Global Gaussian Estimates for the Heat Kernel of Homogeneous Sums of Squares

Abstract

Let ${\mathscr{H}}={\sum }_{j=1}^{m}{X_{j}^{2}}-\partial _{t}$ be a heat-type operator in $\mathbb {R}^{n+1}$, where X = {X₁,…,X_m} is a system of smooth Hörmander’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 Γ(t,x;s,y) of ${\mathscr{H}}$, in terms of the Carnot-Carathéodory 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 Γ. 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 ${\mathscr{H}}$-Dirichlet problem on an arbitrary bounded domain. Finally, we establish a global scale-invariant Harnack inequality for non-negative solutions of ${\mathscr{H}}u=0$.