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

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.

Hörmander a caractérisé les opérateurs différentiels à coefficients constants sur l’espace des fonctions analytiques réelles sur ℝ 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.

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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, Volume 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

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