Global Gaussian estimates for the heat kernel of homogeneous sums of squares

Let $\mathcal{H}={\sum }_{j=1}^m{X}_j^2-{\mathrm{\partial }}_t$ be a heat-type operator in ${\mathbb{R}}^{n+1}$, where $X=\{X_1,\dots ,X_m\}$ is a system of smooth H"{o}rmander’s vector fields in ${\mathbb{R}}^n$, and every $X_j$ is homogeneous of degree $1$ with respect to a family of non-isotropic dilations in ${\mathbb{R}}^n$, while no underlying group structure is assumed. In this paper we prove global (in space and time) upper and lower Gaussian estimates for the heat kernel $\mathrm{\Gamma }(t,x;s,y)$ of $\mathcal{H}$, in terms of the Carnot-Carath'{e}odory distance induced by $X$ on ${\mathbb{R}}^n$, as well as global upper Gaussian estimates for the $t$- or $X$-derivatives of any order of $\mathrm{\Gamma }$. From the Gaussian bounds we derive the unique solvability of the Cauchy problem for a possibly unbounded continuous initial datum satisfying exponential growth at infinity. Also, we study the solvability of the H-Dirichlet problem on an arbitrary bounded domain. Finally, we establish a global scale-invariant Harnack inequality for non-negative solutions of $\mathcal{H}u=0$.