# ANNALES DE L'INSTITUT FOURIER

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.

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 $P$.

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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},
pages = {619--655},
publisher = {Institut Fourier},
volume = {40},
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mrnumber = {92e:46083},
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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. https://aif.centre-mersenne.org/articles/10.5802/aif.1226/

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