Global estimates for the fundamental solution of homogeneous Hörmander operators

Let $${\mathcal {L}}=\sum _{j=1}^{m}X_{j}^{2}$$ be a Hörmander sum of squares vector fields in $${\mathbb {R}}^{n}$$ , where any $$X_{j}$$ is homogeneous degree 1 with respect to family non-isotropic dilations . Then, {L}}$$ known admit global fundamental solution $$\Gamma (x;y)$$ that can represented as the integral sublaplacian operator on lifting space {R}}^{n}\times {\mathbb {R}}^{p}$$ equipped Carnot group structure. The aim this paper prove pointwise (upper and lower) estimates $$\Gamma$$ terms Carnot–Carathéodory distance induced by $$X=\{X_{1},\ldots ,X_{m}\}$$ well (upper) for X-derivatives order together suitable representations these derivatives. least dimensional case $$n=2$$ presents several peculiarities which are also investigated. Applications potential theory singular-integral kernel $$X_{i}X_{j}\Gamma$$ provided. Finally, most results about extended operators drift $$\sum _{j=1}^{m}X_{j}^{2}+X_{0}$$ $$X_{0}$$ 2-homogeneous $$X_{1},\ldots ,X_{m}$$ 1-homogeneous.