Abstract
This paper establishes a heat semigroup version of Bernstein's theorem, applicable to any unimodular Lie group. The result has an intrinsic geometric content, involving estimates for the norms of the heat kernels for small time and large time. The theorem is stated in terms of certain Lipschitz spaces whose definition incorporates these two geometric features of the group in question. The geometric content is further underlined by showing that, in a certain sence, the theorem is best-possible.
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