Characterization of the linear partial differential operators with constant coefficients that admit a continuous linear right inverse

Taylor, B. A. ; Meise, R. ; Vogt, Dietmar

Annales de l'Institut Fourier, Tome 40 (1990), p. 619-655 / Harvested from Numdam

Résumé

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

Abstract

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

Détail

Publié le : 1990-01-01

MR 92e:46083 | Zbl 0703.46025 | 2 citations dans Numdam

DOI : https://doi.org/10.5802/aif.1226

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