Support theorems for Radon transforms on higher rank symmetric spaces
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- by Fulton Gonzalez and Eric Todd Quinto PDF
- Proc. Amer. Math. Soc. 122 (1994), 1045-1052 Request permission
Abstract:
We prove support theorems for Radon transforms with real-analytic measures on horocycles in higher rank symmetric spaces. The microlocal analysis is more difficult than for rank one, but we prove a generalization of Helgason’s support theorem and a theorem that is new even in the classical case.References
- Jan Boman and Eric Todd Quinto, Support theorems for Radon transforms on real analytic line complexes in three-space, Trans. Amer. Math. Soc. 335 (1993), no. 2, 877–890. MR 1080733, DOI 10.1090/S0002-9947-1993-1080733-8
- Josip Globevnik, A support theorem for the X-ray transform, J. Math. Anal. Appl. 165 (1992), no. 1, 284–287. MR 1151072, DOI 10.1016/0022-247X(92)90079-S
- Allan Greenleaf and Gunther Uhlmann, Nonlocal inversion formulas for the X-ray transform, Duke Math. J. 58 (1989), no. 1, 205–240. MR 1016420, DOI 10.1215/S0012-7094-89-05811-0 V. Guillemin, Some remarks on integral geometry, unpublished, 1975.
- Victor Guillemin, On some results of Gel′fand in integral geometry, Pseudodifferential operators and applications (Notre Dame, Ind., 1984) Proc. Sympos. Pure Math., vol. 43, Amer. Math. Soc., Providence, RI, 1985, pp. 149–155. MR 812288, DOI 10.1090/pspum/043/812288
- Victor Guillemin and Shlomo Sternberg, Geometric asymptotics, Mathematical Surveys, No. 14, American Mathematical Society, Providence, R.I., 1977. MR 0516965
- S. Helgason, Duality and Radon transform for symmetric spaces, Amer. J. Math. 85 (1963), 667–692. MR 158409, DOI 10.2307/2373114
- Sigurđur Helgason, An analogue of the Paley-Wiener theorem for the Fourier transform on certain symmetric spaces, Math. Ann. 165 (1966), 297–308. MR 223497, DOI 10.1007/BF01344014
- Sigurđur Helgason, A duality for symmetric spaces with applications to group representations, Advances in Math. 5 (1970), 1–154 (1970). MR 263988, DOI 10.1016/0001-8708(70)90037-X
- Sigurdur Helgason, The surjectivity of invariant differential operators on symmetric spaces. I, Ann. of Math. (2) 98 (1973), 451–479. MR 367562, DOI 10.2307/1970914
- Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 514561
- Sigurdur Helgason, Groups and geometric analysis, Pure and Applied Mathematics, vol. 113, Academic Press, Inc., Orlando, FL, 1984. Integral geometry, invariant differential operators, and spherical functions. MR 754767 L. Hörmander, Fourier integral operators. I, Acta Math. 127 (1971), 79-183. —, The analysis of linear partial differential operators. I, Springer, New York, 1983.
- A. Kaneko, Introduction to hyperfunctions, Mathematics and its Applications (Japanese Series), vol. 3, Kluwer Academic Publishers Group, Dordrecht; SCIPRESS, Tokyo, 1988. Translated from the Japanese by Y. Yamamoto. MR 1026013
- Eric Todd Quinto, The dependence of the generalized Radon transform on defining measures, Trans. Amer. Math. Soc. 257 (1980), no. 2, 331–346. MR 552261, DOI 10.1090/S0002-9947-1980-0552261-8
- Eric Todd Quinto, Real analytic Radon transforms on rank one symmetric spaces, Proc. Amer. Math. Soc. 117 (1993), no. 1, 179–186. MR 1135080, DOI 10.1090/S0002-9939-1993-1135080-8
- Mikio Sato, Takahiro Kawai, and Masaki Kashiwara, Microfunctions and pseudo-differential equations, Hyperfunctions and pseudo-differential equations (Proc. Conf., Katata, 1971; dedicated to the memory of André Martineau), Lecture Notes in Math., Vol. 287, Springer, Berlin, 1973, pp. 265–529. MR 0420735
- C. D. Sogge and E. M. Stein, Averages over hypersurfaces. Smoothness of generalized Radon transforms, J. Analyse Math. 54 (1990), 165–188. MR 1041180, DOI 10.1007/BF02796147
- François Trèves, Introduction to pseudodifferential and Fourier integral operators. Vol. 2, University Series in Mathematics, Plenum Press, New York-London, 1980. Fourier integral operators. MR 597145
- Peter D. Lax and Ralph S. Phillips, A local Paley-Wiener theorem for the Radon transform of $L_{2}$ functions in a non-Euclidean setting, Comm. Pure Appl. Math. 35 (1982), no. 4, 531–554. MR 657826, DOI 10.1002/cpa.3160350404
Additional Information
- © Copyright 1994 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 122 (1994), 1045-1052
- MSC: Primary 44A12; Secondary 43A85, 58G15
- DOI: https://doi.org/10.1090/S0002-9939-1994-1205492-3
- MathSciNet review: 1205492