The existence of a global fundamental solution for homogeneous H"ormander operators via a global lifting method

We prove the existence of a global fundamental solution Γ(x; y) (with pole x) for any Hormander operator L = sum_{i = 1}^m Xi^2 on Rn which is δλ-homogeneous of degree 2. Here homogeneity is meant with respect to a family of non-isotropic diagonal maps δλ of the form δλ(x)=(λ^{σ_1}x_1,...,λ^{σ_n}x_n), with 1 = σ_1...σ_n. Due to a global lifting method for homogeneous operators proved by Folland [Comm. Partial Differential Equations 2 (1977) 161–207], there exists a Carnot group G and a polynomial surjective map π : G → Rn such that L is π-related to a sub-Laplacian L_G on G. We show that it is always possible to perform a (global) change of variable on G such that the lifting map π becomes the projection of G ≡ R^n imes R^p onto R^n . We prove that an integration argument over the (non-compact) fibers of π provides a fundamental solution for L. Indeed, if Γ_G(x, x′; y, y′) (x, y ∈ Rn; x′, y′ ∈ Rp) is the fundamental solution of L_G, we show that ΓG(x, 0; y, y') is always integrable with respect to y' in R^p , and its y-integral is a fundamental solution for L.