A partial differential operator which is surjective on Gevrey classes $Γ^{d}(ℝ³)$ with 1 ≤ d < 2 and d ≥ 6 but not for 2 ≤ d < 6

Rüdiger W. Braun

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Title

A partial differential operator which is surjective on Gevrey classes $Γ^{d} (ℝ ³)$ with 1 ≤ d < 2 and d ≥ 6 but not for 2 ≤ d < 6

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Mathematisches Institut, Heinrich-Heine-Universität, Universitätsstrasse 1, D-40225 Düsseldorf, Germany

Abstract

It is shown that the partial differential operator

P (D) = \partial \frac{⁴}{\partial} x ⁴ - \partial \frac{²}{\partial} y ² + i \frac{\partial}{\partial} z : Γ^{d} (ℝ ³) \to Γ^{d} (ℝ ³)

is surjective if 1 ≤ d < 2 or d ≥ 6 and not surjective for 2 ≤ d < 6.

L. V. Ahlfors, Conformal Invariants, McGraw-Hill, New York, 1973.
R. W. Braun, R. Meise and B. A. Taylor, Ultradifferentiable functions and Fourier analysis, Resultate Math. 17 (1990), 206-237.
R. W. Braun, R. Meise and D. Vogt, Applications of the projective limit functor to convolution and partial differential equations, in: Advances in the Theory of Fréchet Spaces, Proc. Istanbul 1987, T. Terzioğlu (ed.), NATO Adv. Sci. Inst. Ser. C 287, Kluwer, 1989, 29-46.
R. W. Braun, R. Meise and D. Vogt, Characterization of the linear partial differential operators with constant coefficients which are surjective on non-quasianalytic classes of Roumieu type on $ℝ^{N}$ , preprint.
L. Cattabriga, Solutions in Gevrey spaces of partial differential equations with constant coefficients, in: Analytic Solutions of Partial Differential Equations, Proc. Trento 1981, L. Cattabriga (ed.), Astérisque 89/90 (1981), 129-151.
L. Cattabriga, On the surjectivity of differential polynomials on Gevrey spaces, in: Atti del Convegno: 'Linear Partial and Pseudodifferential Operators', Rend. Sem. Mat. Univ. Politec. Torino, fascicolo speziale, 1983, 81-89.
E. De Giorgi e L. Cattabriga, 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.
L. Hörmander, On the existence of real-analytic solutions of partial differential equations with constant coefficients, Invent. Math. 21 (1973), 151-183.
R. Meise, B. A. Taylor and D. Vogt, Characterization of the linear partial differential operators with constant coefficients that admit a continuous linear right inverse, Ann. Inst. Fourier (Grenoble) 40 (1990), 619-655.
L. C. Piccinini, Non surjectivity of the Cauchy-Riemann operator on the space of the analytic functions on $R^{n}$ . Generalization to the parabolic operators, Boll. Un. Mat. Ital. (4) 7 (1973), 12-28.
G. Zampieri, An application of the Fundamental Principle of Ehrenpreis to the existence of global Gevrey solutions of linear partial differential equations, ibid. (6) 5-B (1986), 361-392.

Title

A partial differential operator which is surjective on Gevrey classes ³Γd(ℝ³) with 1 ≤ d < 2 and d ≥ 6 but not for 2 ≤ d < 6

Affiliations

Abstract

Bibliography

A partial differential operator which is surjective on Gevrey classes $Γ^{d} (ℝ ³)$ with 1 ≤ d < 2 and d ≥ 6 but not for 2 ≤ d < 6