Prerequisite

The listener should be acquainted with basics of real analysis, functional analysis and, for some topics, should have some exposure to probability theory (martingales), harmonic function theory and complex analysis.

Introduction

The interaction between probability and analysis, in particular harmonic analysis, can be traced back to the formative days of both fields. In fact, one can say that it predates the mathematical "codification" of probability realized by Kolmogorov's axioms.

Early on this connection was rather implicit, however in the second half of the last century it was studied and developed by many famous researchers (Burkholder, Gundy, Fefferman, Stein, McKean, Makarov, Banuelos, Peres, just to name a few) resulting in many groundbreaking advances. In addition to a new language to describe analytic phenomena they provided an abundance of deep techniques and ideas that were instrumental in the solution of many problems of "classical" analysis.

The goal of this course is to elucidate several instances of this relationship and provide a demonstration of the symbiosis enjoyed by probability and (harmonic) analysis. This particular field is vast and extensive, and it continues to grow in many different directions. Therefore the aim is to concentrate on the most simple (and in a way classical) examples of this kind, thus, essentially, restricting to the discrete approaches. More specifically, the topics discussed will cover the representation of functions by dyadic martingales, the interplay

between the behaviour of various maximal functions, laws of iterated logarithm, the boundary behaviour of harmonic functions, in particular the properties of the harmonic measure.

The course is divided into two parts. In the first one we are focusing on collecting the necessary knowledge about dyadic martingales, harmonic functions and their relationship.

Syllabus

i. Introduction: setting, models and origins

ii. Tree structure and martingales

iii. Quadratic variation and convergence

iv. A few words about wavelets

v. Good-lambda inequalities and the Law of the Iterated Logarithm (LIL) for martingales

vi. The Muckenhoupt-Wheeden-Wolff inequality

vii. A few words about harmonic functions

viii. Approximating harmonic functions by martingales, pt. I: Bloch functions

ix. Approximating harmonic functions by martingales, pt. II: the rest of the story

Lecturer Intro

Pavel Mozolyako is an associate professor at St. Petersburg State University. He leads PhD program in mathematics at the department of Mathematics and Computer Science. He got his PhD degree in 2009, at St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences. He was a postdoc at Norwegian University of Science and Technology, University of Bologna, and a visiting professor at Michigan State University. His research considers mostly boundary behaviour of harmonic functions and discrete models in potential theory.