5-2013

Dissertation

#### Degree Name

Doctor of Philosophy in Mathematics (PhD)

#### Department

Mathematical Sciences

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 $q\in(0,1)$.

COinS