Gradient estimates, Harnack inequalities and estimates for heat kernels of the sum of squares of vector fields
type: Item
instance of: scholarly article
title: Gradient estimates, Harnack inequalities and estimates for heat kernels of the sum of squares of vector fields
zbMATH Open document ID: 0808.58037
DOI: 10.1007/BF02571441
author: Huai-Dong Cao
author: Shing-Tung Yau
published in: Mathematische Zeitschrift
publication date: 1993-03-03T00:00:00Z
full work available at URL: https://eudml.org/doc/174454
review text: 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 \(L - {\partial \over \partial t}\).
Mathematics Subject Classification ID: 58J35
zbMATH DE Number: 126409
zbMATH Keywords: heat kernels
zbMATH Keywords: gradient estimates
zbMATH Keywords: vector fields
zbMATH Keywords: Harnack inequalities
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 126409