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Gradient estimates, Harnack inequalities and estimates for heat kernels of the sum of squares of vector fields. (English) Zbl 0808.58037

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 \(L - {\partial \over \partial t}\).

MSC:

58J35 Heat and other parabolic equation methods for PDEs on manifolds
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