北京雁栖湖应用数学研究院

Discrete models of classical ‘continuous’ problems have lately become a standard method of investigation. Usually it allows to highlight the geometric and combinatorial nature of a problem at hand singling out the core properties and issues. While sometimes the transition from the continuous medium to a discrete one can either result in a loss of information or drastically change key features of the objects in consideration, nevertheless, these approaches are immensely useful and interesting, both by themselves and as an instrument for continuous analysis.
This course aims to present an in-depth introduction to one such discrete model. On the one hand, it is one of the simpler ones, dealing mostly with trees and their variants, and, on the other hand, it already provides a significant insight into the behavior of harmonic functions on the unit disc or on a poly-disc. We will discuss potential theory, linear and non-linear, in the context of certain types of graphs, and demonstrate some of the applications to the more classical problems.
The key part of the model is the so-called weighted Hardy operator, and its embedding properties. We cover three main cases of the underlying graphs – trees, lattices and their products. Applications of the method include Carleson and trace measures, capacitary estimates and multi-parametric potentials.