Date of Award

Summer 7-16-2021

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

Advisor(s)

Yuan, Yuan

Second Advisor

Gursoy, M. Cenk

Keywords

Bergman Projection, Canonical Solution Operator, Compactness, D-bar Neumann Operator, D-bar Problem

Subject Categories

Mathematics | Physical Sciences and Mathematics

Abstract

We begin this thesis by a brief introduction to the $\bar{\partial}$-problem in several complex variables and the classical $L^2$ Theorem of $\bar{\partial}$. We then introduce the $\bar{\partial}$-Neumann operator on pseudoconvex domains and give a description of the relation between the $\bar{\partial}$-Neumann operator, the canonical solution operator, the Bergman projection and the Hankel operator. Meanwhile, the study of these operators is deeply related to the regularity of the $\bar{\partial}$-problem. In this thesis, we focus on the regularity, compactness and boundedness of these operators.

In chapter 2, we study the weakly pseudoconvex points on the boundary of a class of Hartogs domains. On that class of domains, we show that Property $(P)$ of the boundary, the compactness of the $\bar{\partial}$-Neumann operators $N_1$, and the compactness of the Hankel operator are equivalent.

In chapter 3, we apply the Bekoll\'e-Bonami estimate for the (positive) Bergman projection on the weighted $L^p$ spaces on the unit disk. As a consequence, we obtain the boundedness of the Bergman projection on weighted Sobolev space on the symmetrized bidisk. We also improve previous results on the boundedness of the Bergman projection on unweighted $L^p$ spaces on the symmetrized bidisk.

In chapter 4, we define the Hankel operators with symbols of forms. We show the following statements are equivalent: (1) the compactness of $H _{\phi} ^{q}$ on $K _{( 0, q )} ^{2} ( \Om )$, (2) the compactness of the canonical solution operator on $K _{( 0, q + k + 1 )} ^{2} ( \Om )$, and (3) the compactness of $N _{q + k + 1}$ on $L _{( 0, q + k + 1 )} ^{2} ( \Om )$, for $q \geq 1$. A sufficient condition and a necessary condition of the compactness of the Hankel operators are also given. Furthermore, we prove the localization theorem of the compactness of these Hankel operators.

In chapter 5, we study Shaw's question about Sobolev regularity of solution operators on the bidisk. We show that the recent development of the integral operator implies that the canonical solution operator of $\bar\partial$ has loss of $n-2$ derivatives on the polydisk. In particular, it is exact regular on the bidisk.

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