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

Rüdiger Braun

Studia Mathematica (1993)

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

Abstract

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It is shown that the partial differential operator P ( D ) = / x - ² / y ² + i / z : Γ d ( ³ ) Γ d ( ³ ) is surjective if 1 ≤ d < 2 or d ≥ 6 and not surjective for 2 ≤ d < 6.

How to cite

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Braun, 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

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  1. [1] L. V. Ahlfors, Conformal Invariants, McGraw-Hill, New York, 1973. 
  2. [2] R. W. Braun, R. Meise and B. A. Taylor, Ultradifferentiable functions and Fourier analysis, Resultate Math. 17 (1990), 206-237. Zbl0735.46022
  3. [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. [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 N , preprint. Zbl0848.35023
  5. [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. [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. [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. [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. [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. [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. [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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