Non-divergence operators structured on homogeneous Hörmander vector fields: heat kernels and global Gaussian bounds

Abstract

Let $X_1,...,X_m$ be a family of real smooth vector fields defined in ${\mathbb{R}}^n$, $1$-homogeneous with respect to a nonisotropic family of dilations and satisfying Hörmander's rank condition at $0$ (and therefore at every point of ${\mathbb{R}}^n$). The vector fields are not assumed to be translation invariant with respect to any Lie group structure. Let us consider the nonvariational evolution operator $$\mathcal{H}:=\sum _{i,j=1}^m{a}_{i,j}(t,x)X_i{X}_j-{\partial }_t$$ where $(a_{i,j}(t,x){)}_{i,j=1}^m$ is a symmetric uniformly positive $m\times m$ matrix and the entries $a_{ij}$ are bounded Hölder continuous functions on ${\mathbb{R}}^{1+n}$, with respect to the "parabolic" distance induced by the vector fields. We prove the existence of a global heat kernel $\Gamma (\cdot ;s,y)\in C_{X,\mathrm{loc}}^{2,\alpha }({\mathbb{R}}^{1+n}\setminus \{(s,y)\})$ for $\mathcal{H}$, such that $\Gamma$ satisfies two-sided Gaussian bounds and ${\partial }_t\Gamma ,X_i\Gamma ,X_i{X}_j\Gamma$ satisfy upper Gaussian bounds on every strip $[0,T]\times {\mathbb{R}}^n$. We also prove a scale-invariant parabolic Harnack inequality for $\mathcal{H}$, and a standard Harnack inequality for the corresponding stationary operator $$\mathcal{L}:=\sum _{i,j=1}^m{a}_{i,j}(x)X_i{X}_j.$$ with Hölder continuos coefficients.