Hardy inequalities on graphs and Dirichlet spaces.

在这个项目中，我们研究了图和狄利克雷空间上的hardy型不等式。重点不仅在于常数的最优性，而且在于所涉及的哈迪权值的最优渐近性。虽然哈代不等式最初是在离散环境下提出的，但过去几十年的研究主要围绕连续统模型展开。这个项目的方法是解决离散领域的开放性问题，最终目标是找到一个统一的治疗方法。这与近年来的研究进展是一致的，在狄利克雷形式的框架内系统地研究了离散模型和连续模型之间的强类比。项目包括三个部分：(A) hardy型不等式、谱理论及相关不等式。(B)图的群和特定类的最优Hardy权值。(C) Dirichlet和Schrödinger形式的临界理论和Hardy不等式。(A)部分致力于研究图上加权Schrödinger算子的最优Hardy不等式。本文首先讨论一般$ p $的$ \ell^{p} $-Hardy不等式及其与$ p=2情况下的Rellich不等式、Agmon估计和基本谱理论问题的联系。Part (B)的目的是寻找关于具体例子的渐近性和最优常数的明确定量信息。这些例子包括$ Z^{d} $的子集、树和某些Cayley图。除了对这些例子本身的具体兴趣之外，它们还可以作为研究大类离散群的一般方法的玩具模型。(C)部分试图在Dirichlet和Schrödinger形式的一般框架中统一和推进离散和连续模型中已知的临界理论和最优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.