Abstract

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 result either in a loss of information, or it can drastically change key features of the objects in consideration. Nevertheless these approaches is immensely useful and interesting, by itself and as a instrument for continuous analysis. This course aims to present an in-depth introduction to one of such discrete models. It is one of the more simpler ones, dealing mostly with trees and their variants, on the other hand it already provides a significant insight into the behavior of harmonic functions in the unit disc or 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.

Audience

Graduate, Undergraduate

Prerequisite

No previous exposure to potential theory is assumed. The listener should be acquainted with basics of real analysis, functional analysis and, for some topics, should have some exprosure to probability theory (martingales) and complex analysis.

Syllabus

i. Introduction: setting, models and origins

ii. Axiomatic nonlinear potential theory

iii. Important examples: Riesz-Bessel kernels and graph potentials

iv. Sobolev spaces vs. potential spaces

v. The tree as a metric space, Hausdorff measures.

vi. Frostman lemma

vii. Strong capacitary inequality according to Maz'ya and according to Adams

viii. Hardy (trace) inequality on trees: characterizations involving capacity and energy

ix. The Muckenhoupt-Wheeden-Wolff inequality

x. Applications: trace inequalities for Sobolev spaces

xi. Applications: Carleson measures, multipliers, and boundary values for the holomorphic Dirichlet space

xii. Hardy embedding on the lattice.

xiii. The bitree and the failure of the maximum principle

xiv. The bitree: small energy majorization

xv. The bitree: capacity of exceptional sets and strong capacitary inequality

xvi. Hardy embedding on the bitree: capacitary characterization

xvii. Hardy embedding on the bitree: energy characterization

xviii. Hardy embedding on the bitree: single boxes

xix: Hardy embedding: tri-tree and d-trees

xx. Counterexamples on a d-tree

xxi. Applications: Carleson measures for Hardy-Sobolev spaces on the polydisc

Reference

1. D. R. Adams, L. I. Hedberg, Function Spaces and Potential Theory, Grundlehren der Mathematischen Wissenschaften 314, Springer-Verlag, Berlin, 1996, xii+366 pp.

2. N. Arcozzi, P. Mozolyako, K.-M. Perfekt, G. Sarfatti, Bi-parameter Potential theory and Carleson measures for the Dirichlet space on the bidisc, arXiv:1811.04990

3. N. Arcozzi, R. Rochberg, E.T. Sawyer, B.D. Wick, Potential theory on trees, graphs and Ahlfors regular metric spaces, Potential Analysis 41 (2), 2014, 317-366

4 . N. Aronszajn, Theory of Reproducing Kernels, Transactions of the American Mathematical Society, 68 (3), 1950, 337-404

5. K. Falconer, The Geometry of Fractal Sets, Cambridge University Press, 1985, 184 pp.

6. R. Lyons, Y. Peres, Probability on Trees and Networks, Cambridge University Press, New York, 2016, xv+699, available at \url{https://rdlyons.pages.iu.edu/

7. P. Mozolyako, G. Psaromiligkos, A. Volberg, P. Zorin-Kranich, Carleson embedding on tri-tree and on tri-disc, Revista Matematica Iberoamericana, 38, (7), 2022, 2069-2116.