Slice正则函数论

发布时间：2018-02-25 08:30

本文关键词： 四元数 Slice正则函数 Bieberbach猜测增长定理 Bloch-Landau定理 Bernstein不等式 Schwarz引理 Bloch空间测不准原理　出处：《中国科学技术大学》2017年博士论文　论文类型：学位论文

【摘要】：本文主要研究复分析在高维非交换代数上的推广,其中包括以下三个方面:(1)slice正则函数的几何函数论;(2)slice正则函数的函数空间论;(3)四元数Hilbert空间中的测不准原理.全文共分为五章.第一章是绪论,介绍了本论文的研究背景和所取得的成果.第二章给出了本论文中常用的符号、概念和结论.第三章主要研究了 slice正则函数的几何函数论.本章首先在四元数slice正则函数中定义了 slice星形函数,slice近凸函数,slice螺形函数,证明了 Bieberbach猜测对slice近凸函数是成立的,对slice星形函数建立了 Fekete-Szego不等式、增长定理、掩盖定理和偏差定理.其次,本章研究了.类交错代数上slice正则函数的增长定理和偏差定理.然后,针对四元数slice正则函数建立了三类Bloch-Landau型定理并推广了经典的Bernstein不等式.最后,本章围绕Schwarz引理在高维中的推广.特别地,研究了 slice Clifford分析以及多次调和函数中的Schwarz引理及其边界行为.第四章研究了 α-Bloch函数在高维空间中的两类推广.一方面,研究了无限维Hilbert空间单位球上的全纯α-Bloch函数,定义了四种范数并证明了其等价性.作为应用,建立了无限维Hilbert空间中的Hardy-Littlewood定理.另一方面,研究了四元数单位球上的正则α-Bloch函数,建立了相应的Forelli-Rudin估计,Hardy-Littlewood定理,并对其对偶空间进行了研究.第五章建立了四元数Hilbert空间中的测不准原理.
[Abstract]:In this paper, we mainly study the generalization of complex analysis on high dimensional noncommutative algebras. It includes the following three aspects: geometric function theory of slice regular function and function space theory of 2slice regular function. The uncertainty principle in Hilbert space of quaternion is discussed. The whole paper is divided into five chapters. Chapter one is an introduction. The research background and achievements of this thesis are introduced. In chapter 2, the commonly used symbols in this paper are given. In chapter 3, the geometric function theory of slice regular function is studied. In this chapter, the slice star function is defined in the quaternion slice regular function. It is proved that Bieberbach conjecture is true for slice near-convex functions, and Fekete-Szego inequality, growth theorem, concealment theorem and deviation theorem are established for slice star functions. In this chapter, we study the growth theorems and deviation theorems of slice regular functions on quasi-staggered algebras. Then, we establish three Bloch-Landau type theorems for quaternion slice regular functions and generalize the classical Bernstein inequalities. This chapter focuses on the generalization of Schwarz Lemma in higher dimensions. In particular, we study the Schwarz Lemma and its boundary behavior in slice Clifford analysis and multiharmonic functions. In Chapter 4th, we study two generalizations of 伪 -Bloch functions in high dimensional spaces. In this paper, the holomorphic 伪 -Bloch function on the unit sphere of infinite dimensional Hilbert space is studied, four kinds of norms are defined and its equivalence is proved. As an application, the Hardy-Littlewood theorem in infinite dimensional Hilbert space is established. In this paper, we study the regular 伪 -Bloch function on the unit sphere of quaternions, establish the corresponding Forelli-Rudin estimate Hardy-Littlewood theorem, and study its dual space. Chapter 5th establishes the uncertainty principle in the quaternion Hilbert space.
【学位授予单位】：中国科学技术大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O174.5

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