Siberian Mathematical Journal

, Volume 59, Issue 6, pp 1094–1099 | Cite as

The Rao–Reiter Criterion for the Amenability of Homogeneous Spaces

  • Ya. A. KopylovEmail author


We prove that a homogeneous space G/H, with G a locally compact group and H a closed subgroup of G, is amenable in the sense of Eymard–Greenleaf if and only if the quasiregular action πΦ of G on the unit sphere of the Orlicz space LΦ(G/H) for some N-function Φ ∈ Δ2 satisfies the Rao–Reiter condition (PΦ).


locally compact group homogeneous space amenability N-function Orlicz space Δ2-condition 


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  1. 1.
    Eymard P., Moyennes invariantes et représentations unitaires, Springer-Verlag, Berlin etc. (1972) (Lect. Notes Math.; vol. 300).CrossRefzbMATHGoogle Scholar
  2. 2.
    Greenleaf F. P., “Amenable actions of locally compact groups,” J. Funct. Anal., vol. 4, 295–315 (1969).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Stegeman J. D., “On a property concerning locally compact groups,” Nederl. Akad. Wet., Proc., Ser. A, vol. 68, 702–703 (1965).MathSciNetzbMATHGoogle Scholar
  4. 4.
    Reiter H. and Stegeman J. D., Classical Harmonic Analysis and Locally Compact Groups. 2nd ed., Clarendon Press, Oxford (2000).zbMATHGoogle Scholar
  5. 5.
    Rao M. M., “Convolutions of vector fields. III: Amenability and spectral properties,” in: Real and Stochastic Analysis. New Perspectives, Birkhäuser, Boston, MA, 2004, 375–401.CrossRefGoogle Scholar
  6. 6.
    Kopylov Ya., “Amenability of closed subgroups and Orlicz spaces,,” Sib. Élektron. Mat. Izv., vol. 10, 183–190 (2013).MathSciNetzbMATHGoogle Scholar
  7. 7.
    Rao M. M. and Ren Z. D., Theory of Orlicz Spaces, Marcel Dekker, Inc., New York etc. (1991) (Pure Appl. Math.; vol. 146).Google Scholar
  8. 8.
    Rao M. M., Measure Theory and Integration, Marcel Dekker, New York (2004) (Pure Appl. Math.; vol. 265).Google Scholar
  9. 9.
    Bourbaki N., Intégration. Chapitres VII and VIII, Hermann, Paris (1963) (Act. Sci. Ind.; no. 1306).Google Scholar
  10. 10.
    Krasnoselskii M. A. and Rutitskii Ya. B., Convex Functions and Orlicz Spaces, P. Noordhoff Ltd., Groningen (1961).Google Scholar
  11. 11.
    Rao M. M. and Ren Z. D., Applications of Orlicz Spaces, Marcel Dekker, New York (2002) (Pure Appl. Math.; vol. 250).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirsk State UniversityNovosibirskRussia

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