Analysis of Linear Partial Differential Operators. I, Springer-Verlag, Berlin (1983).Google Scholar
  11. 11.
    Epstein C. L., “Incoming and Outgoing Waves,” J. Differential Equations, 60, 337–362 (1985).Google Scholar
  12. 12.
    Zalcman L., “Mean values and differential equations,” Israel J. Math., 14, 339–352 (1973)Google Scholar

Copyright information

© Plenum Publishing Corporation 2004

Authors and Affiliations

  • M. L. Agranovsky
    • 1
  • E. K. Narayanan
    • 2
  1. 1.Bar-Ilan UniversityRamat Gan
  2. 2.Indian Institute of ScienceBangalore

Personalised recommendations