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Solvability of differential equations with linear coefficients of nilpotent type. (English) Zbl 0541.35010

Let L be the vector field on \({\mathbb{R}}^ n\) associated with a real nilpotent \(n\times n\)-matrix. It is shown that L regarded as a differential operator defines a surjective mapping of the space \({\mathcal S}'\) of tempered distributions onto itself; i.e. \(LS'({\mathbb{R}}^ n)=S'({\mathbb{R}}^ n).\) Replacing \({\mathcal S}'\) by the space \({\mathcal D}'\) of ordinary distributions, this is not true in general.

MSC:

35D05 Existence of generalized solutions of PDE (MSC2000)
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
58J99 Partial differential equations on manifolds; differential operators
35G05 Linear higher-order PDEs
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