Extension Phenomena of Integrable Holomorphic Functions in Reinhardt Domains of Holomorphy

Abstract

Holomorphic functions of several complex variables showcase many interesting extension phenomena which have historically motivated much of the development of the discipline. The purpose of this thesis is to explore the extension phenomena of integrable holomorphic functions, an important subclass of the holomorphic functions. We give two classification theorems for two-dimensional Reinhardt L⅟ₕ-domains of holomorphy, as well as two partial results towards classifying n-dimensional Reinhardt L⅟ₕ-domains of holomorphy. Both classification theorems for the two-dimensional domains are geometric classifications in terms of elementary Reinhardt domains. The first gives a classification in terms of monomial inequality representations of elementary Reinhardt domains, while the second gives a classification in terms of a parameterization of such domains by points on the unit circle. While we did not achieve a complete classification of n-dimensional domains, we demonstrate that all bounded Reinhardt domains of holomorphy are themselves L⅟ₕ -domains of holomorphy. Furthermore, while fat L⅟ₕ-domains of holomorphy have been characterized via functional analysis in the past, we provide a geometric characterization of such domains in terms of elementary Reinhardt domains.