Hardy inequalities on graphs and Dirichlet spaces.

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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.