Let S be a Damek–Ricci space and Δ be the Laplace–Beltrami operator of S. We explore the behaviour of heat propagator in S in large time to illustrate the differences with the corresponding results in ${\mathbb{R}}^n$. In particular we study the relation between the limiting behaviour of the ball-averages as radius tends to ∞ and that of the heat propagator as time goes to ∞ and use this relation for characterization of eigenfunctions of Δ.

中文翻译：

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传热器的长时间行为

令S为Damek-Ricci空间，而Δ为S的Laplace-Beltrami算子。我们长时间地探索了传热剂在S中的行为，以说明与结果对应的差异。${\mathbb{[R}}^{ñ}$。特别地，我们研究了当半径趋于∞时，平均球的极限行为与随着时间趋于∞时，传热器的极限行为之间的关系，并将此关系用于表征特征函数Δ。