# A partial differential operator which is surjective on Gevrey classes ${\Gamma}^{d}\left(\mathbb{R}\xb3\right)$ with 1 ≤ d < 2 and d ≥ 6 but not for 2 ≤ d < 6

Studia Mathematica (1993)

- Volume: 107, Issue: 2, page 157-169
- ISSN: 0039-3223

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topBraun, Rüdiger. "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." Studia Mathematica 107.2 (1993): 157-169. <http://eudml.org/doc/216027>.

@article{Braun1993,

abstract = {It is shown that the partial differential operator $P(D) = ∂⁴/∂x⁴ - ∂²/∂y² + i∂/∂z : Γ^d(ℝ³) → Γ^d(ℝ³)$ is surjective if 1 ≤ d < 2 or d ≥ 6 and not surjective for 2 ≤ d < 6.},

author = {Braun, Rüdiger},

journal = {Studia Mathematica},

language = {eng},

number = {2},

pages = {157-169},

title = {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},

url = {http://eudml.org/doc/216027},

volume = {107},

year = {1993},

}

TY - JOUR

AU - Braun, Rüdiger

TI - 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

JO - Studia Mathematica

PY - 1993

VL - 107

IS - 2

SP - 157

EP - 169

AB - It is shown that the partial differential operator $P(D) = ∂⁴/∂x⁴ - ∂²/∂y² + i∂/∂z : Γ^d(ℝ³) → Γ^d(ℝ³)$ is surjective if 1 ≤ d < 2 or d ≥ 6 and not surjective for 2 ≤ d < 6.

LA - eng

UR - http://eudml.org/doc/216027

ER -

## References

top- [1] L. V. Ahlfors, Conformal Invariants, McGraw-Hill, New York, 1973.
- [2] R. W. Braun, R. Meise and B. A. Taylor, Ultradifferentiable functions and Fourier analysis, Resultate Math. 17 (1990), 206-237. Zbl0735.46022
- [3] 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.
- [4] 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 ${\mathbb{R}}^{N}$, preprint. Zbl0848.35023
- [5] 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. Zbl0496.35018
- [6] 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.
- [7] 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.
- [8] L. Hörmander, On the existence of real-analytic solutions of partial differential equations with constant coefficients, Invent. Math. 21 (1973), 151-183. Zbl0282.35015
- [9] 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. Zbl0703.46025
- [10] 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. Zbl0264.35003
- [11] 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. Zbl0624.35011

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