Date of Award

June 2015

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

Advisor(s)

Evgeny A. Poletsky

Keywords

Hardy Spaces

Subject Categories

Physical Sciences and Mathematics

Abstract

The holomorphic functions on the unit disk $\mathbb{D}$ in the complex plane $\mathbb{C}$ have a remarkable property: to know the values of a holomorphic function on $\mathbb{D}$ it suffices to know only its values on the unit circle $\mathbb{T}$. However not all holomorphic functions on $\mathbb{D}$ are defined on $\mathbb{T}$ and the major problem of establishing such values (called boundary values) led to the appearance of Hardy spaces $H^p(\mathbb{D})$, $p\ge1$. If a function lies in a Hardy space then its boundary values can be defined and its values on $\mathbb{D}$ can be obtained using standard Cauchy or Poisson formulas.

The theory of Hardy spaces $H^p(\mathbb{D})$ was well developed in the last century and the spaces became the fundamental ground for complex analysis. To create analogous spaces in higher dimensions Poletsky and Stessin introduced new spaces on hyperconvex domains in $\mathbb{C}^n$ in \cite{PS}. We call these spaces the Poletsky--Stessin Hardy spaces. Poletsky and Stessin used them to study composition operators but did not look at their detailed properties.

In this thesis we fill this gap studying Poletsky--Stessin Hardy spaces on the unit disk $\mathbb{D}$. As in \cite{PS} for their definition we use subharmonic exhaustion functions $u$ and denote these spaces by $H^p_u(\mathbb{D})$. It was mentioned in \cite{PS} that the classical Hardy spaces form a subclass of Poletsky--Stessin Hardy spaces. Our work begins with producing an example that shows that there are subharmonic exhaustion functions $u$ on $\mathbb{D}$ for which the Poletsky--Stessin Hardy spaces $H^p_u(\mathbb{D})$ are different from classical Hardy spaces $H^p(\mathbb{D})$. Thus we have an abundance of new function spaces to be explored.

We show that the theory of boundary values for functions in Poletsky--Stessin Hardy spaces is analogous to the classical theory of Hardy spaces and the most of the classical properties stay true for these new spaces. Since by \cite{PS} the space $H^p_u(\mathbb{D})$ lies in $H^p(\mathbb{D})$ we can use the classical boundary values for functions in $H^p_u(\mathbb{D})$. This allows us to redefine Poletsky--Stessin Hardy spaces as spaces whose boundary values belong to weighted $L^p$ spaces on $\mathbb{T}$ and we completely characterize the weights that produce Poletsky--Stessin Hardy spaces $H^p_u(\mathbb{D})$.

Many problems in complex analysis ask for existence of a bounded function in some class. Usually it is easier to find a function in $H^p_u(\mathbb{D})$ but they are not necessarily bounded. As an application of Poletsky--Stessin Hardy spaces we provide a reduction of such problems to the existence of a function in $H^p_u(\mathbb{D})$ and use it to give shortcuts in the proofs of the famous interpolation theorem and corona problem.

At the end of the thesis we also study the boundary behavior of functions in the Hardy spaces on the polydisk and discuss the intersection of Poletsky--Stessin spaces on bidisk.

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