We study the existence of
global fundamental solution of certain invariant linear differential operators on the
semi-direct product G = V ⋉ 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 A. Cerezo and F. Rouviere’s
method (“Solution élémentaire d’un operateur différentiel invariant sur
un group de Lie compact”, Ann. Sci. Ec. Norm. Sup 4, séri t; 1969, pp.
561-581). We prove that a left invariant differential operator P on G and rightinvariant 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 a bi-invariant differential operator P on the Cartan’s motion
group.
