Gradient estimates, Harnack inequalities and estimates for heat kernels of the sum of squares of vector fields
We study the equations \[ \begin{aligned} \left( L - {\partial \over \partial t} \right) u(x,t) & = 0 \quad \text{ and }\tag{1.1} \\ Lu(x) & = 0 \tag{1.2}\end{aligned} \] associated to the operator \(L = \sum_ i X^ 2_ i - X_ 0\) on a compact manifold \(M\) with a positive measure \(\mu\), where \(X_ 1, X_ 2, \dots, X_ m\) are smooth vector fields on \(M\) and \(X_ 0 = \sum_ i c_ iX_ i\). Our main purpose is to prove (Theorem 3.1 and Theorem 3.2) Harnack inequalities for positive solutions of Eq. (1.1) and Eq. (1.2) and to derive (Theorem 4.1) an upper estimate for the fundamental solution of the operator \(L - {\partial \over \partial t}\).
scientific article; zbMATH DE number 126409
scientific article