On Pseudo-Differential Operators on Group SU(2)

  • Michael Ruzhansky
  • Ville Turunen
Part of the Operator Theory: Advances and Applications book series (OT, volume 189)


In this paper we will outline elements of the global calculus of pseudo-differential operators on the group SU(2). This is a part of a more general approach to pseudo-differential operators on compact Lie groups that will appear in [17].


Pseudo-differential operator Lie groups SU(2) 

Mathematics Subject Classification (2000)

35S05 22E30 


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  1. [1]
    M.S. Agranovich, Spectral properties of elliptic pseudodifferential operators on a closed curve, Funktsional. Anal. i Prilozhen 13 (1979), 54–56 (in Russian).zbMATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    M.S. Agranovich, On elliptic pseudodifferential operators on a closed curve, Trans. Moscow Math. Soc. 47 (1985), 23–74.Google Scholar
  3. [3]
    M.S. Agranovich, Elliptic operators on closed manifolds, Itogi Nauki i Tehniki, Ser. Sovrem. Probl. Mat. Fund. Napravl, 63 (1990), 5–129 (in Russian). (English translation in Encyclopaedia Math. Sci. 63 (1994), 1–130.)zbMATHMathSciNetGoogle Scholar
  4. [4]
    B.A. Amosov, On the theory of pseudodifferential operators on the circle, Uspekhi Mat. Nauk 43 (1988), 169–170 (in Russian). (English translation in Russian Math. Surveys 43 (1988), 197–198.)MathSciNetGoogle Scholar
  5. [5]
    R. Beals, Characterization of pseudo-differential operators and applications, Duke Math. J. 44 (1977), 45–57.zbMATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    H.O. Cordes, On pseudodifferential operators and smoothness of special Lie group representations, Manuscripta Math. 28 (1979), 51–69.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    J. Dunau, Fonctions d’un opérateur elliptique sur une variété compacte, J. Math. Pures et Appl. 56 (1977), 367–391.zbMATHMathSciNetGoogle Scholar
  8. [8]
    F. Geshwind and N.H. Katz, Pseudodifferential operators on SU(2), J. Fourier Anal. Appl. 3 (1997), 193–205.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    L. Hörmander, Pseudo-differential operators, Comm. Pure Appl. Math. 18 (1965), 501–517.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    L. Hörmander, The Analysis of Linear Partial Differential Operators III–IV, Berlin: Springer-Verlag, 1985.Google Scholar
  11. [11]
    J.J. Kohn and L. Nirenberg, On the algebra of pseudo-differential operators, Comm. Pure Appl. Math. 18 (1965), 269–305.zbMATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    H. Kumano-go, Pseudo-Differential Operators, MIT Press, Cambridge, Mass.-London, 1981.Google Scholar
  13. [13]
    W. McLean, Local and global description of periodic pseudodifferential operators, Math. Nachr. 150 (1991), 151–161.zbMATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    M. Ruzhansky and V. Turunen, On the Fourier analysis of operators on the torus, in Modern Trends in Pseudo-Differential Operators, Editors: J. Toft, M.W. Wong and H. Zhu, Operator Theory: Advances and Applications 172, Birkhäuser, 2007, 87–105.Google Scholar
  15. [15]
    M. Ruzhansky and V. Turunen, On pseudo-differential and Fourier integral operators on the torus, in preparation.Google Scholar
  16. [16]
    M. Ruzhansky and V. Turunen, Pseudo-differential operators on spheres, in preparation.Google Scholar
  17. [17]
    M. Ruzhansky and V. Turunen, Pseudo-Differential Operators and Symmetries, Birkhäuser, to appear.Google Scholar
  18. [18]
    Yu. Safarov, Pseudodifferential operators and linear connections, Proc. London Math. Soc. 74 (1997), 379–416.zbMATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    J. Saranen and G. Vainikko, Periodic Integral and Pseudodifferential Equations with Numerical Approximation, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2002.Google Scholar
  20. [20]
    J. Saranen and W.L. Wendland, The Fourier series representation of pseudodifferential operators on closed curves, Complex Variables, Theory Appl. 8 (1987), 55–64.zbMATHMathSciNetGoogle Scholar
  21. [21]
    V.A. Sharafutdinov, Geometric symbol calculus for pseudodifferential operators I Siberian Adv. Math. 15 (2005), 81–125. [Translation of Mat. Tr. 7 (2004), 159–206].MathSciNetGoogle Scholar
  22. [22]
    H. Stetkær, Invariant pseudo-differential operators, Math. Scand. 28 (1971), 105–123.zbMATHMathSciNetGoogle Scholar
  23. [23]
    R. Strichartz, Invariant pseudo-differential operators on a Lie group, Ann. Scuola Norm. Sup. Pisa 26 (1972), 587–611.zbMATHMathSciNetGoogle Scholar
  24. [24]
    M.E. Taylor, Noncommutative Microlocal Analysis. Mem. Amer. Math. Soc. 52 (1984).Google Scholar
  25. [25]
    M.E. Taylor, Beals-Cordes-type characterizations of pseudodifferential operators, Proc. Amer. Math. Soc. 125 (1997), 1711–1716.zbMATHCrossRefMathSciNetGoogle Scholar
  26. [26]
    V. Turunen, Commutator characterization of periodic pseudodifferential operators, Z. Anal. Anw. 19 (2000), 95–108.zbMATHMathSciNetGoogle Scholar
  27. [27]
    V. Turunen, Pseudodifferential Calculus on Compact Lie Groups and Homogeneous Spaces, Ph.D. Thesis, Helsinki University of Technology, 2001.Google Scholar
  28. [28]
    V. Turunen, Pseudodifferential calculus on the 2-sphere, Proc. Estonian Acad. Sci. Phys. Math. 53(3) (2004), 156–164.zbMATHMathSciNetGoogle Scholar
  29. [29]
    V. Turunen and G. Vainikko, On symbol analysis of periodic pseudodifferential operators, Z. Anal. Anw. 17 (1998), 9–22.zbMATHMathSciNetGoogle Scholar
  30. [30]
    G. Vainikko, An integral operator representation of classical periodic pseudo-differential operators, Z. Anal. Anw. 18 (1999), 687–699.zbMATHMathSciNetGoogle Scholar
  31. [31]
    N. Vilenkin, Special Functions and the Theory of Group Representations, Trans. Math. Monographs 22, Amer. Math. Soc., 1968.Google Scholar
  32. [32]
    H. Widom, A complete symbolic calculus for pseudodifferential operators, Bull. Sci. Math. 104 (1980), 19–63.zbMATHMathSciNetGoogle Scholar
  33. [33]
    D. Zelobenko, Compact Lie groups and their Representations. Trans. Math. Monographs 40, Amer. Math. Soc., 1973.Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2008

Authors and Affiliations

  • Michael Ruzhansky
    • 1
  • Ville Turunen
    • 2
  1. 1.Department of MathematicsImperial CollegeLondonUK
  2. 2.Institute of MathematicsHelsinki University of TechnologyFinland

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