The existence of a global fundamental solution for homogeneous Hörmander operators via a global lifting method

Abstract

Abstract. We prove the existence of a global fundamental solution Γ(x; y) (with pole x) for any Hörmander operator L = m i=1 X 2 i on R n which is δ λ -homogeneous of degree 2. Here homogeneity is meant with respect to a family of non-isotropic diagonal maps δ λ of the form δ λ (x) = (λ σ 1 x 1 , . . . , λ σn xn), with 1 = σ 1 ≤ · · · ≤ σn. Due to a global lifting method for homogeneous operators proved by Folland in [On the Rothschild-Stein lifting theorem, Comm. PDEs, 1977], there exists a Carnot group G and a polynomial surjective map π : G → R n 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 ×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, x ′ ∈ R n ; y, y ′ ∈ R p ) is the fundamental solution of L G , we show that Γ G (x, 0; y, y ′ ) is always integrable w.r.t. y ′ ∈ R p , and its y ′ -integral is a fundamental solution for L.

Riesz-type representation formulas for subharmonic functions in sub-Riemannian settings

“…In fact, the validity of assumptions (S)-to-(HY) is proved in [10,Sec. 7] as a direct consequence of (H1)-(H2); as for assumption (FS), it follows from the results contained in [6,10].…”

“…As regards the homogeneous Hörmander sums of squares, instead, the validity of assumption (1) follows again from Hörmander's theorem, while the validity of assumption (2) is proved in [6,10]. Starting from the results in [6], a global theory for this kind of homogeneous operators, has been developed in [5,8,9,11,12].…”

Potential Theory Results for a Class of PDOs Admitting a Global Fundamental Solution

“…Other references related to the (difficult) problem of obtaining explicit/integrally-represented fundamental solutions are the following ones: [2,10,11,12,13,15,18,19,28,32,56]. See [16] for a wider list.…”

On the Baker-Campbell-Hausdorff Theorem: non-convergence and prolongation issues

“…The example in (2) above is obtained by combining an abstract result in [20], together with a class of examples of Lie algebras of vector fields contained in [12]. The problem of the convergence of Z(X, Y ) in Lie algebras of vector fields is of independent interest in the analysis of Hörmander operators (see e.g., [5,6,10,12]), and we shall return to it in a future investigation.…”