连续生成阿贝尔群上的多项式增长调和函数
Polynomial Growth Harmonic Functions on Finitely Generated Abelian Groups
摘要：
本文利用几何分析方法研究了多项式增长的调和函数。我们开发了一个几何分析证明的经典海尔布隆定理和最近的Nayar定理的多项式增长调和函数格$\mathds{Z}^n$，不使用一个表示公式的调和函数。我们还计算了多项式阿贝尔群的调和函数空间的精确维数。Cayley图不仅依赖于阿贝尔群，而且依赖于生成集的选择，我们发现这个维数只依赖于群本身。
英文摘要：
In the present paper, we develop geometric analytic techniques on Cayley graphs of finitely generated abelian groups to study the polynomial growth harmonic functions. We develop a geometric analytic proof of the classical Heilbronn theorem and the recent Nayar theorem on polynomial growth harmonic functions on lattices $\mathds{Z}^n$ that does not use a representation formula for harmonic functions. We also calculate the precise dimension of the space of polynomial growth harmonic functions on finitely generated abelian groups. While the Cayley graph not only depends on the abelian group, but also on the choice of a generating set, we find that this dimension depends only on the group itself.
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