Date of Graduation
5-2013
Document Type
Dissertation
Degree Name
Doctor of Philosophy in Mathematics (PhD)
Degree Level
Graduate
Department
Mathematical Sciences
Advisor/Mentor
Loredana Lanzani
Committee Member
Phillip S. Harrington
Second Committee Member
Daniel H. Luecking
Keywords
Pure sciences, Applied sciences, Cauchy kernel, Complex variables, Hardy spaces, Smirnov space
Abstract
This work is based on a paper by Edgar Lee Stout, where it is shown that for every strictly pseudoconvex domain $D$ of class $C^2$ in $\mathbb{C}^N$, the Henkin-Ram\'irez Kernel Function belongs to the Smirnov class, $E^q(D)$, for every $q\in(0,N)$.
The main objective of this dissertation is to show an analogous result for the Cauchy Kernel Function and for any strictly convex bounded domain in the complex plane. Namely, we show that for any strictly convex bounded $D\subset\mathbb{C}$ of class $C^2$ if we fix $\zeta$ in the boundary of $D$ and consider the Cauchy Kernel Function
\mathcal{K}(\zeta,z)=\frac{1}{2\pi i}\frac{1}{\zeta-z}
as a function of $z$, then the Cauchy Kernel Function belongs to the Smirnov class $E^q(D)$ for every $q\in(0,1)$.
Citation
Espinosa Lucio, B. (2013). Hardy Space Properties of the Cauchy Kernel Function for a Strictly Convex Planar Domain. Graduate Theses and Dissertations Retrieved from https://scholarworks.uark.edu/etd/766