#### Graduation Year

2013

#### Document Type

Dissertation

#### Degree

Ph.D.

#### Degree Granting Department

Mathematics and Statistics

#### Major Professor

Dmitry Khavinson

#### Keywords

Hardy classes, Neuwirth- Newman's Theorem, real boundary values, Smirnov classes, Smirnov domains

#### Abstract

This thesis concerns the classes of analytic functions on bounded, n-connected domains known as the Smirnov classes E^{p}, where p > 0. Functions in these classes satisfy a certain growth condition and have a relationship to the more well known classes of functions known as the Hardy classes H^{p}. In this thesis I will show how the geometry of a given domain will determine the existence of non-constant analytic functions in Smirnov classes that possess real boundary values. This is a phenomenon that does not occur among functions in the Hardy classes.

The preliminary and background information is given in Chapters 1 and 3 while the main results of this thesis are presented in Chapters 2 and 4. In Chapter 2, I will consider the case of the simply connected domain and the boundary characteristics that allow non-constant analytic functions with real boundary values in certain Smirnov classes. Chapter 4 explores the case of an n-connected domain and the sufficient conditions for which the aforementioned functions exist. In Chapter 5, I will discuss how my results for simply connected domains extend Neuwirth-Newman's Theorem and finish with an open problem for n-connected domains.

#### Scholar Commons Citation

De Castro, Lisa, "Analytic Functions with Real Boundary Values in Smirnov Classes E^{p}" (2013). *Graduate Theses and Dissertations.*

http://scholarcommons.usf.edu/etd/4661