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

Let $\mathcal{H}=\sum_{j=1}^{m}X_{j}^{2}-\partial_{t}$ be a heat-type operator in $\mathbb{R}^{n+1}$, where $X=\{X_{1},\ldots,X_{m}\}$ 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 $\Gamma(t,x;s,y)$ 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 $\Gamma$. 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 $\mathcal{H}u=0$.