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Radon transforms and finite type conditions


Author: Andreas Seeger
Journal: J. Amer. Math. Soc. 11 (1998), 869-897
MSC (1991): Primary 35S30; Secondary 47G10, 32F40, 44A12
DOI: https://doi.org/10.1090/S0894-0347-98-00280-X
MathSciNet review: 1623430
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Abstract: We prove regularity of Radon type integral operators in $L^{p}$-Sobolev spaces.


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  • 1. M. Christ, Hilbert transforms along curves, I. Nilpotent groups, Ann. Math. 122 (1985), 575-596. MR 87f:42039a
  • 2. -, Failure of an endpoint estimate for integrals along curves, Fourier analysis and partial differential equations, ed. by J. Garcia-Cuerva, E. Hernandez, F. Soria and J. L. Torrea, CRC Press, 1995. MR 97e:44007
  • 3. M. Christ, A. Nagel, E. M. Stein and S. Wainger, Singular and maximal Radon transforms: analysis and geometry, preprint, 1997 (to appear).
  • 4. A. Greenleaf and A. Seeger, Fourier integral operators with fold singularities, J. reine ang. Math. 455 (1994), 35-56. MR 95h:58130
  • 5. -, Fourier integral operators with simple cusps, Amer. J. Math. (to appear).
  • 6. A. Greenleaf and G. Uhlmann, Composition of some singular Fourier integral operators and estimates for the X-ray transform, I, Ann. Inst. Fourier (Grenoble) 40 (1990), 443-466. MR 91k:58126
  • 7. -, Composition of some singular Fourier integral operators and estimates for the X-ray transform, II, Duke Math. J. 64 (1991), 413-419. MR 93b:58146
  • 8. V. Guillemin, Cosmology in $(2+1)$-dimensions, cyclic models and deformations of $M_{2,1}$, Ann. of Math. Studies 121, Princeton Univ. Press, Princeton, 1989. MR 91k:58140
  • 9. L. Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147-171. MR 36:5526
  • 10. -, Fourier integral operators I, Acta Math. 127 (1971), 79-183. MR 52:9299
  • 11. J. J. Kohn, Boundary behavior of $\overline{\partial }$ on weakly pseudo-convex manifolds of dimension two, J. Diff. Geom. 6 (1972), 523-542. MR 48:727
  • 12. R. Melrose and M. Taylor, Near peak scattering and the correct Kirchhoff approximation for a convex obstacle, Adv. in Math. 55 (1985), 242-315. MR 86m:35095
  • 13. D. H. Phong, Singular integrals and Fourier integral operators, Essays on Fourier analysis in honor of Elias M. Stein, edited by C. Fefferman, R. Fefferman and S. Wainger, Princeton University Press, 1995. MR 95k:42026
  • 14. D. H. Phong and E. M. Stein, Hilbert integrals, singular integrals and Radon transforms I, Acta Math. 157 (1986), 99-157. MR 88i:42028a
  • 15. -, Radon transforms and torsion, International Mathematics Research Notices (1991), 49-60. MR 93g:58144
  • 16. -, Models of degenerate Fourier integral operators and Radon transforms, Ann. Math. 140 (1994), 703-722. MR 96c:35206
  • 17. -, The Newton polyhedron and oscillatory integral operators, Acta Math. 179 (1997), 146-177. CMP 98:05
  • 18. L. P. Rothschild and E. M. Stein, Hypoelliptic operators and nilpotent groups, Acta Math. 137 (1976), 247-320. MR 55:9171
  • 19. A. Seeger, Degenerate Fourier integral operators in the plane, Duke Math. J. 71 (1993), 685-745. MR 94h:35292
  • 20. H. Smith and C. D. Sogge, $L^{p}$ regularity for the wave equation with strictly convex obstacles, Duke Math. J. 73 (1994), 97-153. MR 95c:35048
  • 21. C. D. Sogge and E. M. Stein, Averages of functions over hypersurfaces: smoothness of generalized Radon transforms, J. Analyse Math. 54 (1990), 165-188. MR 91i:58145
  • 22. E. M. Stein, Harmonic analysis: Real variable methods, orthogonality and oscillatory integrals, Princeton Univ. Press, 1993. MR 95c:42002
  • 23. S. Sternberg, Differential Geometry, second edition, Chelsea, 1983. MR 88f:58001

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Additional Information

Andreas Seeger
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
Email: seeger@math.wisc.edu

DOI: https://doi.org/10.1090/S0894-0347-98-00280-X
Keywords: Averaging operators, Radon transforms, finite type conditions
Received by editor(s): October 28, 1997
Additional Notes: The author’s research was supported in part by an NSF grant.
Article copyright: © Copyright 1998 American Mathematical Society

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