×

Pompeiu transforms on geodesic spheres in real analytic manifolds. (English) Zbl 0791.58017

The author’s abstract: “We prove a support theorem for Pompeiu transforms integrating on geodesic spheres of fixed radius \(r>0\) on real analytic manifolds when the measures are real analytic and nowhere zero. To avoid pathologies, we assume that \(r\) is less than the injectivity radius at the center of each sphere being integrated over. The proof of the main result is local and it involves the microlocal properties of the Pompeiu transform and a theorem of Hörmander, Kawai, and Kashiwara on microlocal singularities”.
Reviewer: W.Mozgawa (Lublin)

MSC:

58C35 Integration on manifolds; measures on manifolds
53C35 Differential geometry of symmetric spaces
43A90 Harmonic analysis and spherical functions
32C05 Real-analytic manifolds, real-analytic spaces
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Berenstein, C. A.; Gay, R., A local version of the two circles theorem, Israel J. Math., 55, 267-288 (1986) · Zbl 0624.31002
[2] Berenstein, C. A.; Gay, R., Le Probléme de Pompeiu local, J. Analyse Math., 52, 133-166 (1989) · Zbl 0668.30037
[3] Berenstein, C. A.; Guy, R.; Yger, A., Inversion of the local Pompeiu transform, J. Analyse Math., 54, 259-287 (1990) · Zbl 0723.44002 · doi:10.1007/BF02796152
[4] Berenstein, C. A.; Zalcman, L., Pompeiu’s problem on symmetric spaces, Comment. Math. Helv., 55, 593-621 (1980) · Zbl 0452.43012 · doi:10.1007/BF02566709
[5] Boman, J.; Quinto, E. T., Support theorems for real analytic Radon transforms, Duke Math. J., 55, 943-948 (1987) · Zbl 0645.44001 · doi:10.1215/S0012-7094-87-05547-5
[6] Boman, J.; Quinto, E. T., Support theorems for real analytic Radon transforms on line complexes in ℝ^3, Trans. Amer. Math. Soc., 335, 877-890 (1993) · Zbl 0767.44001 · doi:10.2307/2154410
[7] Delsarte, J.; Lions, J. L., Moyennes généralisées, Comment. Math. Helv., 33, 59-69 (1959) · Zbl 0084.09404 · doi:10.1007/BF02565907
[8] V. Guillemin,Some remarks on integral geometry, unpublished, 1975.
[9] Guillemin, V.; Sternberg, S., Geometric Asymptotics (1977), Providence, RI: Amer. Math. Soc., Providence, RI · Zbl 0364.53011
[10] Helgason, S., Differential operators on homogeneous spaces, Acta Math., 102, 239-299 (1959) · Zbl 0146.43601 · doi:10.1007/BF02564248
[11] Hörmander, L., The Analysis of Linear Partial Differential Operators I (1983), New York: Springer, New York · Zbl 0521.35001
[12] John, F., Plane Waves and Spherical Means (1966), New York: Interscience, New York
[13] Kaneko, A., Introduction to Hyperfunctions (1989), New York: Kluwer, New York
[14] Kobayashi, S.; Nomizu, K., Foundations of Differential Geometry (1963), New York: Interscience, New York · Zbl 0119.37502
[15] Quinto, E. T., The dependence of the generalized Radon transform on defining measures, Trans. Amer. Math. Soc., 257, 331-346 (1980) · Zbl 0471.58022 · doi:10.2307/1998299
[16] Quinto, E. T., The invertibility of rotation invariant Radon transforms, J. Math. Anal. Appl., 91, 510-522 (1983) · Zbl 0517.44009 · doi:10.1016/0022-247X(83)90165-8
[17] Schneider, R., Functions on a sphere with vanishing integrals over certain subspheres, J. Math. Anal. Appl., 26, 381-384 (1969) · Zbl 0167.32703 · doi:10.1016/0022-247X(69)90160-7
[18] Sato, M.; Kawai, T.; Kashiwara, M., Hyperfunctions and pseudodifferential equations, 265-529 (1973), New York: Springer-Verlag, New York
[19] Shahshahani, M.; Sitaram, A., The Pompeiu problem in exterior domains in symmetric spaces, Contemporary Math., 63, 267-277 (1987) · Zbl 0615.53060
[20] Sunada, T., Spherical means and geodesic chains on a Riemannian manifold, Trans. Amer. Math. Soc., 267, 483-501 (1981) · Zbl 0514.58037 · doi:10.2307/1998666
[21] Treves, F., Introduction to Pseudodifferential and Fourier Integral Operators I (1980), New York: Plenum Press, New York · Zbl 0453.47027
[22] Tsujishita, T., Spherical means on Riemannian manifolds, Osaka J. Math., 13, 591-597 (1976) · Zbl 0355.58015
[23] Yosida, K., Lectures on Differential and Integral Equations (1960), New York: Interscience, New York · Zbl 0090.08401
[24] Zalcman, L., Analyticity and the Pompeiu problem, Arch. Rat. Mech. Anal., 47, 237-254 (1972) · Zbl 0251.30047 · doi:10.1007/BF00250628
[25] Zalcman, L., Offbeat integral geometry, Amer. Math. Monthly, 87, 161-175 (1980) · Zbl 0433.53048 · doi:10.2307/2321600
[26] Zalcman, L.; Fuglede, B.; Goldstein, M.; Haussmann, W.; Hayman, W. K.; Rogge, L., A bibliographic survey of the Pompeiu problem, Approximation of Solutions of Partial Differential Equations, 185-194 (1992), Boston: Kluwer Academic, Boston · Zbl 0830.26005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.