Journal d'Analyse Mathématique

, Volume 109, Issue 1, pp 81–162 | Cite as

Fourier-integral-operator product representation of solutions to first-order symmetrizable hyperbolic systems

  • Jérôme Le RousseauEmail author


We consider the first-order Cauchy problem
$$ \begin{gathered} \partial _z u + a(z,x,D_x )u = 0,0 < z \leqslant Z, \hfill \\ u|_{z = 0} = u_0 , \hfill \\ \end{gathered} $$
with Z > 0 and a(z, x,D x ) a k × k matrix of pseudodifferential operators of order one, whose principal part a 1 is assumed symmetrizable: there exists L(z, x, ξ) of order 0, invertible, such that
$$ a_1 (z,x,\xi ) = L(z,x,\xi )( - i\beta _1 (z,x,\xi ) + \gamma _1 (z,x,\xi ))(L(z,x,\xi ))^{ - 1} , $$
where β 1 and γ 1 are hermitian symmetric and γ 1 ≥ 0. An approximation Ansatz for the operator solution, U(z′, z), is constructed as the composition of global Fourier integral operators with complex matrix phases. In the symmetric case, an estimate of the Sobolev operator norm in L((H (s)(R n )) k , (H (s)(R n )) k ) of these operators is provided, which yields a convergence result for the Ansatz to U(z′, z) in some Sobolev space as the number of operators in the composition goes to ∞, in both the symmetric and symmetrizable cases. We thus obtain a representation of the solution operator U(z′, z) as an infinite product of Fourier integral operators with matrix phases.


Cauchy Problem Phase Function Hyperbolic System Pseudodifferential Operator Scalar Case 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    K. Asada and D. Fujiwara, On some oscillatory integral transformations in L 2(R n), Japan. J. Math. (N.S.) 4 (1978), 299–361.MathSciNetGoogle Scholar
  2. [2]
    J.-M. Bony, Sur l’inégalité de Fefferman-Phong, Sémin. Equ. Dériv. Partielles, Ecole Polytech., Palaiseau, France, 1999.Google Scholar
  3. [3]
    A. Boulkhemair, Estimations L 2 précisées pour des intégrales oscillantes, Comm. Partial Differential Equations 22 (1997), 165–184.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    J. Chazarain and A. Piriou, Introduction to the Theory of Linear Partial Differential Equations, North-Holland, Amsterdam, 1982.zbMATHGoogle Scholar
  5. [5]
    M. V. de Hoop, Microlocal analysis of seismic inverse scattering, in Inside Out, Inverse Problems and Applications (G. Uhlmann, ed.), CambridgeUniversity Press, Cambridge, 2003, pp. 219–296.Google Scholar
  6. [6]
    M. V. de Hoop and A. T. de Hoop, Elastic wave up/down decomposition in inhomogeneous and anisotropic media: an operator approach and its approximations, Wave Motion 20 (1994), 57–82.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    M. V. de Hoop, J. Le Rousseau and B. Biondi, Symplectic structure of wave-equation imaging: A path-integral approach based on the double-square-root equation, Geophys. J. Internat. 153 (2003), 52–74.CrossRefGoogle Scholar
  8. [8]
    M. V. de Hoop, J. Le Rousseau and R.-S. Wu, Generalization of the phase-screen approximation for the scattering of acoustic waves, Wave Motion 31 (2000), 43–70.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    J. J. Duistermaat, Fourier Integral Operators, Birkhäuser Boston, Boston, 1996.zbMATHGoogle Scholar
  10. [10]
    K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, Berlin, 1999.Google Scholar
  11. [11]
    D. Fujiwara and N. Kumano-go, Smooth functional derivatives in Feynman path integrals by time slicing approximation, Bull. Sci. Math. 129 (2005), 57–79.zbMATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    L. Hörmander, Fourier integral operators I, Acta Math. 127 (1971), 79–183.zbMATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    L. Hörmander, L 2 estimate for Fourier integral operators with complex phase, Ark. Mat. 21 (1983), 283–307.zbMATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    L. Hörmander, The Analysis of Linear Partial Differential Operators, Vol. IV, Springer-Verlag, Berlin, 1985.Google Scholar
  15. [15]
    L. Hörmander, The Analysis of Linear Partial Differential Operators, Vol. III, Springer-Verlag, Berlin, 1985; second printing, 1994.Google Scholar
  16. [16]
    L. Hörmander, The Analysis of Linear Partial Differential Operators, Vol. I, second ed., Springer-Verlag, Berlin, 1990.Google Scholar
  17. [17]
    T. Ichinose and H. Tamura, Note on the norm convergence of the unitary Trotter product formula, Lett. Math. Phys. 70 (2004), 65–81.zbMATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    T. Kato, Linear evolution equations of “hyperbolic” type, J. Fac. Sci. Univ. Tokyo Sect. I 17 (1970), 241–258.zbMATHMathSciNetGoogle Scholar
  19. [19]
    T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1980.zbMATHGoogle Scholar
  20. [20]
    H. Kitada and H. Kumano-go, A family of Fourier integral operators and the fundamental solultion for a Schrödinger equation, Osaka J. Math. 18 (1981), 291–360.zbMATHMathSciNetGoogle Scholar
  21. [21]
    H. Kumano-go, Algebras of pseudo-differential operators, J. Fac. Sci. Univ. Tokyo Sect. I 17 (1970), 31–50.zbMATHMathSciNetGoogle Scholar
  22. [22]
    H. Kumano-go, A calculus of Fourier integral operators on R N and the fundamental solution for an operator of hyperbolic type, Comm. Partial Differential Equations 1 (1976), 1–44.CrossRefMathSciNetGoogle Scholar
  23. [23]
    H. Kumano-go, Pseudo-Differential Operators, MIT Press, Cambridge, 1981.Google Scholar
  24. [24]
    H. Kumano-go and K. Taniguchi, Fourier integral operators of multi-phase and the fundamental solution for a hyperbolic system, Funkcial. Ekvac. 22 (1979), 161–196.zbMATHMathSciNetGoogle Scholar
  25. [25]
    H. Kumano-go, K. Taniguchi and Y. Tozaki, Multi-products of phase functions for Fourier integral operators with an application, Comm. Partial Differential Equations 3 (1978), 349–380.zbMATHCrossRefMathSciNetGoogle Scholar
  26. [26]
    N. Kumano-go, A construction of the fundamental solution for Schrödinger equations, J. Math. Sci. Univ. Tokyo 2 (1995), 441–498.zbMATHMathSciNetGoogle Scholar
  27. [27]
    N. Kumano-go, Feynman path integrals as analysis on path space by time slicing approximation, Bull. Sci. Math. 128 (2004), 197–251.zbMATHCrossRefMathSciNetGoogle Scholar
  28. [28]
    P. D. Lax and L. Nirenberg, On stability for difference schemes: A sharp form of Gårding’s inequality, Comm. Pure Appl. Math. 19 (1966), 473–492.zbMATHCrossRefMathSciNetGoogle Scholar
  29. [29]
    J. Le Rousseau, Fourier-integral-operator approximation of solutions to pseudodifferential firstorder hyperbolic equations I: convergence in Sobolev spaces, Comm. Partial Differential Equations 31 (2006), 867–906.zbMATHCrossRefMathSciNetGoogle Scholar
  30. [30]
    J. Le Rousseau, On the convergence of some products of Fourier integral operators, Asymptot. Anal. 51 (2007), 189–207.zbMATHMathSciNetGoogle Scholar
  31. [31]
    J. Le Rousseau and M. V. de Hoop, Generalized-screen approximation and algorithm for the scattering of elastic waves, Quart. J. Mech. Appl. Math. 56 (2003), 1–33.zbMATHCrossRefMathSciNetGoogle Scholar
  32. [32]
    J. Le Rousseau and G. Hörmann, Fourier-integral-operator approximation of solutions to pseudodifferential first-order hyperbolic equations II: microlocal analysis, J. Math. Pures Appl. 86 (2006), 403–426.zbMATHMathSciNetGoogle Scholar
  33. [33]
    G. Métivier and K. Zumbrun, Hyperbolic boundary value problems for symmetric systems with variable multiplicities, J. Differential Equations 211 (2005), 61–134.zbMATHCrossRefMathSciNetGoogle Scholar
  34. [34]
    A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.zbMATHGoogle Scholar
  35. [35]
    M. Ruzhansky and M. Sugimoto, Global L 2-boundedness theorems for a class of Fourier integral operators., Comm. Partial Differential Equations 31 (2006), 547–569.zbMATHCrossRefMathSciNetGoogle Scholar
  36. [36]
    C. C. Stolk, A pseudodifferential equation with damping for one-way wave propagation in inhomogeneous media, Wave Motion 40 (2004), 111–121.zbMATHCrossRefMathSciNetGoogle Scholar
  37. [37]
    C. C. Stolk, Parametrix for a hyperbolic intial value problem with dissipation in some region, Asymptot. Anal. 43 (2005), 151–169.zbMATHMathSciNetGoogle Scholar
  38. [38]
    C. C. Stolk and M. V. de Hoop, Microlocal analysis of seismic inverse scattering in anisotropic, elastic media, Comm. Pure Appl. Math. 55 (2002), 261–301.zbMATHCrossRefMathSciNetGoogle Scholar
  39. [39]
    M. E. Taylor, Pseudodifferential Operators, Princeton University Press, Princeton, N.J., 1981.zbMATHGoogle Scholar
  40. [40]
    R. Vaillancourt, A simple proof of Lax-Nirenberg Theorems, Comm. Pure Appl.Math. 23 (1970), 151–163.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Hebrew University Magnes Press 2009

Authors and Affiliations

  1. 1.Laboratoire d’Analyse, Topologie Probabilités CNRS UMR 6632Universités d’Aix-Marseille Université de ProvenceMarseille cedex 13France

Personalised recommendations