We prove the existence of a global fundamental solution Gamma(x; y) (with pole x) for any Hormander operator L = sum_{i=1}^m X_i^2 on R^n which is delta_lambda-homogeneous of degree 2. Here homogeneity is meant with respect to a family of non-isotropic diagonal maps delta_lambda of the form delta_lambda(x) = (lambda^{sigma_1}x_1,...,lambda^{sigma_n}x_n), with 1 = sigma_1 <=... <= sigma_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 pi: G -> R^n such that L is pi-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 pi becomes the projection of G=R^n x R^p onto R^n. We prove that an integration argument over the (non-compact) fibers of pi provides a fundamental solution for L. Indeed, if Gamma_G(x, x'; y, y') (x, y are elements of R^n ; x' , y' are elements of R^p) is the fundamental solution of L_G, we show that Gamma_G(x, 0; y, y') is always integrable with respect to y' in R^p , and its y'-integral is a fundamental solution for L.

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

BONFIGLIOLI, ANDREA;BIAGI, STEFANO
2017

Abstract

We prove the existence of a global fundamental solution Gamma(x; y) (with pole x) for any Hormander operator L = sum_{i=1}^m X_i^2 on R^n which is delta_lambda-homogeneous of degree 2. Here homogeneity is meant with respect to a family of non-isotropic diagonal maps delta_lambda of the form delta_lambda(x) = (lambda^{sigma_1}x_1,...,lambda^{sigma_n}x_n), with 1 = sigma_1 <=... <= sigma_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 pi: G -> R^n such that L is pi-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 pi becomes the projection of G=R^n x R^p onto R^n. We prove that an integration argument over the (non-compact) fibers of pi provides a fundamental solution for L. Indeed, if Gamma_G(x, x'; y, y') (x, y are elements of R^n ; x' , y' are elements of R^p) is the fundamental solution of L_G, we show that Gamma_G(x, 0; y, y') is always integrable with respect to y' in R^p , and its y'-integral is a fundamental solution for L.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11585/627189
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