Smooth operators for the regular representation on homogeneous spaces

Severino Melo

Studia Mathematica (2000)

  • Volume: 142, Issue: 2, page 149-157
  • ISSN: 0039-3223

Abstract

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A necessary and sufficient condition for a bounded operator on L 2 ( 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.

How to cite

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Melo, Severino. "Smooth operators for the regular representation on homogeneous spaces." Studia Mathematica 142.2 (2000): 149-157. <http://eudml.org/doc/216794>.

@article{Melo2000,
abstract = {A necessary and sufficient condition for a bounded operator on $L^2(M)$, 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.},
author = {Melo, Severino},
journal = {Studia Mathematica},
keywords = {Riemannian manifold; Fourier multipliers; Fourier multipliers with variable coefficients; signature; Riemannian compact homogeneous space; regular representation},
language = {eng},
number = {2},
pages = {149-157},
title = {Smooth operators for the regular representation on homogeneous spaces},
url = {http://eudml.org/doc/216794},
volume = {142},
year = {2000},
}

TY - JOUR
AU - Melo, Severino
TI - Smooth operators for the regular representation on homogeneous spaces
JO - Studia Mathematica
PY - 2000
VL - 142
IS - 2
SP - 149
EP - 157
AB - A necessary and sufficient condition for a bounded operator on $L^2(M)$, 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.
LA - eng
KW - Riemannian manifold; Fourier multipliers; Fourier multipliers with variable coefficients; signature; Riemannian compact homogeneous space; regular representation
UR - http://eudml.org/doc/216794
ER -

References

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  1. [1] M. S. Agranovich, On elliptic pseudodifferential operators on a closed curve, Trans. Moscow Math. Soc. 47 (1985), 23-74. Zbl0584.35078
  2. [2] R. Beals, Characterization of pseudodifferential operators and applications, Duke Math. J. 44 (1977), 45-57; Correction, ibid. 46 (1979), 215. Zbl0353.35088
  3. [3] H. O. Cordes, On pseudodifferential operators and smoothness of special Lie-group representations, Manuscripta Math. 28 (1979), 51-69. Zbl0415.35083
  4. [4] H. O. Cordes, The Technique of Pseudodifferential Operators, Cambridge Univ. Press, 1995. Zbl0828.35145
  5. [5] H. O. Cordes and S. T. Melo, Smooth operators for the action of SO(3) on L 2 ( 2 ) , Integral Equations Oper. Theory 28 (1997), 251-260. Zbl0909.35162
  6. [6] G. B. Folland, A Course in Abstract Harmonic Analysis, Stud. Adv. Math, CRC Press, 1995. Zbl0857.43001
  7. [7] S. Gallot, D. Hulin and J. Lafontaine, Riemannian Geometry, Springer, 1993. 
  8. [8] B. Gramsch, Relative Inversion in der Störungstheorie von Operatoren und Ψ*-Algebren, Math. Ann. 269 (1984), 27-71. 
  9. [9] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, 1978. 
  10. [10] L. Hörmander, The Analysis of Linear Partial Differential Operators III, Sprin- ger, 1985. 
  11. [11] W. McLean, Local and global descriptions of periodic pseudodifferential operators , Math. Nachr. 150 (1991), 151-161. Zbl0729.35149
  12. [12] S. T. Melo, Characterizations of pseudodifferential operators on the circle, Proc. Amer. Math. Soc. 125 (1997), 1407-1412. Zbl0870.47030
  13. [13] K. R. Payne, Smooth tame Fréchet algebras and Lie groups of pseudodifferential operators, Comm. Pure Appl. Math. 44 (1991), 309-337. Zbl0763.47022
  14. [14] M. E. Taylor, Noncommutative Harmonic Analysis, Math. Surveys Monographs 22, Amer. Math. Soc., 1986. Zbl0604.43001
  15. [15] M. E. Taylor, Beals-Cordes-type characterizations of pseudodifferential operators, Proc. Amer. Math. Soc. 125 (1997), 1711-1716. Zbl0868.35149

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