Composition of some singular Fourier integral operators and estimates for restricted X-ray transforms
Annales de l'Institut Fourier, Tome 40 (1990) no. 2, pp. 443-466.

Nous établissons un calcul de composition pour les opérateurs intégraux de Fourier associés à une classe de relations canoniques lisses C(T * X0)×(T * Y0). Ces relations canoniques, qui se présentent en géométrie intégrale sont telles que π : CT * Y est un pli de Whitney et ρ : CT * X est une application blow-down. Si AI m (C), BI m (C t ), alors BAI m+m ,0 (Δ,Λ) qui est une classe d’opérateurs pseudodifférentiels avec des symboles singuliers. Il s’ensuit que A est borné sur L 2 avec une perte de dérivée d’un 1/4.

We establish a composition calculus for Fourier integral operators associated with a class of smooth canonical relations C(T * X0)×(T * Y0). These canonical relations, which arise naturally in integral geometry, are such that π : CT * Y is a Whitney fold and ρ : CT * X is a blow-down mapping. If AI m (C), BI m (C t ), then BAI m+m ,0 (Δ,Λ) a class of pseudodifferential operators with singular symbols. From this follows L 2 boundedness of A with a loss of 1/4 derivative.

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     author = {Greenleaf, Allan and Uhlmann, Gunther},
     title = {Composition of some singular {Fourier} integral operators and estimates for restricted $X$-ray transforms},
     journal = {Annales de l'Institut Fourier},
     pages = {443--466},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {40},
     number = {2},
     year = {1990},
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Greenleaf, Allan; Uhlmann, Gunther. Composition of some singular Fourier integral operators and estimates for restricted $X$-ray transforms. Annales de l'Institut Fourier, Tome 40 (1990) no. 2, pp. 443-466. doi : 10.5802/aif.1220. http://www.numdam.org/articles/10.5802/aif.1220/

[1] J. Antoniano and G. Uhlmann, A functional calculus for a class of pseudodifferential operators with singular symbols, Proc. Symp. Pure Math, 43 (1985), 5-16. | MR | Zbl

[2] A. Besse, Manifolds all of Whose Geodesics are Closed, Springer-Verlag, New York, 1978. | MR | Zbl

[3] L. Boutet De Monvel, Hypoelliptic equations with double characteristics and related pseudodifferential operators, Comm. Pure Appl. Math., 27 (1974), 585-639. | MR | Zbl

[4] A. P. Calderón and R. Vaillancourt, A class of bounded pseudodifferential operators, Proc. Nat. Acad. Sci. USA, 69 (1972), 1185-1187. | MR | Zbl

[5] M. Christ, Estimates for the k-plane transform, Indiana Univ. Math. Jour., 33 (1984), 891-910. | MR | Zbl

[6] S. Drury, Lp estimates for the X-ray transform, Illinois Jour. Math., 27 (1983), 125-129. | MR | Zbl

[7] S. Drury, Generalizations of Riesz potentials and Lp estimates for certain k-plane transforms, Illinois Jour. Math., 28 (1984), 495-512. | MR | Zbl

[8] J. J. Duistermaat, Fourier Integral Operators, Courant Institute, New York, 1973. | MR | Zbl

[9] J. J. Duistermaat and V. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics, Inv. Math., 29 (1975), 39-79. | MR | Zbl

[10] C. Fefferman and E. M. Stein, Hp spaces of several variables, Acta Math., 129 (1972), 137-193. | MR | Zbl

[11] I. M. Gelfand, M. I. Graev and N. Ya. Vilenkin, Generalized Functions, V, Academic Press, New York, 1966. | Zbl

[12] A. Greenleaf and G. Uhlmann, Nonlocal inversion formulas for the X-ray transform, Duke Math. Jour., 58 (1989), 205-240. | MR | Zbl

[13] A. Greenleaf and G. Uhlmann, Estimates for singular Radon transforms and pseudodifferential operators with singular symbols, Jour. Func. Anal., 89 (1990), 202-232. | MR | Zbl

[14] V. Guillemin, On some results of Gelfand in integral geometry, Proc. Symp. Pure Math., 43 (1985), 149-155. | MR | Zbl

[15] V. Guillemin, Cosmology in (2 + 1)-dimensions, Cyclic Models, and Deformations, Princeton University Press, Princeton, 1989. | Zbl

[16] V. Guillemin and G. Uhlmann, Oscillatory integrals with singular symbols, Duke Math. Jour., 48 (1981), 251-267. | MR | Zbl

[17] S. Helgason, The Radon Transform, Birkhaüser, Boston, 1980. | Zbl

[18] L. Hörmander, Fourier integral operators, I, Acta Math., 127 (1971), 79-183. | MR | Zbl

[19] L. Hörmander, The Analysis of Linear Partial Differential Operators, IV, Springer-Verlag, New York, 1985. | Zbl

[20] R. Melrose, Equivalence of glancing hypersurfaces, Inv. Math., 37 (1976), 165-191. | MR | Zbl

[21] R. Melrose, Equivalence of glancing hypersurfaces, II, Math. Ann., 255 (1981), 159-198. | MR | Zbl

[22] R. Melrose, Transformation of boundary problems, Acta Math., 147 (1981), 149-236. | MR | Zbl

[23] R. Melrose, The wave equation for a hypoelliptic operator with symplectic characteristics of codimension two, Jour. d'Analyse Math., 44 (1984-1985), 134-182. | MR | Zbl

[24] R. Melrose, Marked lagrangians, notes of lectures at Max Planck Institut, 1987, paper in preparation.

[25] R. Melrose and M. Taylor, Near peak scattering and the corrected Kirchhoff approximation for a convex obstacle, Adv. in Math., 55 (1985), 242-315. | MR | Zbl

[26] R. Melrose and G. Uhlmann, Lagrangian intersection and the Cauchy problem, Comm. Pure Appl. Math., 32 (1979), 482-519. | MR | Zbl

[27] D. Oberlin and E. M. Stein, Mapping Properties of the Radon transform, Indiana Univ. Math. Jour., 31 (1982), 641-650. | MR | Zbl

[28] K. T. Smith and D. C. Solmon, Lower dimensional integrability of L2 functions, Jour. Math. Anal. Appl., 51 (1975), 539-549. | MR | Zbl

[29] R. Strichartz, The Hardy space H1 on manifolds and submanifolds, Canad. Jour. Math., 24 (1972), 915-925. | MR | Zbl

[30] R. Strichartz, Lp estimates for Radon transforms in euclidian and non-euclidian spaces, Duke Math. Jour., 48 (1981), 699-737. | MR | Zbl

[31] H.-T. Wang, Lp estimates for the restricted X-ray transform, Ph. D. thesis, Univ. of Rochester, June 1987.

[32] A. Weinstein, On Maslov's quantization condition, in Fourier Integral Operators and Partial Differential Equations, J. Chazarain, ed., Springer-Verlag, 1975. | MR | Zbl

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