Injectivity of the Spherical Mean Operator on the Conical Manifolds of Spheres
- 32 Downloads
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.
Unable to display preview. Download preview PDF.
- 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
- 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.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.Helgason S., Groups and Geometric Analysis, Academic Press, Orlando etc. (1984).Google Scholar
- 7.Globevnik J., A Decomposition of Functions with Zero Means on Circles [Preprint].Google Scholar
- 8.Courant R. and Hilbert D., Methods of Mathematical Physics. Vol. 2, Wiley, Interscience, New York (1962).Google Scholar
- 9.Folland G. B., Introduction to Partial Differential Equations, Princeton Univ. Press, Princeton, NY (1995).Google Scholar
- 10.Hörmander L., The Analysis of Linear Partial Differential Operators. I, Springer-Verlag, Berlin (1983).Google Scholar
- 11.Epstein C. L., “Incoming and Outgoing Waves,” J. Differential Equations, 60, 337–362 (1985).Google Scholar
- 12.Zalcman L., “Mean values and differential equations,” Israel J. Math., 14, 339–352 (1973)Google Scholar