Theory Of Slice Regular Functions

This Ph.D dissertation mainly studies the generalization from complex analysis to the setting of noncommutative algebra in higher dimensions which includes the follow-ing three aspects.(1)geometry function theory for slice regular functions;(2)theory of function spaces of slice regular functions;(3)uncertainty principles for quaternionic Hilbert spaces.This paper is divided into five chapters.Chapter 1 is the introduction where we introduce the background and main results of this dissertation.In chapter 2,we give some basic notation,terminologies and known results.Chapter 3 is mainly devoted to studying the geometry function theory for slice regular functions.Firstly,we introduce concepts of slice starlike functions,slice almost convex functions,and slice spirallike functions for slice regular functions over quater-nions and show that the Bieberbach conjecture holds for slice close-to-convex functions and then established the Fekete-Szego inequality,growth theorem,distortion theorem and covering theorem for slice starlike functions.Secondly,the growth theorem and distortion theorem for a subclass of slice regular functions on real alternative algebras are established.Thirdly,we establish three Bloch-Landau type theorems for slice regu-lar functions over quaternions and generalized the classical Bernstein inequality to slice regular polynomias.Finally,we focus on the generalization of Schwarz lemma in high-er dimensions.In particular,we investigate Schwarz lemma and its boundary behaviour for slice Clifford analysis and pluriharmonic functions.In chapter 4,we study two generalizations of holomorphic ?-Bloch functions in higher dimensions.On one hand,we initiate the study of ?-Bloch functions on the unit ball of an infinite dimensional Hilbert space among which we define four norms of the?-Bloch space and show their equivalences.As an application,the Hardy-Littlewood theorem in infinite dimensional Hilbert spaces are established.On the another hand,we introduce the concept of regular ?-Bloch functions on the unit ball of quaternions where we establish the corresponding Forelli-Rudin estimates.Hardy-Littlewood theorem,and then study the dual space for regular ?-Bloch functions.In chapter 5,we establish the uncertainty principle for quaternionic Hilbert spaces.