Résolubilité semi-globale des opérateurs différentiels invariants sur les groupes de déplacements. (Semi-global solubility of invariant differential operators on motion groups) (Q1119251)

We study the existence of global fundamental solution of certain invariant linear differential operators on the semidirect product \(G=V\ltimes K\) where V is a real vector space of finite dimension and K a connected compact Lie group which acts on V as a linear group. Using the scalar Fourier transform on G and the method of \textit{A. Cerezo} and \textit{F. Rouvière} [Ann. Sci. Ec. Norm. Supér., IV. Sér. 2, 561- 581 (1969; Zbl 0191.438)] we prove that a left invariant differential operator P on G and right invariant by K admits a fundamental solution on G if and only if its partial Fourier coefficients satisfy a condition of slow growth. Hence we deduce an explicit necessary and sufficient condition for the existence of a fundamental solution for the biinvariant differential operator P on Cartan's motion group.