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

Let X1,…..,Xm be a family of real smooth vector Fields de- Fined in R^n), 1-homogeneous with respect to a nonisotropic family of dilations and satisfying Hormander's rank condition at 0 (and therefore at every point of 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 (formula Presented)