Discrete Hardy inequalities and related problems

• Applicants: Professor Dr. Matthias Keller; Professor Dr. Felix Pogorzelski
• Subject Area: Mathematics
• Term: since 2025
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 560796250

Hardy inequalities are fascinating mathematical objects which are also of major interest in the sciences. Most notably, they give a quantitative description of the Heisenberg uncertainty principle. On top of it, there is a vast body of literature on Hardy weights in a wide range of geometric and operator theoretic settings. Consequently, it is also no surprise that they appear in various mathematical fields such as partial differential equations, geometric analysis, mathematical physics, operator theory and probability. The first Hardy inequality was discrete and goes back to the 1920's when Hardy was investigating Hilbert's inequality for double sums. Two outstanding mathematical features of this inequality are sharpness of the explicit constant and, furthermore, non-existence of a minimizer. When this inequality was later studied in the continuum setting, these two features were also the main focus of interest. Here, the Hardy weight is often a multiple of the reciprocal of the squared distance to a closed set (which is typically a point or the boundary of a set). Classically, given the distance function, one is interested in determining the best constant such that the Hardy inequality holds. In a different recent approach going back to a question of Agmon, one determines an optimal Hardy weight. The notion of optimality includes both the best constant and non-existence of a minimizer. This approach inspired a lot of research in the subsequent years, as it allows to deal with much more general settings than before, for example, graphs. In this project we intend to expand on the previous research of Hardy inequalities on graphs. The project is divided into three parts: (A) Optimal Hardy weights in the linear case. (B) Hardy inequalities for quasilinear operators. (C) Agmon's conjecture on asymptotic formulas for the heat kernel.

• DFG Programme: Research Grants
• International Connection: Israel
• Partner Organisation: The Israel Science Foundation
• Cooperation Partner: Professor Yehuda Pinchover, Ph.D.