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Dedicated to the memory of my mother 0. Introduction. Statement of the theorems. In what follows G will denote a real connected Lie group and = ; P n j=1 X 2 j + X 0 will denote some subelliptic left invariant Laplacian (cf. 1]). This, for us here, will mean that X 0 X 1 : : : X n are left invariant elds on G (i.e. X f g = (X f ) g , f g (x) = f(gx)) and that X 1 : : : X n are generators of the Lie algebra of G (i.e. together with all their successive brackets they span the Lie algebra of Gdoi:10.4171/rmi/215 fatcat:aheshajx65cbdgf4py3e2wn4ve