MATJ5113 MA1: Introduction to Harmonic Measure (JSS32) (2 op)

Harmonic measure is an essential tool for the solution of the Dirichlet problem for Laplace equation, as well as for the study of many problems in complex analysis. The objective of the course is to study the properties of harmonic measure.
The main topics are the following:

• Harmonic functions
• The Dirichlet problem for the Laplace equation
• Harmonic measure
• The Green function
• Behavior of harmonic measure under conformal maps in the complex plane
• The Riesz brother's theorem for Jordan domains
• Dahlberg's theorem for harmonic measure in Lipschitz domains
• Hausdorff measures and the dimension of a measure
• Survey of advanced results:
  - The theorems of Jones-Wolff, Wolff, and Bourgain about the dimension of harmonic measure
  - Harmonic measure and rectifiability
  - L^p solvability of the Dirichlet problem

The study of the metric and geometric properties of harmonic measure is a topic of fundamental importance in analysis, which combines techniques from complex analysis, geometric analysis, and PDE’s. This course will be introductory in this field.

Basic functional analysis and measure theory. Familiarity with elliptic PDE's will be also useful but not necessary to follow the course.