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On heat kernels on Lie groups. (English) Zbl 0792.22007

Let \(p_ t\) be the heat kernel of a convolution semigroup generated by a sub-Laplacian \(L = \sum U^ 2_ j\), where \(U_ j\) are left invariant vector fields on a unimodular Lie group \(G\) which generate the Lie algebra of \(G\). The paper is concerned with proving upper and lower pointwise estimates of the heat kernel \(p_ t\) in the case when \(G\) is a Lie group of polynomial growth or \(G\) is a unimodular compact extension of a solvable Lie group of exponential growth. Some of the results were obtained previously by N. Varopoulos [J. Funct. Anal. 76, 346-410 (1988; Zbl 0634.22008)] and L. Saloff-Coste [Ark. Mat. 28, 315-331 (1990; Zbl 0715.43009)] and in those cases new proofs of the results were supplied, while some others are new. The results obtained provide estimates for both \(p_ t(e)\) and \(p_ t(x)\), \(x \neq e\) with \(t\) small and large.

MSC:

22E25 Nilpotent and solvable Lie groups
58J35 Heat and other parabolic equation methods for PDEs on manifolds
43A80 Analysis on other specific Lie groups
22E30 Analysis on real and complex Lie groups
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References:

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