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ArticleOriginal scientific text

Title

Smooth operators for the regular representation on homogeneous spaces

Authors 1

Affiliations

  1. Instituto de Matemática e Estatí stica, Universidade de São Paulo, Caixa Postal 66281, São Paulo 05389-970, Brazil

Abstract

A necessary and sufficient condition for a bounded operator on L2(M), M a Riemannian compact homogeneous space, to be smooth under conjugation by the regular representation is given. It is shown that, if all formal 'Fourier multipliers with variable coefficients' are bounded, then they are also smooth. In particular, they are smooth if M is a rank-one symmetric space.

Bibliography

  1. M. S. Agranovich, On elliptic pseudodifferential operators on a closed curve, Trans. Moscow Math. Soc. 47 (1985), 23-74.
  2. R. Beals, Characterization of pseudodifferential operators and applications, Duke Math. J. 44 (1977), 45-57; Correction, ibid. 46 (1979), 215.
  3. H. O. Cordes, On pseudodifferential operators and smoothness of special Lie-group representations, Manuscripta Math. 28 (1979), 51-69.
  4. H. O. Cordes, The Technique of Pseudodifferential Operators, Cambridge Univ. Press, 1995.
  5. H. O. Cordes and S. T. Melo, Smooth operators for the action of SO(3) on L2(^2), Integral Equations Oper. Theory 28 (1997), 251-260.
  6. G. B. Folland, A Course in Abstract Harmonic Analysis, Stud. Adv. Math, CRC Press, 1995.
  7. S. Gallot, D. Hulin and J. Lafontaine, Riemannian Geometry, Springer, 1993.
  8. B. Gramsch, Relative Inversion in der Störungstheorie von Operatoren und Ψ*-Algebren, Math. Ann. 269 (1984), 27-71.
  9. S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, 1978.
  10. L. Hörmander, The Analysis of Linear Partial Differential Operators III, Sprin- ger, 1985.
  11. W. McLean, Local and global descriptions of periodic pseudodifferential operators , Math. Nachr. 150 (1991), 151-161.
  12. S. T. Melo, Characterizations of pseudodifferential operators on the circle, Proc. Amer. Math. Soc. 125 (1997), 1407-1412.
  13. K. R. Payne, Smooth tame Fréchet algebras and Lie groups of pseudodifferential operators, Comm. Pure Appl. Math. 44 (1991), 309-337.
  14. M. E. Taylor, Noncommutative Harmonic Analysis, Math. Surveys Monographs 22, Amer. Math. Soc., 1986.
  15. M. E. Taylor, Beals-Cordes-type characterizations of pseudodifferential operators, Proc. Amer. Math. Soc. 125 (1997), 1711-1716.
Studia Mathematica
2000/142/2
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Pages:
149-157
Main language of publication
English
Received
1999-09-13
Published
2000
Exact and natural sciences