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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

##### MSC:
 58J35 Heat and other parabolic equation methods for PDEs on manifolds 35K05 Heat equation
##### Keywords:
heat kernel; heat operator
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