Characterization of the linear partial differential operators with constant coefficients that admit a continuous linear right inverse
Annales de l'Institut Fourier, Volume 40 (1990) no. 3, p. 619-655

Solving a problem of L. Schwartz, those constant coefficient partial differential operators $P\left(D\right)$ are characterized that admit a continuous linear right inverse on $ℰ\left(\Omega \right)$ or ${𝒟}^{\prime }\left(\Omega \right)$, $\Omega$ an open set in ${\mathbf{R}}^{n}$. For bounded $\Omega$ with ${C}^{1}$-boundary these properties are equivalent to $P\left(D\right)$ being very hyperbolic. For $\Omega ={\mathbf{R}}^{n}$ they are equivalent to a Phragmen-Lindelöf condition holding on the zero variety of the polynomial $P$.

Nous résolvons complètement un problème de L. Schwartz sur la caractérisation des opérateurs différentiels aux dérivées partielles $P\left(D\right)$, à coefficients constants sur un ouvert $\Omega$ de ${\mathbf{R}}^{n}$, qui admettent un inverse à droite linéaire continu sur $ℰ\left(\Omega \right)$ ou ${𝒟}^{\prime }\left(\Omega \right)$. Si $\Omega$ est borné à frontière de classe ${C}^{1}$, ces propriétés sont équivalentes à une hyperbolicité très forte de $P\left(D\right)$. Si $\Omega ={\mathbf{R}}^{n}$, elles sont équivalentes à la validité d’un principe du type de Phragmén-Lindelöf sur la variété des zéros du polynôme $P$.

@article{AIF_1990__40_3_619_0,
author = {Taylor, B. A. and Meise, R. and Vogt, Dietmar},
title = {Characterization of the linear partial differential operators with constant coefficients that admit a continuous linear right inverse},
journal = {Annales de l'Institut Fourier},
publisher = {Imprimerie Louis-Jean},
address = {Gap},
volume = {40},
number = {3},
year = {1990},
pages = {619-655},
doi = {10.5802/aif.1226},
zbl = {0703.46025},
mrnumber = {92e:46083},
language = {en},
url = {http://www.numdam.org/item/AIF_1990__40_3_619_0}
}

Taylor, B. A.; Meise, R.; Vogt, Dietmar. Characterization of the linear partial differential operators with constant coefficients that admit a continuous linear right inverse. Annales de l'Institut Fourier, Volume 40 (1990) no. 3, pp. 619-655. doi : 10.5802/aif.1226. http://www.numdam.org/item/AIF_1990__40_3_619_0/

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