On extendability of invariant distributions

Bogdan Ziemian

Annales Polonici Mathematici (2000)

  • Volume: 74, Issue: 1, page 13-25
  • ISSN: 0066-2216

Abstract

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In this paper sufficient conditions are given in order that every distribution invariant under a Lie group extend from the set of orbits of maximal dimension to the whole of the space. It is shown that these conditions are satisfied for the n-point action of the pure Lorentz group and for a standard action of the Lorentz group of arbitrary signature.

How to cite

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Ziemian, Bogdan. "On extendability of invariant distributions." Annales Polonici Mathematici 74.1 (2000): 13-25. <http://eudml.org/doc/208362>.

@article{Ziemian2000,
abstract = {In this paper sufficient conditions are given in order that every distribution invariant under a Lie group extend from the set of orbits of maximal dimension to the whole of the space. It is shown that these conditions are satisfied for the n-point action of the pure Lorentz group and for a standard action of the Lorentz group of arbitrary signature.},
author = {Ziemian, Bogdan},
journal = {Annales Polonici Mathematici},
keywords = {Hausdorff partition; foliation; invariant distribution; distributions; -dimensional Hausdorff analytic manifold; orbit space of action; connected Lie group; semianalytic sets and functions; hyperbolic sets and orbits; invariant distributions; convergence; Lorentz group; pure Lorentz group; -point Lorentz invariant distributions},
language = {eng},
number = {1},
pages = {13-25},
title = {On extendability of invariant distributions},
url = {http://eudml.org/doc/208362},
volume = {74},
year = {2000},
}

TY - JOUR
AU - Ziemian, Bogdan
TI - On extendability of invariant distributions
JO - Annales Polonici Mathematici
PY - 2000
VL - 74
IS - 1
SP - 13
EP - 25
AB - In this paper sufficient conditions are given in order that every distribution invariant under a Lie group extend from the set of orbits of maximal dimension to the whole of the space. It is shown that these conditions are satisfied for the n-point action of the pure Lorentz group and for a standard action of the Lorentz group of arbitrary signature.
LA - eng
KW - Hausdorff partition; foliation; invariant distribution; distributions; -dimensional Hausdorff analytic manifold; orbit space of action; connected Lie group; semianalytic sets and functions; hyperbolic sets and orbits; invariant distributions; convergence; Lorentz group; pure Lorentz group; -point Lorentz invariant distributions
UR - http://eudml.org/doc/208362
ER -

References

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  1. [1] O. V. Besov et al., Integral Representations of Functions and Imbedding Theorems, Nauka, Moscow, 1975 (in Russian). 
  2. [2] G. E. Bredon, Introduction to Compact Transformation Groups, Academic Press, 1972. Zbl0246.57017
  3. [3] A. Cerezo, Equations with constant coefficients invariant under a group of linear transformations, Trans. Amer. Math. Soc. 204 (1975), 267-298. Zbl0301.35009
  4. [4] V. Edén, Disributions invariant under the group of complex orthogonal transformations, Math. Scand. 14 (1964), 75-89. 
  5. [5] C. Herz, Invariant distributions, in: Proc. Sympos. Pure Math. 35, Part 2, Amer. Math. Soc., 1979, 361-373. 
  6. [6] P. Jeanquartier, Distributions et opérateurs différentiels homogènes et inva- riants, Comment. Math. Helv. 39 (1965), 205-252. Zbl0197.40602
  7. [7] S. Łojasiewicz, Ensembles semi-analytiques, IHES, 1965. 
  8. [8] B. Malgrange, Ideals of Differentiable Functions, Oxford Univ. Press, 1966. 
  9. [9] P.-D. Methée, Sur les distributions invariantes dans le groupe des rotations de Lorentz, Comment. Math. Helv. 28 (1954), 225-269. Zbl0055.34101
  10. [10] R. Narasimhan, Analysis on Real and Complex Manifolds, Masson, Paris, 1968. Zbl0188.25803
  11. [11] A. I. Oksak, On invariant and covariant Schwartz distributions in the case of a compact linear group, Comm. Math. Phys. 46 (1976), 269-287. Zbl0331.46028
  12. [12] G. de Rham, Sur la division de formes et de courants par une forme linéaire, Comment. Math. Helv. 28 (1954), 346-352. Zbl0056.31601
  13. [13] L. Schwartz, Séminaire 1954/55, Exposé n°7. 
  14. [14] G. Schwarz, Smooth functions invariant under the action of a compact Lie group, Topology 14 (1975), 63-68. Zbl0297.57015
  15. [15] A. Tengstrand, Distributions invariant under an orthogonal group of arbitrary signature, Math. Scand. 8 (1960), 201-218. Zbl0104.33402
  16. [16] H. Weyl, The Classical Groups, Princeton Univ. Press, 1946. Zbl1024.20502
  17. [17] B. Ziemian, On G-invariant distributions, J. Differential Equations 35 (1980), 66-86. Zbl0423.58019
  18. [18] B. Ziemian, Distributions invariant under compact Lie groups, Ann. Polon. Math. 42 (1983), 175-183. Zbl0541.58007
  19. [19] Yu. M. Zinoviev, On Lorentz invariant distributions, Comm. Math. Phys. 47 (1976), 33-42. Zbl0317.46032

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