Pseudodifferential Operators

  • Michael E. Taylor
Part of the Applied Mathematical Sciences book series (AMS, volume 116)

Abstract

In this chapter we discuss the basic theory of pseudodifferential operators as it has been developed to treat problems in linear PDE. We define pseudodifferential operators with symbols in classes denoted S ρ,δ m introduced by L. Hörmander. In §2 we derive some useful properties of their Schwartz kernels. In §3 we discuss adjoints and products of pseudodifferential operators. In §4 we show how the algebraic properties can be used to establish the regularity of solutions to elliptic PDE with smooth coefficients. In §5 we discuss mapping properties on L 2 and on the Sobolev spaces H s . In §6 we establish Gårding’s inequality.

Keywords

Pseudodifferential Operator Integral Kernel Solution Operator Layer Potential Principal Symbol 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [ADN]
    S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic differential equations satisfying general boundary conditions, CPAM 12(1959), 623–727; II, CPAM 17(1964), 35–92.MathSciNetMATHGoogle Scholar
  2. S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic differential equations satisfying general boundary conditions, CPAM 12(1959), 623–727; II, CPAM 17(1964), 35–92.MathSciNetMATHGoogle Scholar
  3. [Be]
    R. Beals, A general calculus of pseudo-differential operators, Duke Math. J. 42(1975), 1–42.MathSciNetMATHCrossRefGoogle Scholar
  4. [BF]
    R. Beals and C. Fefferman, Spatially inhomogeneous pseudodifferential operators, Comm. Pure Appl. Math. 27(1974), 1–24.MathSciNetMATHCrossRefGoogle Scholar
  5. [BGM]
    M. Berger, P. Gauduchon, and E. Mazet, Le Spectre d’une Variété Riemannienne, LNM #194, Springer-Verlag, New York, 1971.MATHGoogle Scholar
  6. [BJS]
    L. Bers, F. John, and M. Schechter, Partial Differential Equations, Wiley, New York, 1964.MATHGoogle Scholar
  7. [Ca1]
    A. P. Calderon, Uniqueness in the Cauchy problem of partial differential equations, Amer. J. Math. 80(1958), 16–36.MathSciNetMATHCrossRefGoogle Scholar
  8. [Ca2]
    A. P. Calderon, Singular integrals, Bull. AMS 72(1966), 427–465.MathSciNetMATHCrossRefGoogle Scholar
  9. [Ca3]
    A. P. Calderon, Cauchy integrals on Lipschitz curves and related operators, Proc. NAS, USA 74(1977), 1324–1327.MathSciNetMATHCrossRefGoogle Scholar
  10. [CV]
    A. P. Calderon and R. Vaillancourt, A class of bounded pseudodifferential operators, Proc. NAS, USA 69(1972), 1185–1187.MathSciNetMATHCrossRefGoogle Scholar
  11. [CZ]
    A. P. Calderon and A. Zygmund, Singular integral operators and differential equations, Amer. J. Math. 79(1957), 901–921.MathSciNetMATHCrossRefGoogle Scholar
  12. [CMM]
    R. Coifman, A. Mcintosh, and Y. Meyer, Uintegrale de Cauchy definit un operateur borne sur L 2 pour les courbes Lipsehitziennes, Ann. of Math. 116(1982), 361–388.MathSciNetMATHCrossRefGoogle Scholar
  13. [Cor]
    H. O. Cordes, On compactness of commutators of multiplications and convolutions, and boundedness of pseudo-differential operators, J. F une. Anal. 18(1975), 115–131.MathSciNetMATHCrossRefGoogle Scholar
  14. [Cor2]
    H. O. Cordes, Elliptic Pseudodifferential Operators—An Abstract Theory, LNM #756, Springer-Verlag, New York, 1979.Google Scholar
  15. [Cor3]
    H. O. Cordes, Spectral Theory of Linear Differential Operators and Comparison Algebras, London Math. Soc. Lecture Notes #70, Cambridge Univ. Press, London, 1987.MATHCrossRefGoogle Scholar
  16. [CH]
    H. O. Cordes and E. Herman, Gelfand theory of pseudo-differential operators, Amer. J. Math. 90(1968), 681–717.MathSciNetMATHCrossRefGoogle Scholar
  17. [DK]
    B. Dahlberg and C. Kenig, Hardy spaces and the Neumann problem in L p for Laplace’s equation in Lipschitz domains, Ann. of Math. 125(1987), 437–465.MathSciNetMATHCrossRefGoogle Scholar
  18. [Dui]
    J. J. Duistermaat, Fourier Integral Operators, Courant Institute Lecture Notes, New York, 1974.Google Scholar
  19. [Eg]
    Y. Egorov, On canonical transformations of pseudo-differential operators, Uspehi Mat. Nauk. 24(1969), 235–236.MATHGoogle Scholar
  20. [FJR]
    E. Fabes, M. Jodeit, and N. Riviere, Potential techniques for boundary problems in C 1 domains, Acta Math. 141(1978), 165–186.MathSciNetMATHCrossRefGoogle Scholar
  21. [F]
    C. Fefferman, The Uncertainty Principle, Bull. AMS 9(1983), 129–266.MathSciNetMATHCrossRefGoogle Scholar
  22. [FP]
    C. Fefferman and D. Phong, On positivity of pseudo-differential operators, Proc. NAS, USA 75(1978), 4673–4674.MathSciNetMATHCrossRefGoogle Scholar
  23. [Fo2]
    G. Folland, Harmonic Analysis on Phase Space, Princeton Univ. Press, Princeton, N. J., 1989.Google Scholar
  24. [Gå]
    L. Gårding, Dirichlet’s problem for linear elliptic partial differential equations, Math. Scand. 1(1953), 55–72.MathSciNetMATHGoogle Scholar
  25. [GT]
    D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer-Verlag, New York, 1983.MATHCrossRefGoogle Scholar
  26. [Gr]
    P. Greiner, An asymptotic expansion for the heat equation, Arch. Rat. Mech. Anal. 41(1971), 163–218.MathSciNetMATHCrossRefGoogle Scholar
  27. [GLS]
    A. Grossman, G. Loupias, and E. Stein, An algebra of pseudodifferential operators and quantum mechanics in phase space, Ann. Inst. Fourier 18(1969), 343–368.CrossRefGoogle Scholar
  28. [Ho1]
    L. Hörmander, Pseudo-differential operators, Comm. Pure Appl. Math, 18(1965), 501–517.MathSciNetMATHCrossRefGoogle Scholar
  29. [Ho2]
    L. Hörmander, Pseudodifferential operators and hypoelliptic equations, Proc. Symp. Pure Math. 10(1967), 138–183.CrossRefGoogle Scholar
  30. [Ho3]
    L. Hörmander, Fourier integral operators I, Acta Math. 127(1971), 79–183.MathSciNetMATHCrossRefGoogle Scholar
  31. [Ho4]
    L. Hörmander, The Weyl calculus of pseudodifferential operators, Comm. Pure Appl. Math. 32(1979), 355–443.CrossRefGoogle Scholar
  32. [Ho5]
    L. Hörmander, The Analysis of Linear Partial Differential Operators, Vols. 3 and 4, Springer-Verlag, New York, 1985.Google Scholar
  33. [How]
    R. Howe, Quantum mechanics and partial differential equations, J. Func. Anal. 38(1980), 188–254.MathSciNetMATHCrossRefGoogle Scholar
  34. [K]
    T. Kato, Boundedness of some pseudo-differential operators, Osaka J. Math. 13(1976), 1–9.MathSciNetMATHGoogle Scholar
  35. [Keg]
    O. Kellogg, Foundations of Potential Theory, Dover, New York, 1954.Google Scholar
  36. [KN]
    J. J. Kohn and L. Nirenberg, An algebra of pseudo-differential operators, Comm. Pure Appl. Math. 18(1965), 269–305.MathSciNetMATHCrossRefGoogle Scholar
  37. [Kg]
    H. Kumano-go, Pseudodifferential Operators, MIT Press, Cambridge, Mass., 1981.Google Scholar
  38. [LM]
    J. Lions and E. Magenes, Non-homogeneous Boundary Problems and Applications I, II, Springer-Verlag, New York, 1972.CrossRefGoogle Scholar
  39. [MS]
    H. McKean and I. Singer, Curvature and the eigenvalues of the Laplacian, J. Diff. Geom. 1(1967), 43–69.MathSciNetMATHGoogle Scholar
  40. [Mik]
    S. Mikhlin, Multidimensional Singular Integral Equations, Pergammon Press, New York, 1965.MATHGoogle Scholar
  41. [Miz]
    S. Mizohata, The Theory of Partial Differential Equations, Cambridge Univ. Press, Cambridge, 1973.Google Scholar
  42. [Mus]
    N. Muskhelishvilli, Singular Integral Equations, P. Nordhoff, Groningen, 1953.Google Scholar
  43. [Ni]
    L. Nirenberg, Lectures on Linear Partial Differential Equations, Reg. Conf. Ser. in Math., no. 17, AMS, Providence, R. I., 1972.Google Scholar
  44. [Pal]
    R. Palais, Seminar on the Atiyah-Singer Index Theorem, Princeton Univ. Press, Princeton, N. J., 1965.Google Scholar
  45. [Po]
    J. Polking, Boundary value problems for parabolic systems of partial differential equations, Proc. Symp. Pure Math. 10(1967), 243–274.MathSciNetCrossRefGoogle Scholar
  46. [RS]
    M. Reed and B. Simon, Methods of Mathematical Physics, Academic Press, New York, Vols. 1,2, 1975; Vols. 3,4,1978.MATHGoogle Scholar
  47. [Se1]
    R. Seeley, Refinement of the functional calculus of Calderon and Zygmund, Proc. Acad. Wet. Ned. Ser. A 68(1965), 521–531.MathSciNetMATHGoogle Scholar
  48. [Se2]
    R. Seeley, Singular integrals and boundary value problems, Amer. J. Math. 88(1966), 781–809.MathSciNetMATHCrossRefGoogle Scholar
  49. [Se3]
    R. Seeley, Complex powers of an elliptic operator, Proc. Symp. Pure Math 10(1967), 288–307.MathSciNetCrossRefGoogle Scholar
  50. [So]
    S. Sobolev, Partial Differential Equations of Mathematical Physics, Dover, New York, 1964.MATHGoogle Scholar
  51. [St]
    E. Stein, Singular Integrals and the Differentiability of Functions, Princeton Univ Press, Princeton, N. J., 1972.Google Scholar
  52. [St2]
    E. Stein, Harmonic Analysis, Princeton Univ. Press, Princeton, N. J., 1993.MATHGoogle Scholar
  53. [SW]
    E. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Space, Princeton Univ. Press, Princeton, N. J., 1971.Google Scholar
  54. [T1]
    M. Taylor, Pseudodifferential Operators, Princeton Univ. Press, Princeton, N. J., 1981.MATHGoogle Scholar
  55. [T2]
    M. Taylor, Noncommutative Microlocal Analysis, Memoirs AMS, no. 313, Providence, R. I., 1984.Google Scholar
  56. [T3]
    M. Taylor, Noncommutative Harmonic Analysis, AMS, Providence, R. I., 1986.MATHGoogle Scholar
  57. [T4]
    M. Taylor, Pseudodifferential Operators and Nonlinear PDE, Birkhauser, Boston, 1991.MATHCrossRefGoogle Scholar
  58. [Tre]
    F. Treves, Introduction to Pseudodijferential Operators and Fourier Integral Operators, Plenum, New York, 1980.Google Scholar
  59. [Ver]
    G. Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace ‘s equation in Lipschitz domains, J. Func. Anal. 39(1984), 572–611.MathSciNetCrossRefGoogle Scholar
  60. [Wey]
    H. Weyl, The Theory of Groups and Quantum Mechanics, Dover, New York, 1931.MATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1996

Authors and Affiliations

  • Michael E. Taylor
    • 1
  1. 1.Department of MathematicsUniversity of North CarolinaChapel HillUSA

Personalised recommendations