Hardy inequalities on graphs and Dirichlet spaces.

图和狄利克雷空间上的 Hardy 不等式。

• 批准号: 422487706
• 负责人: Professor Dr. Matthias Keller
• 金额: --
• 依托单位: Institut für Mathematik
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2019
• 资助国家: 德国
• 起止时间: 2018-12-31 至 2023-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/422487706?language=en
• 关键词: Hardy; inequalities; graphs; Dirichlet; spaces.

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.

在这个项目中，我们研究图和Dirichlet空间上的Hardy型不等式。特别的重点在于不仅是常数的最优性，但也对所涉及的哈代权重的最优渐近。 虽然哈代不等式最初是在离散环境中制定的，但过去几十年的研究都集中在连续模型上。该项目的方法是解决离散领域的开放问题，最终目标是找到一个统一的治疗方法。这与近年来在Dirichlet形式的框架下系统地研究离散模型和连续模型之间的强类比的研究进展是一致的。该项目包括三个部分：（A）Hardy型不等式，谱理论和相关不等式。(B)群和特定类图的最优哈代权。(C)临界理论和Dirichlet和Schr dinger形式的哈代不等式。（A）部分研究图上加权Schr dinger算子的最优哈代不等式。这涉及到第一$ \ell^{p} $-哈代不等式一般$ p $以及它们的连接到Rellich不等式，Agmon估计和基本的谱理论问题的情况下$ p=2。部分（B）旨在为具体实例找到关于渐近性和最佳常数的明确定量信息。这些例子包括$ Z^{d} $的子集、树和某些凯莱图。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型离散群的一般方法的玩具模型。部分（C）试图统一和推进已知的临界理论和最佳哈代不等式在离散和连续模型的一般框架下的狄利克雷和薛定谔形式。