Some problems in harmonic analysis and probabilistic potential theory

Thesis

In 1917 Borel wrote a monograph entitled "Lessons Monogenic functions". In it he described a class of functions which behave, in many ways, like analytic functions. The one big difference between Cauchy's analytic functions and Borel's monogenic functions is the following. The domain of an analytic function by definition open; the domain of a monogenic function need have no interior. We propose examining a similar phenomenon here.

This thesis studies the properties of finely holomorphic functions. Using methods from fine potential theory, stochastic processes and rational approximation, one can make sense of the notion of holomorphy for functions defined on anfinely open set in C . Such a set need have no interior for the usual metric topology.

In this thesis a weak definition of fine holomorphy is adopted.

[Abstract continues in thesis]

Authors: Lyons, TJ

• DOI: 10.5287/ora-amavzgn42
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English