The surjectivity of a constant coefficient homogeneous differential operator in the real analytic functions and the geometry of its symbol

Rüdiger W. Braun

Annales de l'institut Fourier (1995)

  • Volume: 45, Issue: 1, page 223-249
  • ISSN: 0373-0956

Abstract

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Hörmander has characterized the surjective constant coefficient partial differential operators on the space of all real analytic functions on ℝ N by a Phragmén-Lindelöf condition. Geometric implications of this condition and, for homogeneous operators, of the corresponding condition for Gevrey classes are given.

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Braun, Rüdiger W.. "The surjectivity of a constant coefficient homogeneous differential operator in the real analytic functions and the geometry of its symbol." Annales de l'institut Fourier 45.1 (1995): 223-249. <http://eudml.org/doc/75115>.

@article{Braun1995,
abstract = {Hörmander has characterized the surjective constant coefficient partial differential operators on the space of all real analytic functions on $\{\Bbb R\}^N$ by a Phragmén-Lindelöf condition. Geometric implications of this condition and, for homogeneous operators, of the corresponding condition for Gevrey classes are given.},
author = {Braun, Rüdiger W.},
journal = {Annales de l'institut Fourier},
keywords = {partial differential operator; singularities; Phragmén-Lindelöf condition},
language = {eng},
number = {1},
pages = {223-249},
publisher = {Association des Annales de l'Institut Fourier},
title = {The surjectivity of a constant coefficient homogeneous differential operator in the real analytic functions and the geometry of its symbol},
url = {http://eudml.org/doc/75115},
volume = {45},
year = {1995},
}

TY - JOUR
AU - Braun, Rüdiger W.
TI - The surjectivity of a constant coefficient homogeneous differential operator in the real analytic functions and the geometry of its symbol
JO - Annales de l'institut Fourier
PY - 1995
PB - Association des Annales de l'Institut Fourier
VL - 45
IS - 1
SP - 223
EP - 249
AB - Hörmander has characterized the surjective constant coefficient partial differential operators on the space of all real analytic functions on ${\Bbb R}^N$ by a Phragmén-Lindelöf condition. Geometric implications of this condition and, for homogeneous operators, of the corresponding condition for Gevrey classes are given.
LA - eng
KW - partial differential operator; singularities; Phragmén-Lindelöf condition
UR - http://eudml.org/doc/75115
ER -

References

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  1. [1] K.G. ANDERSSON, Propagation of analyticity of solutions of partial differential equations with constant coefficients, Ark. Mat., 8 (1971), 277-302. Zbl0211.40502MR45 #8986
  2. [2] J. BOCHNAK, M. COSTE, M.-F. ROY, Géométrie algébrique réelle, Ergebnisse Math. Grenzgebiete 3. Folge 12, Springer, Berlin 1987. Zbl0633.14016MR90b:14030
  3. [3] R.W. BRAUN, Hörmander's Phragmén-Lindelöf principle and irreducible singularities of codimension 1, Boll. Un. Mat. Ital., (7), 6-A (1992), 339-348. Zbl0777.35020MR94b:35012
  4. [4] R.W. BRAUN, Surjektivität partieller Differentialoperatoren auf Roumieu-Klassen, Habilitationsschrift, Düsseldorf, 1993. 
  5. [5] R.W. BRAUN, R. MEISE, B. A. TAYLOR, Ultradifferentiable functions and Fourier analysis, Result. Math., 17 (1990), 206-237. Zbl0735.46022MR91h:46072
  6. [6] R.W. BRAUN, R. MEISE, D. VOGT, Applications of the projective limit functor to convolution and partial differential equations, in Advances in the Theory of Fréchet-Spaces, T. Terzioǧlu (Ed.), Istanbul 1987, NATO ASI Series C, Vol. 287, Kluwer, Dordrecht 1989, 29-46. Zbl0726.46022MR92b:46119
  7. [7] R. W. BRAUN, R. MEISE, D. VOGT, Characterization of the linear partial differential operators with constant coefficients which are surjective on non-quasianalytic classes of Roumieu type on ℝN, Math. Nachrichten, 168 (1994), 19-54. Zbl0848.35023MR95g:35004
  8. [8] L. CATTABRIGA, Solutions in Gevrey spaces of partial differential equations with constant coefficients, in Analytic Solutions of Partial Differential Equations, L. Cattabriga (Ed.), Trento 1981, Astérisque, 89/90 (1981), 129-151. Zbl0496.35018MR84h:35030
  9. [9] L. CATTABRIGA, On the surjectivity of differential polynomials on Gevrey spaces, in Atti del Convegno : “Linear Partial and Pseudodifferential Operators” Rendiconti del Seminario Matematico, Fascicolo Speziale. Torino, Università e Politecnico, 1983, 81-89. Zbl0561.35008MR85f:35046
  10. [10] L. CATTABRIGA, E. De GIORGI, Una dimostrazione diretta dell'esistenza di soluzioni analitiche nel piano reale di equazioni a derivate parziali a coefficienti costanti, Boll. Un. Mat. Ital., (4) 4 (1971), 1015-1027. 
  11. [11] L. HÖRMANDER, On the existence of real analytic solutions of partial differential equations with constant coefficients, Invent. Math., 21 (1973), 151-183. Zbl0282.35015MR49 #817
  12. [12] L. HÖRMANDER, The Analysis of Linear Partial Differential Operators II, Grundlehren 257, Springer, Berlin, 1983. Zbl0521.35002MR85g:35002b
  13. [13] L. HÖRMANDER, An Introduction to Complex Analysis in Several Variables, North Holland, Amsterdam, 1990. Zbl0685.32001
  14. [14] R. MEISE, B.A. TAYLOR, D. VOGT, Characterization of the linear partial differential operators with constant coefficients that admit a continuous linear right inverse, Ann. Inst. Fourier, 40-3 (1990), 619-655. Zbl0703.46025MR92e:46083
  15. [15] R. MEISE, B.A. TAYLOR, D. VOGT, Continuous linear right inverses for partial differential operators with constant coefficients and Phragmén-Lindelöf conditions, in “Functional Analysis”, K.D. Bierstedt, A. Pietsch, W. Ruess, D. Vogt (Eds.), Marcel Dekker, New York 1993, 357-389. Zbl0806.46041
  16. [16] R. MEISE, B.A. TAYLOR, D. VOGT, Phragmén-Lindelöf principles on algebraic varieties, J. Amer. Math. Soc., to appear. Zbl0896.32008
  17. [17] R. NARASIMHAN, Introduction to the Theory of Analytic Spaces, LNM 25, Springer, Berlin, 1966. Zbl0168.06003MR36 #428
  18. [18] R. NEVANLINNA, Eindeutige analytische Funktionen, Grundlehren 46, Springer, Berlin, 1974. Zbl0278.30002MR49 #9165
  19. [19] V.P. PALAMODOV, A criterion for splitness of differential complexes with constant coefficients, in Geometric and Algebraic Aspects in Several Complex Variables, C.A. Berenstein, D.C. Struppa (Eds.), EditEl 1991, 265-291. Zbl1112.58304MR94d:58137
  20. [20] L.C. PICCININI, Non-surjectivity of the Cauchy-Riemann operator on the space of the analytic functions on ℝN, Boll. Un. Mat. Ital., (4) 7 (1973), 12-28. Zbl0264.35003MR48 #4480
  21. [21] H. WHITNEY, Complex Analytic Varieties, Addison-Wesley, Reading (Mass.), 1972. Zbl0265.32008MR52 #8473
  22. [22] G. ZAMPIERI, An application of the fundamental principle of Ehrenpreis to the existence of global Gevrey solutions of linear partial differential equations, Boll. Un. Mat. Ital., (6) 5-B (1986), 361-392. Zbl0624.35011MR88a:35044

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