This thesis consists of eight chapters. Chapter 1 contains the preliminaries: the background, notation and results needed for this work.
In Chapter 2 we study the problem of when P, the set of analytic polynomials, is dense in the Hardy space H<sup>t</sup>(G) or the Bergman space L<sup>t</sup><sub>n</sub>G, where G is a bounded domain and t ∈ [1,∞). Characterizations of special domains are also given.
In Chapter 3 we generalize the definition of a Carleson measure to an arbitrary simply connected domain. Let G be a bounded simply connected domain with harmonic measure ω. We say a positive measure τ on G is a Carleson measure if there exists a positive constant c such that for each t ∈ [1, ∞) and each polynomial p we have
⎮⎮p⎮⎮<sub>L¹(τ)</sub>≤ c ⎮⎮p⎮⎮ <sub>Lᵗ(ω)</sub>,
We characterize all Carleson measures on a normal domain-definition: a domain G where P is dense in H¹(G). It turns out that P is dense in Hᵗ(G) for all t when G is normal.
In Chapter 4 we describe some special simply connected domains and describe how they are related to each other via various types of polynomial approximation.
In Chapter 5 we study the various equivalence classes of subnormal operators under the relations of unitary equivalence, similarity and quasi similarity under the assumption that G is a normal domain.
In Chapter 6 we characterize the Carleson measures on a finitely connected domain. We are able to push our techniques in the latter setting to characterize those subnormal operators similar to the shift on the closure of R(K) in L²(σ) when R(K) is a hypo dirichlet algebra.
In Chapter 7 we illustrate our results by looking at their implications when G' is a crescent. Several interesting function theory problems are studied.
In Chapter 8 we study arc length and harmonic measures. Let G be a Dirichlet domain with a countable number of boundary components. Let ω be the harmonic measure of G. We show that if J is a rectifiable curve and E ⊂ ∂G ∩ J is a subset with ω(E) > 0, then E has positive length. / Ph. D.
| Identifer | oai:union.ndltd.org:VTETD/oai:vtechworks.lib.vt.edu:10919/38499 |
| Date | 06 June 2008 |
| Creators | Qiu, James Zhijan |
| Contributors | Mathematics, Olin, Robert F., Thomson, James E., Holub, James R., McCoy, Robert A., Linnell, Peter A. |
| Publisher | Virginia Tech |
| Source Sets | Virginia Tech Theses and Dissertation |
| Language | English |
| Detected Language | English |
| Type | Dissertation, Text |
| Format | vii, 118 leaves, BTD, application/pdf, application/pdf |
| Rights | In Copyright, http://rightsstatements.org/vocab/InC/1.0/ |
| Relation | OCLC# 29985306, LD5655.V856_1993.Q58.pdf |
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