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Estimates for the heat kernel for a sum of squares of vector fields. (English) Zbl 0639.58026
Let \(L=\sum^{m}_{j=1} X\) \(2_ j\) be a sum of squares of vector fields on a \(C^{\infty}\), compact manifold M. Suppose that \(X_ j\) satisfy Hörmander’s condition: the vector fields \(X_ j\) and their commutators up to some finite order p span the tangent space of M at each point of M. The authors investigate the behaviour of the heat kernel h(t;x,y) associated to L and to a given \(C^{\infty}\) non-vanishing measure \(\mu\) on M. They obtain lower and upper bounds of h(t;x,y) by functions of the form \[ \mu (B(x,t^{1/2}))^{-1}\exp (-Cd(x,y)^ 2/t) \] where C is a constant, d(x,y) is a distance associated to L and \(B(x,r)=\{y\in M\); d(x,y)\(\leq r\}\).
Reviewer: G.Popov

58J35 Heat and other parabolic equation methods for PDEs on manifolds
35K05 Heat equation
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