Skip to main content
SpringerLink
Log in

Theory of Pseudo-differential Operators

  • Chapter
  • First Online:
Semigroups, Boundary Value Problems and Markov Processes

Part of the book series: Springer Monographs in Mathematics ((SMM))

  • 1934 Accesses

Abstract

In this chapter we present a brief description of the basic concepts and results from the theory of pseudo-differential operators – a modern theory of potentials –which will be used in the subsequent chapters. The development of the theory of pseudo-differential operators has greatly advanced our understanding of partial differential equations, and the pseudo-differential calculus has become an indispensable tool in contemporary analysis, in particular, in the study of elliptic boundary value problems. The calculus of pseudo-differential operators will be applied to elliptic boundary value problems in Chaps. 10 and 11.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 109.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 139.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 139.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Abraham, R., Marsden, J.E., Ratiu, T.: Manifolds, Tensor Analysis, and Applications. Global Analysis Pure and Applied: Series B, vol. 2. Addison-Wesley, Reading (1983)

    Google Scholar 

  2. Bergh, J., Löfström, J.: Interpolation Spaces: An Introduction. Grundlehren der Mathematischen Wissenschaften. Springer, Berlin/New York (1976)

    Book  MATH  Google Scholar 

  3. Bourdaud, G.: L p-estimates for certain non-regular pseudo-differential operators. Commun. Partial Differ. Equ. 7, 1023–1033 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  4. Boutet de Monvel, L.: Boundary problems for pseudo-differential operators. Acta Math. 126, 11–51 (1971)

    Google Scholar 

  5. Chazarain, J., Piriou, A.: Introduction à la théorie des équations aux dérivées partielles linéaires. Gauthier-Villars, Paris (1981)

    MATH  Google Scholar 

  6. Coifman, R.R., Meyer, Y.: Au-delà des opérateurs pseudo-différentiels. Astérisque, vol. 57. Société Mathématique de France, Paris (1978)

    Google Scholar 

  7. Duistermaat, J.J.: Fourier Integral Operators. Courant Institute of Mathematical Sciences, New York University, New York (1973)

    MATH  Google Scholar 

  8. Duistermaat, J.J., Hörmander, L.: Fourier integral operators II. Acta Math. 128, 183–269 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  9. Èskin, G.I.: Boundary value problems for elliptic pseudodifferential equations. Nauka, Moscow (1973, in Russian); English translation: American Mathematical Society, Providence (1981)

    Google Scholar 

  10. Gel’fand, I.M., Shilov, G.E.: Generalized Functions I. Properties and Operations. Academic, New York/London (1964)

    MATH  Google Scholar 

  11. Grubb, G.: Functional Calculus for Pseudodifferential Boundary Value Problems. Progress in Mathematics, vol. 65, 2nd edn. Birkhäuser, Boston (1996)

    Google Scholar 

  12. Grubb, G., Hörmander, L.: The transmission property. Math. Scand. 67, 273–289 (1990)

    MATH  MathSciNet  Google Scholar 

  13. Hörmander, L.: Pseudo-differential operators and non-elliptic boundary problems. Ann. Math. 83, 129–209 (1966)

    Article  MATH  Google Scholar 

  14. Hörmander, L.: Pseudo-differential operators and hypoelliptic equations. In: Calderón, A.P. (ed.) Proceedings of Symposia in Pure Mathematics: Singular Integrals, vol. X, Chicago, pp. 138–183. American Mathematical Society, Providence (1967)

    Google Scholar 

  15. Hörmander, L.: Fourier integral operators I. Acta Math. 127, 79–183 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  16. Hörmander, L.: The Analysis of Linear Partial Differential Operators III. Pseudo-Differential Operators. Reprint of the 1994 edition, Grundlehren der Mathematischen Wissenschaften. Springer, Berlin/Heidelberg/New York/Tokyo (2007)

    Google Scholar 

  17. Kannai, Y.: Hypoellipticity of certain degenerate elliptic boundary value problems. Trans. Am. Math. Soc. 217, 311–328 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  18. Kumano-go, H.: Pseudodifferential Operators. MIT, Cambridge (1981)

    Google Scholar 

  19. Lang, S.: Differential Manifolds. Addison-Wesley, Reading (1972)

    MATH  Google Scholar 

  20. Munkres, J.R.: Elementary Differential Topology. Annals of Mathematics Studies, vol. 54. Princeton University Press, Princeton (1966)

    Google Scholar 

  21. Nagase, M.: The L p-boundedness of pseudo-differential operators with non-regular symbols. Commun. Partial Differ. Equ. 2, 1045–1061 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  22. Oleĭnik, O.A., Radkevič, E.V.: Second Order Equations with Nonnegative Characteristic Form. Itogi Nauki, Moscow (1971, in Russian); English translation: American Mathematical Society/Plenum Press, Providence, New York/London (1973)

    Google Scholar 

  23. Polking, J.C.: Boundary value problems for parabolic systems of partial differential equations. In: Calderón, A.P. (ed.) Proceedings of Symposia in Pure Mathematics: Singular Integrals, Chicago, vol. X, pp. 243–274. American Mathematical Society, Providence (1967)

    Google Scholar 

  24. Rempel, S., Schulze, B.-W.: Index Theory of Elliptic Boundary Problems. Akademie-Verlag, Berlin (1982)

    MATH  Google Scholar 

  25. Schaefer, H.H.: Topological Vector Spaces. Graduate Texts in Mathematics, Third printing, vol. 3. Springer, New York/Berlin (1971)

    Google Scholar 

  26. Schrohe, E.: A short introduction to Boutet de Monvel’s calculus. In: Gil, J., Grieser, D., Lesch, M. (eds.) Approaches to Singular Analysis. Operator Theory, Advances and Applications, vol. 125, pp. 85–116. Birkhäuser, Basel (2001)

    Chapter  Google Scholar 

  27. Schrohe, E., Schulze, B.-W.: Boundary value problems in Boutet de Monvel’s calculus for manifolds with conical singularities I. In: Michael Demuth, Elmar Schrohe and Bert-Wolfgang Schulze (eds.) Pseudo-Differential Calculus and Mathematical Physics. Mathematical Topics, vol. 5, pp. 97–209. Akademie Verlag, Berlin (1994)

    Google Scholar 

  28. Schrohe, E., Schulze, B.-W.: Boundary value problems in Boutet de Monvel’s calculus for manifolds with conical singularities II. In: Michael Demuth, Elmar Schrohe and Bert-Wolfgang Schulze (eds.) Boundary Value Problems, Schrödinger Operators, Deformation Quantization. Mathematical Topics, vol. 8, pp. 70–205. Akademie Verlag, Berlin (1995)

    Google Scholar 

  29. Schrohe, E., Schulze, B.-W.: Mellin operators in a pseudodifferential calculus for boundary value problems on manifolds with edges. In: I. Gohberg, R. Mennicken and C. Tretter (eds.) Differential and Integral Operators (Regensburg, 1995). Operator Theory: Advances and Applications, vol. 102, pp. 255–285. Birkhäuser, Basel (1998)

    Chapter  Google Scholar 

  30. Schrohe, E., Schulze, B.-W.: A symbol algebra for pseudodifferential boundary value problems for manifolds with edges. In: Michael Demuth and Bert-Wolfgang Schulze (eds.) Differential Equations, Asymptotic Analysis, and Mathematical Physics (Potsdam, 1996). Mathematical Research, vol. 100, pp. 292–324. Akademie Verlag, Berlin (1997)

    Google Scholar 

  31. Schrohe, E., Schulze, B.-W.: Mellin and Green symbols for boundary value problems on manifolds with edges. Integral Equ. Oper. Theory 34, 339–363 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  32. Schulze, B.-W.: Pseudo-Differential Operators on Manifolds with Singularities. Studies in Mathematics and its Applications, vol. 24. North-Holland, Amsterdam (1991)

    Google Scholar 

  33. Schulze, B.-W.: Boundary Value Problems and Singular Pseudo-Differential Operators. Pure and Applied Mathematics (New York). Wiley, Chichester (1998)

    Google Scholar 

  34. Schwartz, L.: Théorie des distributions. Nouvelle édition, Hermann, Paris (1966)

    MATH  Google Scholar 

  35. Seeley, R.T.: Refinement of the functional calculus of Calderón and Zygmund. Nederl. Akad. Wetensch. Proc. Ser. A 68, 521–531 (1965)

    MATH  MathSciNet  Google Scholar 

  36. Seeley, R.T.: Singular integrals and boundary value problems. Am. J. Math. 88, 781–809 (1966)

    Article  MATH  MathSciNet  Google Scholar 

  37. Stein, E.M.: The characterization of functions arising as potentials II. Bull. Am. Math. Soc. 68, 577–582 (1962)

    Article  Google Scholar 

  38. Taibleson, M.H.: On the theory of Lipschitz spaces of distributions on Euclidean n-space I. Principal properties. J. Math. Mech. 13, 407–479 (1964)

    MathSciNet  Google Scholar 

  39. Taira, K.: Boundary Value Problems and Markov Processes. Lecture Notes in Mathematics, vol. 1499, 2nd edn. Springer, Berlin (2009)

    Google Scholar 

  40. Taylor, M.: Pseudodifferential Operators. Princeton Mathematical Series, vol. 34. Princeton University Press, Princeton (1981)

    Google Scholar 

  41. Treves, F.: Topological Vector Spaces, Distributions and Kernels. Academic, New York/London (1967)

    MATH  Google Scholar 

  42. Triebel, H.: Theory of Function Spaces. Monographs in Mathematics, vol. 78. Birkhäuser Verlag, Basel (1983)

    Google Scholar 

  43. Višik, M.I., Èskin, G.I.: Normally solvable problems for elliptic systems of convolution equations. Mat. Sb. (N.S.) 74(116), 326–356 (1967, in Russian)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, University of Tsukuba, Tsukuba, Ibaraki, Japan

    Kazuaki Taira

Authors
  1. Kazuaki Taira
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Taira, K. (2014). Theory of Pseudo-differential Operators. In: Semigroups, Boundary Value Problems and Markov Processes. Springer Monographs in Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43696-7_7

Download citation

Publish with us

Policies and ethics

Search

Navigation