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, Tome 45 (1995) no. 1, pp. 223-249.

Hörmander a caractérisé les opérateurs différentiels à coefficients constants sur l’espace des fonctions analytiques réelles sur N par une condition du type Phragmén-Lindelöf. On donne des conséquences géométriques de cette condition et, pour les opérateurs homogènes, de la condition analogue pour les classes de Gevrey.

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

@article{AIF_1995__45_1_223_0,
     author = {Braun, R\"udiger W.},
     title = {The surjectivity of a constant coefficient homogeneous differential operator in the real analytic functions and the geometry of its symbol},
     journal = {Annales de l'Institut Fourier},
     pages = {223--249},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {45},
     number = {1},
     year = {1995},
     doi = {10.5802/aif.1454},
     zbl = {0816.35007},
     mrnumber = {96e:35025},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1454/}
}
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
SP  - 223
EP  - 249
VL  - 45
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.1454/
DO  - 10.5802/aif.1454
LA  - en
ID  - AIF_1995__45_1_223_0
ER  - 
%0 Journal Article
%A Braun, Rüdiger W.
%T The surjectivity of a constant coefficient homogeneous differential operator in the real analytic functions and the geometry of its symbol
%J Annales de l'Institut Fourier
%D 1995
%P 223-249
%V 45
%N 1
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.1454/
%R 10.5802/aif.1454
%G en
%F AIF_1995__45_1_223_0
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, Tome 45 (1995) no. 1, pp. 223-249. doi : 10.5802/aif.1454. https://aif.centre-mersenne.org/articles/10.5802/aif.1454/

[1] K.G. Andersson, Propagation of analyticity of solutions of partial differential equations with constant coefficients, Ark. Mat., 8 (1971), 277-302. | MR | Zbl

[2] J. Bochnak, M. Coste, M.-F. Roy, Géométrie algébrique réelle, Ergebnisse Math. Grenzgebiete 3. Folge 12, Springer, Berlin 1987. | MR | Zbl

[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. | MR | Zbl

[4] R.W. Braun, Surjektivität partieller Differentialoperatoren auf Roumieu-Klassen, Habilitationsschrift, Düsseldorf, 1993.

[5] R.W. Braun, R. Meise, B. A. Taylor, Ultradifferentiable functions and Fourier analysis, Result. Math., 17 (1990), 206-237. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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] L. Hörmander, On the existence of real analytic solutions of partial differential equations with constant coefficients, Invent. Math., 21 (1973), 151-183. | MR | Zbl

[12] L. Hörmander, The Analysis of Linear Partial Differential Operators II, Grundlehren 257, Springer, Berlin, 1983. | MR | Zbl

[13] L. Hörmander, An Introduction to Complex Analysis in Several Variables, North Holland, Amsterdam, 1990. | Zbl

[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. | Numdam | MR | Zbl

[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. | Zbl

[16] R. Meise, B.A. Taylor, D. Vogt, Phragmén-Lindelöf principles on algebraic varieties, J. Amer. Math. Soc., to appear. | Zbl

[17] R. Narasimhan, Introduction to the Theory of Analytic Spaces, LNM 25, Springer, Berlin, 1966. | MR | Zbl

[18] R. Nevanlinna, Eindeutige analytische Funktionen, Grundlehren 46, Springer, Berlin, 1974. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[21] H. Whitney, Complex Analytic Varieties, Addison-Wesley, Reading (Mass.), 1972. | MR | Zbl

[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. | MR | Zbl

Cité par Sources :