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.
@article{bwmeta1.element.bwnjournal-article-apmv74z1p13bwm,
author = {Ziemian, Bogdan},
title = {On extendability of invariant distributions},
journal = {Annales Polonici Mathematici},
volume = {75},
year = {2000},
pages = {13-25},
zbl = {0964.46023},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-apmv74z1p13bwm}
}
Ziemian, Bogdan. On extendability of invariant distributions. Annales Polonici Mathematici, Tome 75 (2000) pp. 13-25. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-apmv74z1p13bwm/
[000] [1] O. V. Besov et al., Integral Representations of Functions and Imbedding Theorems, Nauka, Moscow, 1975 (in Russian).
[001] [2] G. E. Bredon, Introduction to Compact Transformation Groups, Academic Press, 1972. | Zbl 0246.57017
[002] [3] A. Cerezo, Equations with constant coefficients invariant under a group of linear transformations, Trans. Amer. Math. Soc. 204 (1975), 267-298. | Zbl 0301.35009
[003] [4] V. Edén, Disributions invariant under the group of complex orthogonal transformations, Math. Scand. 14 (1964), 75-89.
[004] [5] C. Herz, Invariant distributions, in: Proc. Sympos. Pure Math. 35, Part 2, Amer. Math. Soc., 1979, 361-373.
[005] [6] P. Jeanquartier, Distributions et opérateurs différentiels homogènes et inva- riants, Comment. Math. Helv. 39 (1965), 205-252. | Zbl 0197.40602
[006] [7] S. Łojasiewicz, Ensembles semi-analytiques, IHES, 1965.
[007] [8] B. Malgrange, Ideals of Differentiable Functions, Oxford Univ. Press, 1966.
[008] [9] P.-D. Methée, Sur les distributions invariantes dans le groupe des rotations de Lorentz, Comment. Math. Helv. 28 (1954), 225-269. | Zbl 0055.34101
[009] [10] R. Narasimhan, Analysis on Real and Complex Manifolds, Masson, Paris, 1968. | Zbl 0188.25803
[010] [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. | Zbl 0331.46028
[011] [12] G. de Rham, Sur la division de formes et de courants par une forme linéaire, Comment. Math. Helv. 28 (1954), 346-352. | Zbl 0056.31601
[012] [13] L. Schwartz, Séminaire 1954/55, Exposé n°7.
[013] [14] G. Schwarz, Smooth functions invariant under the action of a compact Lie group, Topology 14 (1975), 63-68. | Zbl 0297.57015
[014] [15] A. Tengstrand, Distributions invariant under an orthogonal group of arbitrary signature, Math. Scand. 8 (1960), 201-218. | Zbl 0104.33402
[015] [16] H. Weyl, The Classical Groups, Princeton Univ. Press, 1946. | Zbl 1024.20502
[016] [17] B. Ziemian, On G-invariant distributions, J. Differential Equations 35 (1980), 66-86. | Zbl 0423.58019
[017] [18] B. Ziemian, Distributions invariant under compact Lie groups, Ann. Polon. Math. 42 (1983), 175-183. | Zbl 0541.58007
[018] [19] Yu. M. Zinoviev, On Lorentz invariant distributions, Comm. Math. Phys. 47 (1976), 33-42. | Zbl 0317.46032