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Caractérisation des opérateurs linéaires aux dérivées partielles avec coefficients constants sur \({\mathcal E}({\mathbb{R}}^ N)\) admettant un inverse à droite qui est linéaire et continu. (Characterization of the linear partial differential operators with contant coefficients on \({\mathcal E}({\mathbb{R}}^ N)\) which admit a continuous linear right inverse). (French) Zbl 0649.46031
Summary: Let \({\mathcal E}({\mathbb{R}}^ N\)) (resp. \({\mathcal D}'({\mathbb{R}}^ N\))) denote the space of infinitely differentiable functions (resp. distributions) on \({\mathbb{R}}^ N.\) For a complex polynomial P in N variables let P(D) denote the linear partial differential operator induced by P on \({\mathcal E}({\mathbb{R}}^ N\)) (resp. \({\mathcal D}'({\mathbb{R}}^ N\))). It is shown that P(D): \({\mathcal E}({\mathbb{R}}^ N\))\(\to {\mathcal E}({\mathbb{R}}^ N\)) admits a continuous linear right inverse iff P(D): \({\mathcal D}'({\mathbb{R}}^ N\))\(\to {\mathcal D}'({\mathbb{R}}^ N\)) has this property, which is also equivalent to a certain Phragmén-Lindelöf principle holding on the variety V of P.

46F10 Operations with distributions and generalized functions
47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
46F05 Topological linear spaces of test functions, distributions and ultradistributions
47F05 General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX)