In this note we investigate the multi-parameter Potential Theory on the weighted d-tree (Cartesian product of several copies of uniform dyadic tree), which is connected to the discrete models of weighted Dirichlet spaces on the polydisc. We establish some basic properties of the respective potentials, capacities and equilibrium measures (in particular in the case of product polynomial weights). We explore multi-parameter Hardy inequality and its trace measures, and discuss some open problems of potential-theoretic and combinatorial nature.

中文翻译：

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多元树上多参数Hardy算子的迹线不等式的一些性质

在本说明中，我们研究了加权d树（统一二叉树的多个副本的笛卡尔积）的多参数势理论，该理论与多碟上加权Dirichlet空间的离散模型有关。我们建立了各个电位，容量和平衡度量的一些基本属性（尤其是在产品多项式权重的情况下）。我们探索了多参数哈代不等式及其跟踪测度，并讨论了一些潜在的理论和组合性质的开放问题。