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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Equations with constant coefficients invariant under a group of linear transformations


Author: André Cerezo
Journal: Trans. Amer. Math. Soc. 204 (1975), 267-298
MSC: Primary 35E99; Secondary 46F15, 58G15
DOI: https://doi.org/10.1090/S0002-9947-1975-0430501-1
MathSciNet review: 0430501
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Abstract: If $ P$ is a linear differential operator on $ {{\mathbf{R}}^n}$ with constant coefficients, which is invariant under a group $ G$ of linear transformations, it is not true in general that the equation $ Pu = f$ always has a $ G$-invariant solution $ u$ for a $ G$-invariant $ f$. We elucidate here the particular case of a ``big'' group $ G$, and we count the invariant solutions when they exist (see Corollary 28 and Theorems 32, 33). The case, of special interest, of the wave equation and the Lorentz group is covered (Corollary 27). The theory of hyperfunctions provides the frame for the work.


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  • [1] M. F. Atiyah, Resolution of singularities and division of distributions, Comm. Pure Appl. Math. 23 (1970), 145-150. MR 41 #815. MR 0256156 (41:815)
  • [2] R. Harvey, Hyperfunctions and partial differential equations, Proc. Nat. Acad. Sci. U.S.A. 55 (1966), 1042-1046. MR 34 #495. MR 0200604 (34:495)
  • [3] H. Komatsu, An introduction to the theory of hyperfunctions, Hyperfunctions and Pseudo-Differential Equations, Lecture Notes in Math., vol 287, Springer-Verlag, Berlin and New York, 1973. MR 0394190 (52:14994)
  • [4] P. D. Methée, Sur les distributions invariantes dans le groupe des rotations de Lorentz, Comment. Math. Helv. 28 (1954), 225-269. MR 16, 255. MR 0064268 (16:255c)
  • [5] T. Miwa, On the existence of hyperfunctions solutions of linear differential equations of the first order with degenerate real principal symbols, Proc. Japan Acad. 49 (1973), 88-93. MR 0348236 (50:734)
  • [6] M. Rais, Solutions élémentaires des opérateurs bi-invariants sur un groupe de Lie nilpotent, C. R. Acad. Sci. Paris Sér. A-B 273 (1971), A495-A498. MR 44 #6908. MR 0289720 (44:6908)
  • [7] M. Riesz, L'integrale de Riemann-Liouville et le problème de Cauchy, Acta Math. 81 (1949), 1-223. MR 10, 713. MR 0030102 (10:713c)
  • [8] M. Sato, Theory of hyperfunctions. I, II, J. Fac. Sci. Univ. Tokyo 8 (1959), 139-193, 387-437. MR 22 #4951; 24 #A2237. MR 0114124 (22:4951)
  • [9] M. Sato, T. Kawai and M. Kasiwara, Microfunctions and pseudo-differential equations, Hyperfunctions and Pseudo-Differential Equations, Lecture Notes in Math., vol. 287, Springer-Verlag, Berlin and New York, 1973, pp. 265-529. MR 0420735 (54:8747)
  • [10] P. Shapira, Théorie des hyperfonctions, Lecture Notes in Math., vol. 126, Springer-Verlag, Berlin and New York, 1970.
  • [11] G. de Rham, Sur la division de formes et de courants par une forme linéaire, Comment. Math. Helv. 28 (1954), 346-352. MR 16, 402. MR 0065241 (16:402d)
  • [12] P. D. Methée, Transformées de Fourier de distributions invariantes. II, C. R. Acad. Sci. Paris 241 (1955), 684-686. MR 17, 845. MR 0076093 (17:845g)
  • [13] G. de Rham, Solution élémentaire d'opérateurs différentiels du second ordre, Ann. Inst. Fourier (Grenoble) 8 (1958), 337-366. MR 22 #8216. MR 0117437 (22:8216)
  • [14] L. Gårding and J. L. Lions, Functional analysis, Nuovo Cimento (10) 14 (1959), supplemento, 9-66. MR 22 #8321. MR 0117543 (22:8321)
  • [15] A. Tengstrand, Distributions invariant under an orthogonal group of arbitrary signature, Math. Scand. 8 (1960), 201-218. MR 23 #A3450. MR 0126154 (23:A3450)
  • [16] M. Raïs, Les solutions invariantes de l'équation des ondes, C. R. Acad. Sci. Paris 259 (1964), 2169-2170. MR 0173080 (30:3295)
  • [17] V. Edén, Distributions invariant under the group of complex orthogonal transformations, Math. Scand. 14 (1964), 75-89. MR 30 #4151. MR 0173944 (30:4151)

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DOI: https://doi.org/10.1090/S0002-9947-1975-0430501-1
Article copyright: © Copyright 1975 American Mathematical Society

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