Results in Mathematics

, Volume 41, Issue 3–4, pp 361–368 | Cite as

Sharp estimates for a class of hyperbolic pseudo-differential equations



In this paper we consider the Cauchy problem for a class of hyperbolic pseudodifferential operators. The considered class contains constant coefficient differential equations, also allowing the coefficients to depend on time. We establish sharp L p − Lp, Lipschitz, and other estimates for their solutions. In particular, the ellipticity condition for the roots of the principal symbol is eliminated for certain dimensions. We discuss the situation with no loss of smoothness for solutions. In the space R1+n with n ≤ 4 (total dimension ≤ 5), we give a complete list of L p − Lp properties. In particular, this contains the very important case R1+3.

Mathematics Subject Classification (1991)

35A20 35S30 58G15 32D20 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    M. Beals, Lp boundedness of Fourier integrals, Mem. Amer. Math. Soc., 264 (1982).Google Scholar
  2. [2]
    Y. Colin de Verdière, M. Frisch, Régularité Lipschitzienne et solutions de l’équation des ondes sur une viriété Riemannienne compacte, Ann. Scient. Ecole Norm. Sup., 9 (1976), 539–565.MATHGoogle Scholar
  3. [3]
    J.J. Duistermaat, Fourier integral operators, Birkhäuser, Boston, 1996.MATHGoogle Scholar
  4. [4]
    L. Hörmander, The analysis of linear partial differential operators. Vols. III–IV, Springer-Verlag, New York, Berlin, 1985.Google Scholar
  5. [5]
    W. Littman, Lp − Lp-estimates for singular integral operators, Proc. Symp. Pure and Appl. Math. A.M.S., 23 (1973) 479–481.MathSciNetCrossRefGoogle Scholar
  6. [6]
    A. Miyachi, On some estimates for the wave operator in Lp and Hp, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 27 (1980), 331–354.MathSciNetMATHGoogle Scholar
  7. [7]
    J. Peral, Lp estimates for the wave equation, J. Funct. Anal., 36 (1980), 114–145.MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    D.H. Phong, Regularity of Fourier integral operators, Proc. Int. Congress Math., 862–874 (1994), Zürich, Switzerland.Google Scholar
  9. [9]
    M. Ruzhansky, Analytic Fourier integral operators, Monge-Ampère equation and holomorphic factorization, Arch. Mat., 72, 68–76 (1999).MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    M. Ruzhansky, Holomorphic factorization for the solution operators for hyperbolic equations, Int. Series of Num. Math.130, 803–811 (1999).MathSciNetGoogle Scholar
  11. [11]
    M. Ruzhansky, On the sharpness of Seeger-Sogge-Stein orders, Hokkaido Math. J. 28, 357–362 (1999).MathSciNetMATHGoogle Scholar
  12. [12]
    M. Ruzhansky, Sharp estimates for a class of hyperbolic differential equations, preprint, 1999.Google Scholar
  13. [13]
    M.V. Ruzhansky, Singularities of affine fibrations in the regularity theory of Fourier integral operators, Russian Math. Surveys 55, 93–161 (2000).MathSciNetCrossRefGoogle Scholar
  14. [14]
    M. Ruzhansky, Regularity theory of Fourier integral operators with complex phases and singularities of affine fibrations, CWI Tracts, to appear.Google Scholar
  15. [15]
    M. Ruzhansky, On the failure of the factorization condition for non-degenerate Fourier integral operators, to appear in Proc. Amer. Math. Soc.Google Scholar
  16. [16]
    A. Seeger, C.D. Sogge and E.M. Stein, Regularity properties of Fourier integral operators, Ann.of Math., 134 (1991), 231–251.MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    C.D. Sogge, Fourier integrals in classical analysis, Cambridge University Press, 1993.Google Scholar
  18. [18]
    E. M. Stein, Lp boundedness of certain convolution operators, Bull. Amer. Math. Soc., 77 (1971), 404–405.MathSciNetMATHCrossRefGoogle Scholar
  19. [19]
    E.M. Stein, Harmonic analysis, Princeton University Press, Princeton, 1993.MATHGoogle Scholar
  20. [20]
    M. Sugimoto, On some Lp-estimates for hyperbolic equations, Arkiv för Matematik, 30 (1992), 149–162.MathSciNetMATHCrossRefGoogle Scholar
  21. [21]
    M. Sugimoto, A priori estimates for higher order hyperbolic equations, Math. Z., 215 (1994), 519–531.MathSciNetMATHCrossRefGoogle Scholar
  22. [22]
    M. Sugimoto, Estimates for hyperbolic equations with non-convex characteristics, Math. Z., 222 (1996), 521–531.MathSciNetMATHCrossRefGoogle Scholar
  23. [23]
    M. Sugimoto, Estimates for hyperbolic equations of space dimension 3, J. Funct. Anal., 160 (1998), 382–407.MathSciNetMATHCrossRefGoogle Scholar
  24. [24]
    F. Treves, Introduction to pseudodifferential and Fourier integral operators, Vol. 2, Plenum Press, 1982.Google Scholar

Copyright information

© Birkhäuser Verlag, Basel 2002

Authors and Affiliations

  1. 1.Department of MathematicsJohns Hopkins UniversityBaltimore
  2. 2.Mathematics DepartmentImperial CollegeLondonUK

Personalised recommendations