Mathematische Annalen

, Volume 352, Issue 2, pp 293–337 | Cite as

An FIO calculus for marine seismic imaging, II: Sobolev estimates



We establish sharp L2-Sobolev estimates for classes of pseudodifferential operators with singular symbols [Guillemin and Uhlmann (Duke Math J 48:251–267, 1981), Melrose and Uhlmann (Commun Pure Appl Math 32:483–519, 1979)] whose non-pseudodifferential (Fourier integral operator) parts exhibit two-sided fold singularities. The operators considered include both singular integral operators along curves in \({\mathbb R^2}\) with simple inflection points and normal operators arising in linearized seismic imaging in the presence of fold caustics [Felea (Comm PDE 30:1717–1740, 2005), Felea and Greenleaf (Comm PDE 33:45–77, 2008), Nolan (SIAM J Appl Math 61:659–672, 2000)].

Mathematics Subject Classification (2000)

35S05 35S30 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abramowitz M., Stegun I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. Dover, New York (1972)Google Scholar
  2. 2.
    Beals R.: Spatially inhomogeneous pseudodifferential operators, II, Comm. Pure Appl. Math. 27, 161–205 (1974)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Beylkin G.: Imaging of discontinuities in the inverse problem by inversion of a generalized Radon transform. J. Math. Phys. 28, 99–108 (1985)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Casarino V., Secco S.: L pL q boundedness of analytic families of fractional integrals. Stud. Math. 184, 153–174 (2008)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Christ M., Nagel A., Stein E.M., Wainger S.: Singular and maximal Radon transforms: analysis and geometry. Ann. Math. 150, 489–577 (1999)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Comech A.: Optimal regularity of Fourier integral operators with one-sided folds. Comm. P.D.E. 24, 1263–1281 (1999)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Cuccagna S.: Sobolev estimates for fractional and singular Radon transforms. J. Funct. Anal. 139, 94–118 (1996)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Cuccagna S.: L 2 estimates for averaging operators along curves with two-sided k-fold singularities. Duke Math. J. 89, 203–216 (1997)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Duistermaat J.J.: Fourier Integral Operators. Birkhäuser, Boston (1996)MATHGoogle Scholar
  10. 10.
    Felea R.: Composition calculus of Fourier integral operators with fold and blowdown singularities. Comm. P.D.E. 30, 1717–1740 (2005)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Felea R., Greenleaf A.: An FIO calculus for marine seismic imaging: folds and crosscaps. Comm. P.D.E. 33, 45–77 (2008)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Geller D., Stein E.: Estimates for singular convolution operators on the Heisenberg group. Math. Ann. 267, 1–15 (1984)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Golubitsky M., Guillemin V.: Stable Mappings and their Singularities. Springer, New York (1973)MATHGoogle Scholar
  14. 14.
    Grafakos L.: Strong type endpoint bounds for analytic families of fractional integrals. Proc. Am. Math. Soc. 117, 653–663 (1993)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Greenblatt M.: An analogue to a theorem of Fefferman and Phong for averaging operators along curves with singular fractional integral kernel. Geom. Funct. Anal. 17, 1106–1138 (2007)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Greenleaf A., Seeger A.: Fourier integral operators with fold singularities. J. Reine Angew. Math. 455, 35–56 (1994)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Greenleaf, A., Seeger, A.: Oscillatory and Fourier integral operators with degenerate canonical relations. In: Proceedings of 6th International Conference on Harmonic Analysis and Partial Differential Equations, pp. 93–141. (El Escorial, 2000), Publ. Mat. Vol. Extra (2002)Google Scholar
  18. 18.
    Greenleaf A., Uhlmann G.: Estimates for singular Radon transforms and pseudodifferential operators with singular symbols. J. Funct. Anal. 89, 220–232 (1990)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Greenleaf A., Uhlmann G.: Composition of some singular Fourier integral operators and estimates for restricted X-ray transforms. Ann. Inst. Fourier Grenoble 40(2), 443–466 (1990)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Greenleaf A., Uhlmann G.: Composition of some singular Fourier integral operators and estimates for restricted X-ray transforms. II. Duke Math. J. 64(3), 415–444 (1991)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Greenleaf A., Uhlmann G.: Recovering singularities of a potential from singularities of scattering data. Commun. Math. Phys. 157, 549–572 (1993)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Guillemin, V.: On some results of Gelfand in integral geometry. In: Proceedings of Symposia in Pure Mathematics vol. 43, pp. 149–155. Amer. Math. Soc. Providence (1985)Google Scholar
  23. 23.
    Guillemin V., Uhlmann G.: Oscillatory integrals with singular symbols. Duke Math. J. 48(1), 251–267 (1981)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Hörmander L.: Fourier integral operators, I. Acta Math. 127, 79–183 (1971)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Hörmander L.: The Analysis of Linear Partial Differential Operators III Grundlehren math Wiss 274. Springer, Berlin (1985)Google Scholar
  26. 26.
    ten Kroode A., Smit D., Verdel A.: A microlocal analysis of migration. Wave Motion 28, 149–172 (1998)CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Kurylev Y., Lassas M., Uhlmann G.: Rigidity of broken geodesic flow and inverse problems. Am. J. Math. 132, 529–562 (2010)CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Melrose, R.: Marked Lagrangians. Lecture notes Max Planck Institut (1987)Google Scholar
  29. 29.
    Melrose R., Taylor M.: Near peak scattering and the corrected Kirchhoff approximation for a convex obstacle. Adv. Math. 55(3), 242–315 (1985)CrossRefMATHMathSciNetGoogle Scholar
  30. 30.
    Melrose R., Uhlmann G.: Lagrangian intersection and the Cauchy problem. Commun. Pure Appl. Math. 32(4), 483–519 (1979)CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    Mendoza G.: Symbol calculus associated with intersecting Lagrangians. Comm. P.D.E. 7, 1035–1116 (1982)CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    Nagel, A., Stein, E., Wainger, S.: Hilbert transforms and maximal functions related to variable curves. In: Proceedings of Symposia in Pure Mathematics, XXXV Part 1, pp. 95–98. Amer. Math. Soc., Providence RI (1979)Google Scholar
  33. 33.
    Nolan C.: Scattering in the presence of fold caustics. SIAM J. Appl. Math. 61(2), 659–672 (2000)CrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    Nolan C., Symes W.: Anomalous reflections near a caustic. Wave Motion 25, 1–14 (1997)CrossRefMATHMathSciNetGoogle Scholar
  35. 35.
    Nolan C., Symes W.: Global solutions of a linearized inverse problem for the acoustic wave equation. Comm. P.D.E. 22, 919–952 (1997)CrossRefMATHMathSciNetGoogle Scholar
  36. 36.
    Parissis I.: A sharp bound for the Stein-Wainger oscillatory integral. Proc. Am. Math. Soc. 136, 963–972 (2008)CrossRefMATHMathSciNetGoogle Scholar
  37. 37.
    Phong D., Stein E.M.: Hilbert integrals, singular integrals and Radon transforms. Acta Math. 157, 99–157 (1986)CrossRefMATHMathSciNetGoogle Scholar
  38. 38.
    Phong D., Stein E.M.: Singular Radon transforms and oscillatory integrals. Duke Math. J. 58, 347–369 (1989)CrossRefMATHMathSciNetGoogle Scholar
  39. 39.
    Phong D., Stein E.M.: Radon transforms and torsion. Int. Math. Res. Not. (4), 49–60 (1991)Google Scholar
  40. 40.
    Phong D., Stein E.M.: Models of degenerate Fourier integral operators and Radon transforms. Ann. Math. 140(2), 703–722 (1994)CrossRefMATHMathSciNetGoogle Scholar
  41. 41.
    Rakesh : A linearized inverse problem for the wave equation. Comm. P.D.E. 13, 573–601 (1988)CrossRefMATHMathSciNetGoogle Scholar
  42. 42.
    Ricci F., Stein E.M.: Harmonic analysis on nilpotent groups and singular integrals, III Fractional integration along manifolds. J. Funct. Anal. 86, 360–389 (1989)CrossRefMATHMathSciNetGoogle Scholar
  43. 43.
    Seeger A.: Degenerate Fourier integral operators in the plane. Duke Math. J. 71, 685–745 (1993)CrossRefMATHMathSciNetGoogle Scholar
  44. 44.
    Seeger A., Wainger S.: Bounds for singular fractional integrals and related Fourier integral operators. J. Funct. Anal. 199, 48–91 (2003)CrossRefMATHMathSciNetGoogle Scholar
  45. 45.
    Stein E.M.: Harmonic Analysis: Real Variable Methods, Orthogonality and Oscillatory Integrals. Princeton Univ Press, Princeton (1993)MATHGoogle Scholar
  46. 46.
    Stein E.M., Wainger S.: The estimation of an integral arising in multiplier transformations. Stud. Math. 35, 101–104 (1970)MATHMathSciNetGoogle Scholar
  47. 47.
    Stein E.M., Wainger S.: Problems in harmonic analysis related to curvature. Bull. Am. Math. Soc. 84, 1239–1295 (1978)CrossRefMATHMathSciNetGoogle Scholar
  48. 48.
    Stolk C.: Microlocal analysis of a seismic linearized inverse problem. Wave Motion 32, 267–290 (2000)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Raluca Felea
    • 1
  • Allan Greenleaf
    • 2
  • Malabika Pramanik
    • 3
  1. 1.School of of Mathematical SciencesRochester Institute of TechnologyRochesterUSA
  2. 2.Department of MathematicsUniversity of RochesterRochesterUSA
  3. 3.Department of MathematicsUniversity of British ColumbiaVancouverCanada

Personalised recommendations