Hardy inequalities on graphs and Dirichlet spaces.

在这个项目中，我们研究了图和Dirichlet空间上的Hardy型不等式。特别的焦点不仅在于常数的最优性，而且在于所涉及的Hardy权的最优渐近性。虽然哈代的不等式最初是在离散环境下提出的，但过去几十年的研究都集中在连续统模型上。这个项目的方法是解决离散领域中的公开问题，最终目标是找到统一的治疗方法。这与最近几年的研究进展是一致的，其中离散模型和连续模型之间的强相似已经在Dirichlet形式的框架内被系统地研究。该项目由三部分组成：(A)Hardy型不等式、谱理论和相关的不等式。(B)群和特定图类的最优Hardy权。(C)Dirichlet和Schrödinger形式的临界性理论和Hardy不等式。(A)部分研究了图上加权薛定谔算子的最优Hardy不等式。本文首先讨论了一般情况下$p的Hardy-Hardy不等式，以及它们与Rellich不等式、Agmon估计以及在$p=2情形下的基本谱理论问题的联系。$部分(B)的目的是找到关于具体例子的渐近和最优常数的显式定量信息。这些例子包括$Z^{d}$、树和某些Cayley图的子集。除了对这些例子本身的具体兴趣之外，它们还进一步成为研究大类离散群体的一般方法的玩具模型。(C)部分试图在Dirichlet和Schrödinger形式的一般框架下统一和推进离散模型和连续模型中的已知临界性理论和最优Hardy不等式。

In this project we study Hardy-type inequalities on graphs and Dirichlet spaces. The particular focus lies on optimality of not only the constant but also on the optimal asymptotics of the involved Hardy weights. While Hardy's inequality was originally formulated in the discrete setting, the research of the last decades gravitated around continuum models. The approach of this project is to address the open questions in the discrete realm with the ultimate goal to find a unified treatment. This is in line with research developments of recent years where the strong analogies between discrete and continuum models have been systematically studied within the framework of Dirichlet forms.The project consists of three parts: (A) Hardy-type inequalities, spectral theory and related inequalities. (B) Optimal Hardy weights on groups and specific classes of graphs. (C) Criticality theory and Hardy inequalities for Dirichlet and Schrödinger forms. Part (A) is devoted to the study of optimal Hardy inequalities for weighted Schrödinger operators on graphs. This concerns first $ \ell^{p} $-Hardy inequalities for general $ p $ as well as their connection to Rellich inequalities, Agmon estimates and basic spectral theoretic questions in the case $ p=2. $ Part (B) aims at finding explicit quantitative information on the asymptotics and optimal constants for specific examples. These examples include subsets of $ Z^{d} $, trees, and certain Cayley graphs. Beyond the concrete interest in these examples themselves, they serve furthermore as toy model towards a general method of studying large classes of discrete groups. Part (C) seeks to unify and advance the known criticality theory and optimal Hardy inequalities in discrete and continuum models in the general framework of Dirichlet and Schrödinger forms.