On heat kernels on Lie groups
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 \textit{N. Varopoulos} [J. Funct. Anal. 76, 346-410 (1988; Zbl 0634.22008)] and \textit{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.
solvable Lie group of exponential growth
scientific article; zbMATH DE number 108648
scientific article