Date of Graduation

5-2013

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematics (PhD)

Degree Level

Graduate

Department

Mathematical Sciences

Advisor

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 $q\in(0,1)$.

Share

COinS